RAY'S INTELLECTUAL ARITHMETIC
INDUCTION AND ANALYSIS.
.
HINTS TO TEACHERS.
LET the pupils be classified with reference to their attainments and abilities. The recitation should be short and spirited, every pupil being required to give undivided attention to the question before the class. Generally, while reciting, the pupils should be permitted to have their books open before them the test of having properly studied the lesson, being the readiness and accuracy with which the several questions are analyzed and answered. The explanations and operations termed ANALYSIS, are intended as Model Solutions, pointing out to the learner the manner in which the questions in the lessons are to be solved and explained. The pupil should be required to furnish a similar explanation to each of the succeeding questions, and to give, not only a correct answer, but also, the reason for the method by which he obtained it. A method of solving questions in Mental Arithmetic, now much used, is the following, called the "Four Step Method."
ILLUSTRATIONS. First step, James gave 7 cents for apples and 8 cents for peaches; how many cents did he spend? Second step, as many as the sum of 7 cents and 8 cents. Third step, 1 cents and 8 cent1? are 15 cents. Fourth step, hence, if James gave 7 cents for apples, and 8 cents for peaches, he spent 15 cents.
Again: First step, 4-fifths of 25 are how many times 6? Second step, as many times 6 as 6 is contained times in 4-fifths of 25. Third step, 1-fifth of 25 is 5, 4-fifths are 4 times 5, which are 20; 6 in 20 is contained 3 and 2-sixths times. Fourth step, therefore, 4-fifths of 25 are 3 and 2-sixths times 6? As a means of keeping the attention of the class directed to each question, it. will be proper for the instructor occasionally to read an example aloud, and, having allowed sufficient time for the answer, to call upon some one for the result, and then for the analysis. By this means, each one is obliged to solve the example not knowing but that he may be required to answer it.INTELLECTUAL
ARITHMETIC.
SECTION I. NUMERATION.
Pupils who have studied Ray's Arithmetic, First Book, may commence with Lesson IV, on page 14.NUMERATION is naming Numbers.
Learn the name and form of these figures:
One, two, three, four, five, six, seven, eight, nine, ten.
An Illustration of the Increase of Numbers.
L,2,3,4,5,6,7,8,9,10,10RAY'S INTELLECTUAL ARITHMETIC.
SECTION II. ADDITION.
LESSON I.
1. James had one apple, and his brother gave him one more : how many had he ? Why ? Ans. Because 1 and 1 are 2.
2. Henry had two cents, and his sister gave him one more : how many had he ? Why ? Ans. Because 2 and 1 are 3.
3. A boy had one marble, and found three more : how many did he then have ? Why ? Ans. Because 1 and 3 are 4.
4. Thomas had 4 cents, and his mother gave him one more : how many cents had Thomas ? Why ?
5. Samuel had two cakes, and his father gave him two more : how many did he then have ? Why ?
6. Three oranges and two oranges are how many oranges ? Why ?
7. James had three apples, and his brother gave him three more : how many apples had James then ? Why ?
8. John had four plums, and his sister gave him two more : how many did he then have ? Why ?
9. Daniel had three cents ; his brother gave him two, and his sister one : how many did he then have? Why?
10. Mary had four pears, and her brother gave her three more : how many did she then have ? Why ?
11. George recites daily four lessons perfectly, one imperfectly, and is absent from one recitation : how many
daily lessons has he ? Why ?
12. How many fingers have you on one hand ? How many on both? Why?
13. Ida had four cents ; her mother gave her three more at one time, and one at another : how many did she then have? Why?
14. Three cakes, and one cake, and two cakes, are how many, cakes ? Why ?
15. Four cents, and three cents, and two cents, are how many cents ? Why ?
16. Five oranges, and one orange, are how many oranges ? Why ?
17. Henry had five cents, and his mother gave him two more: how many did he then have? Why?
18. Five boys, and one boy, and two boys, are how many boys ? Why ?
19. Oliver has five dollars, Henry three, and Samuel one : how many dollars have all together ? Why ?
20. Three peaches, and six peaches, are how many peaches ? Why ?
21. A lady paid one dollar for gloves, three dollars for a shawl, and five dollars for a dress : how much did she
spend? Why?
22. Four cents, and three cents, and two cents, and one cent, are how many cents ? Why ?
23. If a man buy six pounds of sugar at one time, and four at another, how much does he buy ? Why ?
24. Seven oranges, and one orange, and two oranges, are how many ? Why ?
25. George has two cents, his sister three cents, and his brother five cents : if they were all given to George
how many would he have ? Why ?
26. How many are 4 and 3 and 3?
27. James has 4 cents, and Joseph 2 more than James: how many has Joseph ? How many have both ? Why ?
LESSON II.
1. One and 1 are how many? 1 and 2? 3 and 1? 4 and 1? 1 and 3? 1 and 5? 1 and 6? 6 and 1?
1 and 7? 8 and 1? 9 and 1 ? 1 and 8?
2. Two and 2 are how many? 2 and 3? 2 and 2
and 1? 1 and 2 and 3? 4 and 1 and 2?
3. Four and 2 are how many ? 4 and 3 ? 2 and 4
andl? 4 and land 2? 3 and 4? 5 and 4?
4. Six and 2 are how many? 6 and 1 and 2? 6
and 3? 6 and 4? 6 and 1 and 3? 1 and 2 and 6? 1 and 3 and 6?
5. Eight and 2 are how many? 8 and 3? 8 and 2 and 1? 2 and 8 and 1 ? 1 and 4 and 3? 8 and
4? 7 and 4? 6 and 7? 7 and 1 and 3* 1 and 6 and 2 ? 1 and 8 and 2 and 2 ?
6. Nine and 2 are how many? 9 and 3? 9 and 1 and 2? 8 and 1 and 3 ? 5 and 6 ? 1 and 4 and
5? 2 and 4 and 6? 3 and 4 and 5? 4 and 6 and 2 ? 5 and 4 and 2 and 1 ?
7. How many are 1 and 9? 7 and 3? 4 and 6? 9 and 1? 5 and 5? 6 and 4? 2 and 8 ? 3 and 7?
and 10? 8 and 2? 10 and 0? What two numbers added together make 10?
8. Which is the greater, 7 and 2, or 6 and 4? Why? one and 3 and 6, or 8 and and 2 ? Why ?
9. Begin at 4, and add 2 each time, up to 10.
10. Begin at 1, and add 3 each time, up to 13.
11. What two numbers added together will make 12? What three numbers?
12. Seven and 5 and 2 are how many?
13. One and 7 and 3 and 4 are how many?
14. If 3 be added to 3, and that sum to 4, what will be the result?
15. If you add 3, to 3 and 1 more, and then add 7, what will be the amount?
16. I have in one basket 8 dozen eggs, in another 4 dozen, in another 3 dozen: how many in all?
17. A lady bought 2 yards of tape for 3 cents, some pins for 1 cent, and received 2 cents change : how many
cents had she at first?
18. Two and 1 more, and 3 and 4 more, are together how many ?
19. One and 3 and 4 and 5 are how many? 5 and one and 3 and 4 ?
20. Two and 1 and 3, taken from a certain number, leave 2 : what is that number ?
21. A boy bought 3 cents worth of marbles, and 2 cents worth of candy, and received 5 cents change : how muck money had he ?
22. I bought 3 hams for 8 dollars, and 10 bushels of apples for 3 dollars : how much did I spend ?
23. If 6 yards of cloth will make 2 coats, and 4 yards will make 2 pairs of pants, how many yards must I buy
for 2 coats and 2 pairs of pants?
24. Oliver has 4 cents in one hand, 3 in the other, and 4 in his pocket : how many cents has he ?
RAY'S INTELLECTUAL ARITHMETIC.
3. Five and 7 are how many? 5 and 6? 5 and 9? five and 12? 5 and 10? 5 and 8 ? 5 and 11?
4. Seven and 6 are how many? 7 and 10? 7 and eight? 7 and 12? 7 and 9 ? 7 and 11?
5. Nine and 11 are how many ? 9 and 9 ? 9 and 12 ? nine and 7 ? 9 and 10 ? 9 and 8 ? 9 and 11 ?
6. Ten and 6 are how many? 10 and 8? 10 and ten? 10 and 12? 10 and 11? 10 and 5? 10 and 7?
ten and 9 ?
7. Eleven and 2 are how many ? 11 and 4? 11 and six? 11 and 5? 11 and 7? 11 and 9? 11 and 11?
eleven and 10?
8. Twelve and 5 are how many? 12 and 4? 12 and 6? 12 and 8? 12 and 10? 12 and 7? 12
and 9? 12 and 11? 12 and 12?
9. Thirteen and 4 are how many ? 13 and 6 ? 13 and 5? 13 and 7 ? 13 and 9? 13 and 10 ?
13 and 8? 13 and 11? 13 and 12?
10. Fourteen and four are how many? 14 and 6? 14 and 8? 14 and 5 ? 14 and 7? 14 and 10?
14 and 9? 14 and 11 ? 14 and 12 ?
11. Fifteen and 5 are how many? 15 and 7? 15 and 9? 15 and 4? 15 and 8? 15 and 10? 15 and
12? 15 and 11?
12. Sixteen "and 4 are how many? 16 and 6? 16 and 8? 16 and 5? 16 and 7? 16 and 9? 16 and 11?
16 and 10 ? 16 and 12 ?
13. Seventeen and 6 are how many? 17 and 4? 17 and 7 ? 17 and 5 ? 17 and 9 ? 17 and 8 ? 17 and
10 ? 17 and 12 ? 17 and 11 ?
14. Eighteen and 10 are how many? 18 and 4? 18and 7? 18 and 5? 18 and 8? 18 and 6? 18 and 9?
18 and 11 ? 18 and 12 ?
15. Nineteen and 5 are how many? 19 and 3? 19 and 2? 19 and 7? 19 and 9? 19 and 8? 19 and 10?
19 and 6? 19 and 12 ? 19 and 11?
16. How many are 9 and 2 ? 19 and 2? 29 and 21 49 and 2 ? 6 (J and 2 ? 39 and 2 ? 59 and 2 ? 79 and
2 ? 99 and 2 ?
17. How many are 9 and 3? 3 and 19? 29 and 3? 3 and 49 ? 59 and 3? 3 and 39? 69 and 3? 3 and
79? 3 and 89? 99 and 3?
18. How many are 9 and 7 ? 29 and 7 ? 7 and 49? 39 and 7 ? 7 and 59 ? 79 and 7 ? 7 and 69 ? 89 and
7? 7 and 99?
19. How many are 9 and 8? 29 and 8? 49 and 8? 39 and 8? 8 and 69? . 59 and 8? 79 and 8?
20. How many are 9 and 9? 19 and 9? 9 and 29? 49 and 9? 69 and 9? 59 and 9? 79 and 9? 89 and
9? 9 and 99?
21. How mar- are 8 and 3 ? 28 and 3? 48 and 3? 68 and 3? 88 and 3? 98 and 3?
22. How many are 8 and 7 ? 28 and 7 ? 7 and 38 ? 48 and 7? 68 and 7 ? 58 and 7? 88 and 7?
23. How many are 7 and 7 ? 17 and 7? 27 and 7? 47 and 7 ? 57 and 7 ? 37 and 7 ? 67 and 7 ? 87 and
7 ? 77 and 7 ? 97 and 7 ?
24. How many are 7 and 10? 17 and 10? 27 and 10 ? 47 and 10 ? 37 and 10 ? 57 and 10 ?
25. How many are 6 and 5 ? 16 and 5? 15 and 6?
26 and 5? 25 and 6? 24 and 6? 26 and 4? 36 and 6? 48 and 6? 45 and 6? 57 and 6? 59 and 6?
66 and 6? 75 and 6? 86 and 6?
26. How many are 13 and 8? 17 and 3 ? 23 and 8?
24 and 8? 33 and 8 ? 3 and 37? 8 and 43? 47 and
3? 7 and 53? 58 and 3 ? 67 and 3? 3 and 87 ? 97
and 3? 88 and 3? 3 and 98 ?
27. How many are 14 and 9? 9 and 24? 25 and 9?
9 and 34 ? 36 and 9 ? 9 and 44 ? 9 and 47 ? 54 and
9? 9 and 56? 9 and 64? 74 and 9? 9 and 72? 8*
and 9 ? 86 and 9 ? 94 and 9 ?
28. How many are 10 and 6? 6 and 21 ? 10 and
26? 46 and 10? 10 and 35? 10 and 55? 56 and
10? 10 and 66? 10 and 69? 76 and 10? 10 and
86? 96 and 10?
29. How many are 11 and 6? 11 and 16? 11 and
27? 25 and 11? 11 and 23? 31 and 11? 11 and
35? 37 and 11? 11 and 59? 46 and 11? 11 and
48? 52 and 11? 11 and 63?
30. Fifty-six and 1 are how many? 56 and 3? 56
and 5? 2 and 56? 4 and 56? 7 and 56 ? 56 and 6?
8 and 56 ? 56 and 9 ? 10 and 56 ?
31. Ninety-eight and 1 ? 3 and 98? 5 and 98? 98
and 2 ? 4 and 98 ? 98 and 6 ? 7 and 98 ? 98 and 8 ?
10 and 98? 98 and 9?
32. How many are 1 and 2 and 4 and 5 ? 5 and 1 and
four and 2 ? 2 and 1 and 5 and 4 ?
33. How many are 12 and 8 ? 9 and 2 and 9? 4 and
2 and 9 and 5 ? 4 and 2 and 5 and 5 and 4 ? 8 and 4
and 6 and 2 ? 8 and 9 and 6 ?
34. If you add 3 to 3, and 4 to 4, and 5 to 5, and add
those sums together, what number will you have ?
35. If to 10 you add 6, 7, and 9, how much will you
then have ?
NOTE. The numbers in the following examples should be added aloud; and if preferred, the "Four step method" can be applied to all the questions. Take for example the second question ; Four and 5 and 7 are how many? As many as the sum of 4 and 5 and 7; 4 and 5 are 9, 9 and 7 are 1C ; therefore, 4 and 5 and 7 are 16.
1. Three and 6 and 4 are how many ?
2. Four and 5 and 7 are how many ?
3. Five and 6 and 2 are how many ?
4. Six and 4 and 5 are how many?
5. Seven and 3 and 5 and 2 are how many ?
6. Eight and 2 and 3 and 4 are how many ?
7. Nine and 2 and 4 and 3 are how many ?
8. Two and 9 and 5 and 4 are how many ?
9. Three and 9 and 5 and 4 are how many ?
10. Four and 8 and 3 and 5 and 2 and 6 and 3 and
one are how many ?
11. Five and 7 and 2 and 3 and 4 and 6 and 5 and 2 are
how many ?
12. Two and 4 and 3 and 5 and 6 and 2 and 7 and 4 are
how many ?
13. Three and 2 and 4 and 5 and 4 and 6 and 3 and
seven and 5 are how many ?
14. Four and 3 and 5 and 7 and 6 and 8 and 2 and 4 are
how many ?
15. Four and 9 and 3 and 5 and 6 and 7 and 8 and 9 are
how many ?
16. Five and 8 and 5 and 8 and 5 and 8 and 5 and 8 are
how many ?
17. Six and 8 and 7 and 3 and 5 and 4 and 7 and 1 and
nine are how many ?
18. Seven and 9 and 5 and 4 and 6 and 3 and 8 and
five and 9 are how many ?
19. Eight and 7 and 6 and 5 and 4 and 9 and 3 and
seven and 8 are how many ?
20. Nine and 6 and 7 and 4 and 5 and 3 and 8 and
two and 9 are how many ?
21. Seven and 6 and 5 and 8 and 7 and 9 and 8 and
four and 9 and 8 are how many ?
22. Nine and 8 and 7 and 5 and 8 and 9 and 5 and
four and 7 and 3 and 9 and 8 are how many ?
23. Twelve and 11 and 7 and 4 and 9 are how many ?
24. Thirteen and 10 and 8 and 6 and 4 and 10 are how
many?
25. Fourteen and 16 and 7 and 5 and 9 and 8 and 9 and
six and 4 are how many ?
26. James gave 7 cents for apples, and 8 cents for
peaches : how many cents did he spend ?
27. Seven dollars, and 5 dollars, and 3 dollars, are how
many dollars ?
28. David had 11 books ; he bought 7 more, and his
brother gave him 5 ; how many had he then ?
29. A man gave 13 dollars for a cart, 6 for a plow, and
5 for a harrow : how many dollars did he spend ?
30. James has 8 marbles in one pocket, 5 in another^
6 in another, and 7 in another : how many in all ?
31. If a pound of butter cost 18 cents, of lard 7 cents,
and of cheese 9 cents, how much will all cost?
32. A man owes to one person 8 dollars, to another
5 dollars, to another 3 dollars, and to another 7 dollars:
how much does he owe ?
33. A boy gave 19 cents for a spelling-book, 8 cents
for a slate, and 6 cents for pencils : how many cents did
he spend?
34. A drover bought hogs as follows : of one man 17,
of another 9, of another 7, of another 8 : how many did
he buy ?
35. A little girl gave 10 cents for thread, 7 cents for
pins, 6 cents for needles, and 9 cents for tape : how
many cents did she spend ?
36. William has 7 cents, Thomas 10 cents, David 9
cents, and Moses 8 cents: if the other boys give their
cents to Moses, how many will he have?
37. The age of Thomas is 8 years ; Frank, 5 years ;
and William is as old as both together : what "is the sum
of all their ages?
38. Joseph has 4 marbles, William has 2, and David
has twice as many as Joseph : how many do they all
have?
39. Begin with 2, and count one hundred by adding
2 successively. Thus, 2 and 2 are 4, and 2 are 6, and 2
are 8, and 2 are 10, and so on.
40. Begin with 3, and count ninety-nine by adding 3
successively. Thus, 3 and 3 are 6, and 3 are 9, and 3
are 12, and so on.
41. Begin with 4, and count one hundred by adding
four successively.
42. Begin with 5, and count one hundred by adding
five successively.
43. Begin with 6, and count one hundred and two by
adding 6 successively.
44. Begin wit1
7, and count ninety-eight by adding
seven successive^ .
45. Begin with 8, and count one hundred and four by
adding 4 successively.
46. Begin with 9, and count ninety-nine by adding
nine successively.
47. Begin with 1, and count one hundred by adding
three successively.
48. Begin with 3, and count one hundred and three
by adding 4 successively.
49. Begin with 2, and count one hundred and two by
adding 5 successively.
50. Begin with 5, and count one hundred and seven
by adding 6 successively.
51. Begin with 6, and count one hundred and four by
adding 7 successively.
52. Begin with 7, and count one hundred and three by
adding 8 successively.
53. Begin with 8, and count one hundred and seven by
adding 9 successively.
54. If an apple cost 3 cents, and an orange cost 2 cents
more than an apple, what will be the cost of 2 apples
and one orange?
55. James has 5 marbles ; Henry, 2 more than James ;
and Samuel, as many as both James and Henry : how
many have all ?
56. Mary had a certain number of peaches ; she gave
5 to her sister, 7 to her brother, and then had 10 remaining: how many had she ?
57. A boy bought a sled for 20 cents, and paid 5 cents
for repairing it: what was it then worth?
53. If a hat cost 4 dollars, and a coat as much as
three hats, what do they both cost?
59. How many times must 10 be added to fifteen, to
make fifty-five?
60. A boy's father gave him 5 cents ; his mother, one
cent more than his father ; and his brother, two cents
more than his father: how many did he then have ?
61. What is the exercise of putting numbers together
called? Ans. Addition.
62. What is simple Addition?
Ans. Simple Addition is finding the SUM of two or
more numbers of the same denomination.
63. What is meant by same denomination? Same denomination means all of the same name;
that is all pounds, or all dollars, & cents.SECTION III. SUBTRACTION.
LESSON I.
1. James had 2 apples, and gave 1 to his brother:
how many had he left? Ans. 1. Why? Because 1 and
I are 2.
2. Then 1 from 2 leaves how many ?
3. Joseph had 3 apples and lost 1 : how many did he have ? Ans. 2. Why ? Because 1 and 2 are 3.
4. Then 1 from 3 leaves how many?
5. Thomas had 4 cents, and gave 1 to Frank: how
many had he left? Why?
6. Then 1 from 4 leaves how many ?
7. One from 5 leaves how many? From 6? 7? 8?
9? 10?
8. John had 4 cents and gave his sister 2 : how many
had he left? Why?
9. Then 2 from 4 leaves how many ?
10. James had 5 apples, and gave his brother 2 : how
many had he left ? Why ?
11. Then 2 from 5 leaves how many?
12. Two from 6 leaves how many? From 7? 8? 9?
10? 11?
13. Thomas had 5 cents and lost 3 : how many had he
left? Ans.2. Why?
14. Then 3 from 5 leaves how many ?
15. Three from 6 leaves how many? From 7? 8?
9? 10? 11? 12?
16. Joseph had 9 marbles- and lost 4: how many had
he left ? Why ?
17. Then 4 from 9 leaves how many ?
18. Four from 10 leaves how many? From 11 ? 12?
13? 14? 15?
19. William had 10 apples and gave Joseph 5: how
many had he left ? Why ?
20. Then 5 from 10 leaves how many ?
21. Five from 11 leaves how many ? From 12? 13?
14? 15? 16?
22. James had 11 marbles and lost 6 : how many had
he left? Why?
23. Then 6 from 11 leaves how many ?
SUBTRACTION. 23
24. Six from 12 leaves how many? From 13? 14?
15? 16? 17? 18?
25. William had 12 cents and lost 7 : how many had
he left? Why?
26. Then 7 from 12 leaves how many?
27. Seven from 13 leaves how many? From 14? 15?
16? 17? 18? 19?
28. James had 13 apples and gave his sister 8 : how
many had he left? Why?
29. Then 8 from 13 leaves how many?
30. Eight from 14 leaves how many ? From 15 ? 16?
17? 18? 19? 20?
31. Thomas had 13 apples and gave his sister 9: how
many had he left ? Why?
32. Then 9 from 13 leaves how many?
33. Nine from 14 leaves how many? From 15? 16?
i7? 18? 19? 20?
34. Henry had 14 cents and lost 5 : how many had he
remaining ?
35. Mary is 15 years old, and Anna is 8 : how much
older is Mary than Anna ?
36. Sold a load of corn for 17 dollars; received for it
a barrel of flour worth 6 dollars, and the rest in money :
how much money did I receive ?
37. A boy had 18 marbles and lost 10 : how many had
he then ?
38. Nineteen is 11 more than what number ?
39. Went shopping with 20 dollars ; spent 10 dollars in
one store, and 5 in another : how much had I left ?
In the present edition of this volume, a very few examples have been modified, the better to adapt them to a more perfect gradation. Such are indicated by the character, 0. The omission of these examples in classes having books of the present and former edition, will obviate all confusion..
LESSON II. SUBTRACTION TABLE.
To TEACHERS. Instead of requiring pupils to recite the Subtraction Table regularly, some of the best instructors omit it, and connect the exercises with addition, thus:
1 and 2 are 3 2 and 3 are 5
2 and 1 are 3 3 and 2 are 5
1 from 3 leaves 2 2 from 5 leaves 3
2 from 3 leaves 1 3 from 5 leaves 2
In case the Table should not be omitted, the above exercises will be found very profitable.
1. Two from 7 leaves how many?
2 from 12?
4. Thomas had 18 cents given him by two boys ; one
gave 9 : how many did the other give ?
5. Bought a book for 14 cents, and gave the shopkeeper
20 cents : how much change did he return me ?
6. William has 19 apples ; in one pocket he has 15;
how many are in the other ?
7. A man has 25 miles to travel : when he has gone
nineteen, how many will he still have to travel ?
8. A boy gave 24 cents for a book, and sold it for
sixteen cents : how much did he lose ?
9. James had 24 marbles; he gave 19 to his brother:
how many had he left ?
10. A man bought a horse for 19 dollars, and sold
him for 27 dollars : how much did he gain ?
11. A man owing 26 dollars, paid 18 : how many did
he then owe ?
12. Frank had 26 cents given him by William and
Thomas. William gave him 17 : how many did Thomas
give ? How many more did William give than Thomas ?
13. If you had 10 apples, and should give 2 to John,
and 6 to your sister, how many would you have left?
14. Abel had 36 cents, and his mother gave him
pnough to make 40 : how many did she give him ?
15. George had 40 marbles; he lost 20, and found
five : how many did he then have ?
16. A man had 100 barrels of flour; he sold 50, and
afterward bought 10 : how many did he then have?
17. A farmer had 35 bushels of grain ; a part having
been wasted, he found there were but 22 bushels remaining : how much was wasted ?
18. John's father is 36 years old ; John is 12 : in how
many years will he be as old as his father now is ?
19. I had 65 cents ; spent 20 cents for a book and
ten for a slate : how many had I left ?
SUBTRACTION. 27
. If you take 10 from the sum of two numbers there
be 8 left : what is their sum ?
21. If you take 16 from the difference of two numbers
there will remain 12 : what is their difference ?
22. The sum of two numbers is 20 : what number musl
be added to make their sum 30 ?
23. The sum of two numbers is 16 more -than thei/
difference ; if their difference is 4, what is their sum ?
24. The greater of two numbers is 12, and their difference 5 : what is the less?
25. The sum of two numbers is 21 ; the less number is
eight : what is the greater ?
26. If you take one number from another, what is the
operation called ? Ans. Subtraction.
27. If you add one number to another, what is the
operation called? Ans. Addition.
28. In what respect do Addition and Subtraction differ?
Ans. One is exactly the reverse or opposite of the other.
29. When you take one number from another, what
do you call that which is left?
Ans. Difference or Remainder.
30. The remainder and less number, added together
are always equal to what ? Ans. The greater number.
SECTION IV. REVIEW.
LESSON I.
1. James had 13 marbles; he gave 2 to Henry, and
three to Thomas : how many had he left ?
ANALYSIS. He bad as many left as the difference between
13 marbles, and the sum of 2 marbles and 3 marbles ; the sum
of 2 marbles and 3 marbles is 5 marbles; 5 from 13 leaves 8
marbles ; therefore, if James had 13 marbles, and gave 2 to
Henry and 3 to Thomas, he had 8 left.
28 KAY'S INTELLECTUAL ARITHMETIC.
2. A merchant had 40 barrels of flour; he sold to
one man 9, to another 21 : how many had 2le left?
3. On Christmas day, William had 36 cents given
him; ho spent 6 cents for apples, 9 cents for cakes,
and 10 cents for candy : how many had he left?
4. A man paid 30 dollars for a horse, the keeping
cost 9 dollars, and he sold him for 29 dollars : how many
dollars did he lose ?
5. A man having 34 dollars, bought a barrel of
molasses for 15 dollars, and a bag of coffee for 10 dollars
: how many dollars had he left ?
6. A grocer bought some oranges for 9 dollars, some
lemons for 7 dollars, some prunes for 5 dollars, and some
figs for 9 dollars, and then sold them for 41 dollars:
how much did he gain ?
7. A lady bought a comb for 25 cents, some pins for
10 cents, tape for 7 cents, thread for 6 cents, and toy
books for 5 cents ; she gave 60 cents to the shopkeeper :
how much changae ougojht she to receive ?
8. Two boys commenced playing marbles ; each had
18 when they began ; when they quit, one had 25 : how
many had the other ?
*
9. Thomas has 7 marbles, David 5, and Moses 11;
how many have they altogether ? How many have Moses
and D?vid together more than Thomas?
10. Three boys commence playing marbles : Thomas
had 20, David 10, and Moses 4; when they quit, David
had 6 and Moses 8 : how many had Thomas ?
11. A farmer had 24 sheep: 9 of them were killed by
wolves, 5 of them were stolen, and 6 he sold : how many
had he left?
12. A grocer bought sugar for 12 dollars, flour for
six dollars, and coffee for 5 dollars ; he sold the whole
for 30 dollars : how much did he make ?
13. A lady had 50 cents ; she spent 25 cents for butter,
and 10 cents for eggs : how much had she left?
REVIEW. 29
14. A man is indebted to A, 5 dollars; to B, 6 dollars;
and to C, 10 dollars: he has cash to the amount of 20
dollars, and goods valued at 10 dollars : should he pay
his debts, how much would he be worth ?
LESSON II.
1. Four and 3, less 2, are how many?
ANALYSIS. As many as the difference between 2, and Hit
sum of 4 and 3 : 4 and 3 are 7 j 7 less 2 are 5 ; therefore, 4
and 3 less 2 are 5.
2. Five and 6 and 2, less 8, are how many?
3o Seven and 4 and 3, less 5, are how many ?
4. Eight and 5 and 4, less 3, are how many ?
5. Two and 3 and 5, less 7, are how many ?
6. Six and 4 and 3, less 6, are how many ?
7. Six and 3 and 5, less 7, are how many ?
8. Seven and 4 and 6, less 5, are how many ?
9. Eight and 5 and 4 and 6, less 5, are how many ?
10. Four and 7 and 6 and 5, less 8, are how many ?
11. Five and 8 and 4 and 9, less 7, are how many ?
12. Seven and 5 and 8 and 5, less 6, are how many ?
13. Eight and 4 and 3 and 9, less 5, are how many ?
14. Nine and 5 and 8 and 3 and 1, less 7, are how
many?
15. Eight and 6 and 5 and 2 and 4 and 3, less 4, are
how many?
16. Seven and 4 and 8 and 5 and 6 and 2 and 5, less
eight, are how many?
17. Nine and 7 and 5 and 3 and 6 and 8 and 7, less
six, are how many?
18. Eleven and 4 and 6 and 5 and 7 and 9, less 3, are
how many ?
80 RAY'S INTELLECTUAL ARITHMETIC.
19. Twelve and 5 and 7 and 6 and S and 3 and 4,
less 7, are how many ?
20. Twelve and 7 and 6 and 4 and 5 and 8 and 2,
less 7, are how many ?
21. Eleven and 6 and 5 and 3 and 6 and 8 and 6,
less 9, are how many?
22. Eleven and 7 and 5 and 4 and 5 and 9 and 6,
less 8, are how many ?
23. Eleven and 9 and 8 and 7 and 6 and 5 and 4,
less 3, are how many ?
24. Twelve and 9 and 7 and 6 and 5 and 3, less 7,
are how many ?
25. Thirteen and 4 and 5 and 7 and 8 and 6 and 9,
less 5, are how many?
26. Thirteen and 7 and 3 and 9 and 8 and 6 and 5,
less 8, are how many ?
27. Eighteen and 9 and 10 and 8 and 7, less 9, are
how many ?
28. Twenty-one and 5 and 6 and 7 and 8 and 9 and
10 and 9, less 8, are how many?
29. Seventy, less 10 and 9 and 8 and 6 and 5 and 4 and
3 and 2 and 4 and 5 and 6, are how many ?
LESSON III.
1. Henry had 24 cents, and spent all but 15 : how
many did he spend ?
2. A man bought a cask of wine containing 27 gallons
; after selling 10 gallons he found there were but 9
gallons remaining, the rest having leaked out : how much
did he lose ?
3. If from 20 you take 12 less 3, how many will
remain ?
4. If from the sum of 19 and 10, you take the difference
between 17 and 10, what will be left?
REVIEW. 3j
5. A man owed 60 dollars : lie paid at one time 20
dollars, and at another 30 dollars: he afterward borrowed
5 dollars : how much does he still owe ?
6. A maft paid 38 dollars for a horse, and 20 for a
colt: he afterward sold the colt for 10 dollars, and the
horse for 65 : how much did he make by the transaction ?
7. Twenty-four less 8, and 12 less 5, are together how
much less than 25 ?
8. Engaged to do a piece of work for 60 dollars : had
an assistant 25 days at a dollar a day, and paid 20 dollars
for materials : how much did I clear ?
9. If from the sum of 8 and 9 and 10 and 11, you
take the sum of 4 and 5 and 6 and 7, what will you have
remaining ?
10. A jeweler bought a watch for 40 dollars, a chain
for 15 dollars, and a key for 3 dollars : he sold them for
63 dollars : what did he gain ?
11. A drover bought sheep as follows : of one man 10 ;
of another, 12 ; of another, 5 ; of another, 3 : he sold at
one time 15 ; and at another, 5 : how many were left ?
12. A gentleman having 40 dollars, purchased a suit of
clothes : his pants cost 7 dollars ; vest, 5 dollars ; coat, 25
dollars : how much had he left?
13. What number must be added to 25, to make a
gum 14 less than 45 ?
14. What number must be taken from 62, to give a
result which shall be 12 more than 45 ?
15. If from the sum of 25 and 10 and 12, you take the
difference between 28 and 19, what will remain ?
16. A man bought a horse for 40 dollars : and after
paying 15 dollars for keeping him, sold him for 75 dollars
: how much did he make ?
17. A gentleman engaged in trade with 75 dollars :
after losing at one time 10 dollars, and at another 5, he
gained 20 dollars : how much did he then have ?
32 RAY'S INTELLECTUAL ARITHMETIC.
SECTION V. MULTIPLICATION.
LESSON I.
1. A boy gave 2 cents for one lemon, and 2 cents foi
another : how many cents did he give for both ?
2. How many, then, are 2 times 2? Why? Because
2 and 2 are 4.
3. A boy gave 3 cents for one peach and 3 cents for
another : how many cents did he give for both ?
4. How many, then, are 2 times 3? Why?
5. At 4 cents apiece, what will 2 pears cost?
6. How many are 2 times 4? 4 times 2? Why?
7. At 3 cents apiece, what will 3 peaches cost ?
8. How many are 3 times 3? Why?
9. At 3 cents apiece, what will 4 apples cost?
10. How many are 4 times 3 ? Why ?
11. At 3 cents apiece, what will 5 pears cost? Hov
many are 5 times 3? 3 times 5 ?
12. At 4 cents apiece, what will 4 lemons cost? How
many are 4 times 4 ?
13. At 5 dollars a yard, what will 4 yards of cloth
cost? Htfw many are 4 times 5 ? 5 times 4?
14. At 6 dollars a barrel, what will 4 barrels of flour
cost? How many are 4 times 6 ? 6 times 4?
15. At 5 cents apiece, what will 5 oranges cost? How
many are 5 times 5 ?
16. At 6 cents a yard, what will 5 yards of tape cost?
How many are 5 times 6 ? 6 times 5 ?
17. At 6 cents apiece, what will 6 oranges cost? How
many are 6 times 6 ?
18. At 7 cents a yard, what will 2 yards of tape cost?
How many are 2 times 7 ? 7 times 2 ?
MULTIPLICATION, 33
19. At 7 cents apiece, what will 3 lemons cost? How
many are 3 times 7 ? 7 times 3 ?
20. At 7 cents apiece, what will 4 oranges cost? How
many are 4 times 7 ? 7 times 4 ?
21. If 1 marble is worth 7 apples, how many apples are
worth 5 maizes ? How many are 5 times 7?
22. If 1 peach is worth 8 apples, how many apples are
worth 2 peaches ? How many are 2 times 8 ?
23. If 1 orange cost 8 cents, how many cents will
three oranges cost? How many are 3 times 8?
24. If 1 barrel of flour cost 8 dollars, how many dollars
will 4 barrels cost? How many are 4 times 8?
25. If 1 orange is worth 8 apples, how many apples
are 5 oranges worth ? How many are 5 times 8 ?
26. At 9 cents apiece, what will 2 oranges cost? How
many are 2 times 9 ? 9 times 2 ?
27. At 9 cents a yard, what. will 3 yards of ribbon cost?
flow many are 3 times 9 ? 9 times 3 ?
28. At 10 cents a quart, what will 3 quarts of chestnuts
cost? How many are 3 times 10?
29. If 1 yard of cloth cost 1 1 dollars, what will 3 yards
cost? How many are 3 times 11?
30. At 10 cents a bunch, what will 4 bunches of grapes
cost? How many are 4 times 10?
31. How many are 5 times 10? 10 times 5 ?
SUGGESTION. The exercises given under the subtraction table
can be extended to multiplication. One great use of such exercises
is to aw-iken an interest in the recitation.
Take, for example, the numbers 6 and 6. Five and 6 are 11
6 and 5 are 11 ; 5 from 11 leaves 6 ; 6 from 11 leaves 5; 5 times
$ are 30; 6 times 5 are 30,
RAY'S INTELLECTUAL ARITHMETIC.
LESSON II. MULTIPLICATION TABLE.
1 time 1 is 1 ,
MULTIPLICATION. 35
1 time 10 is 10
36 BAY'S INTELLECTUAL ARITHMETIC,
LESSON III.
1. At 2 cents each, what will 7 oranges cost?
ANALYSIS. Seven oranges will cost 7 times as much as i
orange. If 1 orange cost 2 cents, 7 oranges will cost 7 times
2 cents, which are 14 cents; therefore, 7 oranges at 2 cents
each will cost 14 cents.
2. At 7 cents each, what will 3 melons cost?
3. At 6 cents a dozen, what cost 5 dozen apples?
4. At 6 cents a pound, what cost 7 pounds of beef?
NOTE. The dollar sign, $, will now be used in place of the
word dollar : thus, $-5, $(3 ; read 5 dollars, t> dollars.
5. At $6 a pound, what cost 8 pounds of opium?
6. At $3 a barrel, what cost 9 barrels of cider?
7. At $4 a pair, what cost 7 pairs of boots ?
8. At 8 cents a dozen, what cost 10 dozen pens?
9. What cost 6 yards of cloth at $7 a yard ?
10. What cost 8 barrels of flour at $5 a barrel?
11. If a man travel 7 miles an hour, how far will lie
travel in 8 hours ?
12. On a chessboard are 8 rows of squares, and 8
squares in each row: how many squares on the board?
13. An orchard has 11 rows of trees, and 7 trees in
each row : how many trees in the orchard?
14. In 1 cent are 10 milis ; how many mills are there
in 3 cents? In 4? In 5? 6? 7? 8?
15. In 1 pint are 4 gills ; how many gills are there in
2 pints? In 3? In 4? 5? 6? 7? 8?
16. In 1 bushel are 4 pecks ; how many pecks are
there in 2 bushels? In 3? In 4? 5? 6? 7 ?
17. In 1 peck are 8 quarts ; how many quarts ara
&ere in 2 pecks ? In 3? In 4? 5? 6? 7?
MULTIPLICATION. 37
18. In 1 bushel how many quarts? Why ?
19. What will 9 yards of cloth cost at $6 a yard?
20. What will 9 oranges cost at 8 cents each?
21. Two men start from the same place and travel in
apposite directions : one travels 2 miles an hour, the other
4 miles : how far will they be apart at the end of 1 hour?
At the end of 2 hours ? 3 hours ?
22. If 2 men can do a job of work in 3 days, how
many days will it take 1 man to do it ?
ANALYSIS. It will require 1 man twice as long as 2 men.
If it take 2 men 3 days, it will take 1 man twice 3 days^
which are 6 days; therefore, if 2 men do a job of work in
3 days, 1 man will do it in 6 days.
23. If 3 men can do a piece of work in 4 days, in how
many days can 1 man do it ?
24. If 4 men can do a piece of work in 6 days, in how
many days can 1 man do it?
25. If a quantity of bread serve 8 men 4 days, how
many days will it serve 1 man ?
26. If a man can earn $6 in 1 week, how many dollars
can he earn in 8 weeks ?
27. A person has a job of work which 6 men can ao
in 9 days ; but it is necessary to do it in one day : how
many men must be employed ?
28. If 2 barrels of cider last 6 persons 4 weeks, how
many weeks will it last 1 person ?
29. If $9 worth of provisions last 8 persons 11 days,
how many person^ will it last 1 day ?
30. If the interest of $1 is 6 cents a year, what will be
the interest for 2 years? For 3 .years? For 4? 5?
For 6? 7? 8? 9? 10?
31. I bought 6 barrels of apples at $2 a barrel, and 4
barrels of sugar at $11 a barrel : how much did they
both cost?
2d Bk. 3
38 HAY'S INTELLECTUAL ARITHMETIC.
32. Four and 4 are 8, and 4 are 12, and 4 are 16, and
4 are 20 : here we find that 4 taken five times makes 20.
What is this operation called? Ans. Addition.
33. When we say 5 times 4 make 20, what do we call
the operation? Ans. Multiplication.
34. How, then, would you define Multiplication ? Ans.
Multiplication is a short method of performing several additions
of the same number.
35. The number to be multiplied, 4, is called the multiplicand
; the number you multiply by, 5, the multiplier;
and the answer, 20, the product.
LESSON IV.
1. Bought 2 apples at 2 cents each, 2 pears at 3 cents
each, and an orange for 5 cents : what did they cost ?
2c Two men start from the same place and travel in
the same direction ; one, 5 miles an hour ; the other, 7
miles : how far will they be apart in 10 hours ?
3. If, in the above question, the men travel in opposite
directions, how far will they be apart in 12 hours?
4. A lady went shopping with $15; she bought 4 yards
of cloth at 82 a yard ; 2 pairs of gloves at $1 a pair ;
and a shawl for $2 : what did they cost, and how much
had she left?
5. A man bought 4 peaches at 5 cents each, 3 pears
at 3 cents each, and 2 pints of chestnuts at 5 cents a
pint : how much did they cost ?
6. What will be the sum of 3 and 9 and 7?
less the
sum of 8 and 6 and 1 ?
7. If a man earn 5 shillings a day, and a boy 3 shillings,
how much will both earn in 7 days ?
8. A drover gave $10 and 7 sheep, valued at $4 *
head, for a cow and calf: how much did they cost?
MULTIPLICATION. 39
9. A merchant sold cloth at $7 a yard : a tailo* bought
of this cloth, at one time, 5 yards, and at another, 3 yards:
what was the amount of his bill ?
10 .Two brothers, Henry and Rufus, each received for
their work 3 dimes a day : how much did they both receive
for 6 days' work ?
11. If 12 horses can be sustained in a pasture 10
months, how many horses will it feed 1 month?
12. What is 3 times the difference between 15, and tho
sum of 5 and 2 ?
13. The sum of two numbers is 23; the smaller is 11 :
what is 5 times the larger?
14. The difference between two numbers is 7 : if the
larger be 12, what will 8 times the smaller be ?
15. If a boy buy apples at 1 cent each, and sell them
for 3 cents each, what would he make if he purchase 10
cents' worth of apples?
16. George bought a book for 50 cents and sold it
for $1 : what would he have made, had he bought 2 books,
and sold them at the same rate as the first ?
17 .Albert has 5 times two marbles less than 50, and
Edward has 5 times two more than 50 : how many has
each ?
18. If $1 gain $3 in a year, what will $12 gain in
double the time ?'
19. A man bought a cask of wine containing 20 gallons,
at $1 a gal. ; 5 gal. having leaked out, he sold the remainder
at $2 a gal. : how much did he make ?
20. Two men start from the same place, at the same
time, and tiavel the same way ; if they travel at the same
rate, how far will they be apart at the end of 10 hours?
If one goes 10 miles an hour, and the other 7, how fai
will they be apart in 7 hours ?
If they go in opposite directions, each at the rate of 5
miles per hour, how far will they be apart in 9 hours ?
40 RAY'S INTELLECTUAL ARITHMETIC,
21. A stage starts from a certain town, and travels at
the rate of 8 miles per hour : at the same time, another
starts from the same place, and travels in the same direction,
4 miles per hour : how far will they be apart at the
end of 12 hours ?
22. A grocer bought 10 pounds ot tea at 7 shillings
a pound ; after using 3 pounds, he sold the remainder
at 10 shillings a pound : how much did the 3 pounds
which he used cost him, in the end ?
23. Bought 6 bushels of corn at 5 dimes a bushel ;
sold 4 bushels at 6 dimes a bushel, and 2 bushels at 4
dimes a bushel : how much did I make ?
24. If an orange cost 5 cents, and an apple 2 cents,
what will 2 oranges and 4 apples cost ?
25. If pork is 8 cents, and beef 10 cents a pound, what
cost 7 pounds of pork and 6 pounds of beef?
26. If an orange cost 5 times as much as an apple, how
much more will 6 oranges cost than 25 apples, if an apple
is worth 1 cent?
27. If a pound of sugar cost 5 cents, and a pound of
coffee 3 times as much, less 3 cents, what will be the cost
of 3 pounds of sugar and 2 of coffee ?
*28 .Bought, at one time, 5 yards of muslin at 10 cents
a yard ; at another, 10 yards at 5 cents a yard : how
much did it all cost?
29. When salt is 4 cents a quart, and molasses 3 times
as much lacking 2 cents, what would be the cost of 3
quarts of molasses and 2 of salt ?
30. If a man earn $15 per week, and spend $11 a
week, how much will lae save in 3 weeks? How much
can he save in two months of 4 weeks each ?
* In the present edition of this volume, a very few examples
have been modified, the better to adapt them to a more perfect
gradation. Such are indicated by the character, 0. The omission
of these examples in classes having books of the present and for-
Bier edition, will obviate all confusion.
DIVISION, 41
SECTION VI. DIVISION
LESSON I.
1. At 1 cent each, how many cakes can you buy fot
4 cents ? 1 in 4 how many times ? Why ? Ans. Because
4 times 1 are 4.
2. At 2 cents each, how many apples can you buy
for 4 cents ?
ANALYSIS. You can buy as many apples as 2 cents are
contained times in 4 cents ; 2 cents are contained in 4 cents
2 times ; therefore, at 2 cents each, you can buy 2 apples for
4 cents.
Two in 4 how many times ? Why ? Because 2 times
2 cure 4.
3. Among how many boys can 6 apples be divided,
giving to each boy two apples ? 2 in 6 how many times ?
Why ? Because 3 times 2 are 6.
4. At 2 cents each, how many apples can you buy
for 8 cents ? 2 in 8 how many times ? Why ?
5. At 3 cents each, how many peaches can you buy
for 6 cents ? 3 in 6 how many times ? Why ?
6. At 3 cents each, how many pears can you buy
for 9 cents ? 3 in 9 how many times ? Why ?
7. At 2 cents each, how many cakes can you buy
for 10 cents ? 2 in 10 how many times ? Why ?
8. At 2 cents each, how many marbles can you buy
for 14 cents ? 2 in 14 how many times ? Why ?
9. At 5 cents each, how many lemons can you buy
for 15 cents ? 5 in 15 how many times ? Why?
10. A boy has 16 marbles, and wishes to divide them
into piles of 2 each : how many piles will there be 1
How many twos in 16 ? Why ?
42 RAY'S INTELLECTUAL ARITHMETIC.
11. At 3 cents each, how many peaches can you buy
for 18 cents ? 3 in 18 how many times ?
12. At 5 cents each, how many oranges can you buy
for 20 cents ? 5 in 20 how many times ?
13. At $3 a yard, bow many yards of cloth can yov
buy for $21 ? 3 in 21 how many times ?
NOTE. The dollar sign, $, is used ie. place of the word dollar ;
thus, $4, $7 ; read 4 dollars, 7 dollars.
14. A lady spent 22 cents for tape, at 2 cents a yard:
how many yards did she buy ?
15. At 6 cents each, how many oranges can you buy
for 24 cents ? How many at 8 cents each ? 6 in 24 how
many times ? 8 in 24 how many times ?
16. In an orchard of 25 apple trees there are 5 rows;
how many trees in each row ?
ANALYSIS. One tree in each row requires 5 treec ; hence,
there will be as many rows, as 5 trees are contained times in
25 trees ; 5 trees in 25 trees, 5 times. Ans. 5 rows.
17. If a man can travel 3 miles in an hour, how many
hours will it take him to travel 27 miles ? 3 in 27 how
many times ?
18. A man gave 28 for sheep, at $4 a head : how many
did he buy ? 4 in 28 how many times ?
19. If you had 30 cents, how many marbles could you
buy at 3 cents each ? 3 in 30 how many times ?
20. There are 32 cents on a table, in 4 piles : how many
in eacn pile? 4 in 32 how many times?
21. In an orchard containing 35 apple trees, there are
5 rows : how many trees are there in each row ? 5 in 35
how many times ?
22. Six men receive $36 for a job of work: what is
each man's share ? 6 in 36 how many times ?
DIVISION. 43
23. Four quarts make 1 gallon : how many gallons in
36 quarts ? 4 in 36 how many times ?
24. If a man travel 10 miles in 1 hour, in how many
hours will he travel 40 miles ?
25. Forty-two cents were diyided equally among 6 boys;
how many cents did each boy receive ?
26. Forty-two are how many times 7 ?
27. If you divide 45 apples equally among 9 boys, how
many apples will each boy receive ?
LESSON II -DIVISION TABLE.
2 in 2 1 timo
44 . BAY'S INTELLECTUAL ARITHMETIC.
8 in 8 1 time
DIVISION. 45
3. Seven in 14 how many times ? 7 in 28 ? 7 in
42? 7 in 56? 7 in 63 ? 7 in 84? 8 in 24? 8 in
40 ? 8 in 56 ? 8 in 72 ? 8 in 96 ?
4. Nine in 18 how many times ? 9 in 27 ? 9 in
45? 9 in 54? 9 in 63 ? 9 in 81 ? 9 in 108 ? 10 in
20? 10 in 60? 10 in 90 ? 10 in 100 ?
5. Eleven in 55 how many times? 11 in 77 ? 11
in 99? 11 in 110? 11 in 121 ? 12 in 24 ? 12 in 48 ?
12 in 60? 12 in 72? 12 in 96? 12 in 108? 12 in
120 ? 12 in 144 ?
6. If 12 peaches be divided equally among 3 children,
how many will each have ?
7. Four boys gave their sister 24 apples, each an equal
number : how many did each give ?
8. A mother divided 20 cents equally between her 2
little girls : how many did each receive ?
9. Five books cost 35 cents: how much is that
apiece ?
10. A man has $40 : if he spend $5 a week, how long
will it last ?
11. There are 3 feet in 1 yard: how manv yards are
there in 21 feet? In 27 feet? In 36 feet?
12. Four quarts make 1 gallon : how many gallons in
28 quarts ? In 16 ? In 32 ? In 36 ? 44 ? 48 ?
13. If 5 apples are worth 1 pear, how many pears are
worth 25 apples ? 35 apples ? 45 apples ?
14. If 6 pears are worth an orange, how many oranges
can you get for 30 pears ? For 42 pears? For 54 pears?
For 66 pears?
15. If 1 man do a piece of work in 42 days, how many
days will it take 7 men ?
16. If 1 man eat a certain quantity of provisions in 56
days, how many days will it last 7 men ?
17. If 1 pipe empty a cistern in 63 hours, in how many
hours will 9 pipes of the same size empty it ?
46 BAY'S INTELLECTUAL ARITHMETIC.
18. Eight quarts make a peck : how many pecks in 24
quarts ? In 40 ? In 56 ? In 72 ?
19. If hay is worth S9 a ton, how many tons can be
bought for 27 ? For $45 ? For 54 ? For 63 ?
20. Ten men bought a horse for 60 : how much did
each one pay?
21. If 11 ounces cost 88 cents: what cost 1 ounce?
22. A man paid 108 for 12 Saxony sheep : how inucJ*
was that apiece ?
23. In an orchard there are 120 trees in 10 rows : how
many trees in each row ?
24. A man earns 144 in 12 weeks : how much is that
a week ? How much a day, allowing 6 working days to
the week ?
25. If 6 men earn 84 in 7 days, how much do they
all earn in 1 day?
26. If 9 men earn 108 in 3 days, how much does i
man earn ? How much does each man earn in a day ?
27. If 2 from 6 leaves 4, 2 from 4 leaves 2, and 2 from
2 leaves 0, how many times is 2 taken from 6 ? Ans. 3.
What do you call the operation ? Subtraction.
28. Two is contained in 6, 3 times ; what do you call
the operation? Ans. Division.
29. How would you define Division? Ans. Division
is subtracting the same number several times; or, finding
koto many times one number is contained in another.
30. The number to be divided, 6, is called the dividend;
the number you divide by, 2, the divisor ; and the answer,
8, the quotient.
LESSON IV.
1. Twelve are how many times 2? 3? 4? 6?
2. Twenty-four are how many times 3? 6? 8? 12?
3. Seventy-two are how many times 12? 8? 0? 9?
DIVISION. 47
4. How many oranges at 5 cents each, must be given
for 10 pears at 2 cents each ?
5. A wheel is 10 feet in circumference ; what distance
Will it move in making one revolution ? how many revolutions
will it make in going 120 feet?
6. An orchard contains 10 rows of trees, and 7 trees
in a row ;
if there were but 5 rows, how many trees would
there be in a row ?
7. I have three times as many marbles as the sum
of 1, 2, and 3, is contained times in 60 : how many
have I ?
8. Bought 6 hats at $5 apiece, and 4 yards of cloth
at $3 a yard ; gave in exchange flour at $6 a barrel : how
many barrels did it take ?
9. If a man gain 6 miles in 3 hours, how long will
it take to gain 24 miles ?
10. Two times 6 are contained how many times in the
sum of 36 and 12 ?
11. If 60 be divided by some number, the result will
be 10 : what is that number ?
12. I have a number in my mind which, divided by 3,
gives 2 times 6 : what is the number ?
13 .If I purchase lemons at the rate of 2 for 6 cents,
and sell 7 for 28 cents, how much do I gain ?
14. A man has a job of work which 9 men can perform
in 2 days ; he desires to complete it in 3 days : how
many men must he employ ?
15. Five times the sum of two numbers is equal to 60;
if 7 is one of them, what is the other?
16. Henry has 6 dimes ; Thomas twice as many less 2;
and Samuel 3 times as many as Henry : how many have
they together ?
17 .If to the number of times 4 is contained in 12, you
add 3? and subtract the result from 9, what will remain ?
48 RAY'S INTELLECTUAL ARITHMETIC.
18. Five oranges were sold for 25 cents, and 10 cents
were gained : what did each cost ?
19. What number subtracted from 17, will leave double
the remainder that 5 from 9 leaves ?
20. A boy said that 10 taken from the number of
apples he had, left twice as great a remainder as the difference
between 1 dozen and 8 : how many had he ?
21. A certain number multiplied by 10, is 5 less
than 45 : what is that number ?
22. If you multiply any number, 10, by any other
number, 5, and divide the product by the same number,
5, what will be the result ?
23. If 2 oranges are worth 5 apples, how many apples
are worth 12 oranges ?
ANALYSIS. As many times 5 apples, as 2 oranges are con*
tamed times in 12 oranges.
24. One man goes 10 miles while another goes 7 ;
when the first has gone 90 miles, how far will the
second have gone ?
25. James earns 8 cents while John earns 12; when
John has earned 60, how many has James earned ?
26 .George learns 5 lessons while Charles learns 4:
how many lessons will both have learned when Charle?
has learned 20 ?
27 . A man can earn 9 while a boy earns 5 : how
many dollars will both have earned when the man has
earned $36 ?
28. How many times will the hammer of a common
clock strike, from noon till 6 o'clock in the evening?
29. Two numbers added together make 30 ; if the
greater number was 5 more, and the less 3 more, what
would their sum then be ?
30. William can count 11 while James counts 7: low
many will James count while William is counting 77 ?
PARTS OF NUMBERS. 49
31. When flour was 5 cents a pound and sugar 11
cents, 10 pounds of flour and 5 pounds of sugar were
given for a box of eggs at 5 cents a dozen : how many
dozen were in the box ?
32. The sum of two numbers is 24 ; if they were both
equal to the greater, their sum would be 28 : what are
the numbers?
SEC. VII. PARTS OF NUMBERS.
LESSON I.
If a unit or whole thing is divided into TWO equal parts,
one of the parts is called ONE-HALF.
1. When an apple is divided into two equal parts, what
is one part called? Ans. One-half of an apple.
- 2. How many halves in 1 apple?- Ans. Two halves.
3. How many halves in 2 apples ? In 3 ? In 4 ?
[n5? In 6? In 7? In 8? In 9? In 10?
ANALYSIS. In 2 apples there are two times as many halves
as there are in 1 apple; but in 1 apple there are 2 halves, and
in 2 apples there will be 2 times 2 halves, or 4 halves ; therefore,
there are 4 halves in 2 apples.
4. How many halves in 2 apples and one-half of an
apple ? In 3 apples and one-half? In 4 and one-half?
5. What do you mean by one -half of anything?
If any thing or any number is divided into THREE
equal parts, one of the parts is called ONE-THIRD of the
thing or number ; two parts are called TWO-THIRDS and
three parts, THREE-THIRDS, or the whole.
6. When an apple, or any thing is divided into three
equal parts, what is one part called ? What two parts ?
7. How many thirds in 1 apple ? In 2 ? In 3 ? In
4? In 5? In 6? In 7? In 8? In 9?
50 RAY'S INTELLECTUAL ARITHMETIC.
8. How many thirds in 1 apple and one-third of **
apple ? In one apple and two-thirds of an apple ?
9. How many thirds in two apples and one-third?
In 2 apples and two-thirds ?
10. What do you understand by one -third of any
thing? By two-thirds? Ans. That a whole thing has
been divided into 3 equal parts and 2 of those parts taken.
If any thing is divided into 4 equal parts, 1 of the parts
is called ONE-FOURTH ; two parts are called TWO-FOURTHS,
and so on.
When any thing is divided into 5 equal parts, 1 of the
parts is called ONE-FIFTH; two parts, TWO-FIFTHS; three
.parts, THREE-FIFTHS, and so on.
11. When an apple is divided into four equal parts, what
is one part called ? What two parts ? Ans. Two-fourths.
What three parts ? Am. Three-fourths.
12. How many fourths in 1 apple ? In 2 apples ? In
3 apples? In 4? In 5 ? In 6 ? In 7?
13. How many fourths in one apple and one-fourili?
In 1 and two -fourths ? In 1 and three-fourths ?
14. In 2 apples and one-fourth how many fourths?
In 2 apples and two-fourths ? In 2 and three-fourths ?
15. What are fourths often called ? Ans. Quarter*.
16. What do you understand by one-fourth of any
thing ? By two-fourths ? By three-fourths ?
17. When an apple is divided into 5 equal parts, what
is one part called? What are 2 parts called? What 3
parts? What 4 parts? Ans. Four-fifths.
18. How many fifths in 1 apple? In 2 ? In 3 ?
In 4 ? In 5 ? In 6 ? In 7 ? In 8 ? In 9 ?
19. How many fifths in 1 apple and 1 -fifth of an
apple ? In 1 and 2-fifths? 1 and 3-fifths? 1 and 4-fifths?
20. How many fifths in 2 apples and 1 -fifth ? In 2 and
2-fifths ? 2 and 3-fifths ? 2 and 4-fifths ?
PARTS OF NUMBERS. 51
21. What do you understand by 1 -fifth of any thing?
By 2-fifths ? By 3-fifths ?
22. When an apple is divided into 6 equal parts, what
is 1 part called ? What are 2 parts called ? What 3 parts ?
W^hat 4 parts ? What 5 parts ?
23. How many sixths in 1 apple ? In 2 apples ? In
3 apples? In 4? In 5 ? In 6? In 7? In 8?
24. How many sevenths in 1 apple ? In 2 apples ?
Ln3? In 4? In 5? In 6? In 7? In 8 ?
25. How many eighths in 1 apple? In 2? In 3 ?
In 4? In 5? In 6? In 7? In 8 ? In 9?
26. How many ninths in 1 apple? In 2 ? In 3 ?
In 4? In 5? In 6? In 7 ? In 8? In 9?
LESSON II.
1. If a yard of tape is worth 2 cents, what is onehalf
of it worth ? Ans. One cent.
2. What is one -half of 2 cents ? Ans. One cent.
Why ? Ans. Because if you divide 2 cents into two equal
parts, one of the parts is 1 cent.
3. One is what part of 2 ? Ans. One is the half of 2.
4. If you can buy an apple for 2 cents, how many
can you buy for 3 cents? Ans. One apple and l-half.
5. Three are how many times 2 ? Ans. Once 2 and
l-half of 2.
6. Four are how many times 2 ?
7. If 2 cents will buy 1 yard of tape, how many
yards will 5 cents buy ?
8. Five are how many times 2? -4ns. Two times 2
and l-half of 2.
9. 6 are how many times 2 ? How many 2's ?
10. 7 are how many times 2 ? How many 2's ?
11. 8 are how many times 2? How many 2's?
52 RAY'S INTELLECTUAL ARITHMETIC.
12. 9 are how many times 2 ? How many 2's ?
13. 10 are how many times 2 ? How many 2's ?
14. If an apple is worth 3 cents, what is 1 -third of it
worth ? What are 2-thirds of it worth ?
15. What is 1-third of 3? What are 2-thirds of 3?
16. If an orange is worth 3 cents, what part of it will
1 cent buy ? What part will 2 cents buy ?
17. One is what part of 3? Ans. One is I -third of 3.
18. Two is what part of 3 ? Ans. Two is 2-thirds of 3,
or 2 times I -third of 3.
19. If a yard of cloth cost S3, how much can you buy
for 84 ? How much for $5 ?
20. Four are how many times 3? Ans. Once 3 and
1-third of 3.
21. Five are how many times 3 ? Ans. Once 3 and
2-thirds of 3.
22. 6 are how many times 3 ? 7 are how many 3's ?
23. 8 are how many times 3 ? 9 are how many 3's ?
24. 10 are how many times 3 ? 11 are how many 3's ?
25. If a lemon is worth 4 cents, what is 1 -fourth of it
worth ?
26. What is 1-fourth of 4 ? What are 2-fourths of 4?
What are 3-fourths of 4 ?
27. If you buy a yard of cloth for $4, what part of it
can you buy for 91 ? For 82 ? For 83 ?
28. What part of 4 is 1 ? Ans. 1-fourth of 4.
29. What part of 4 is 2 ? Ans. 2-fourths of 4.
30. -What part of 4 is 3 ? Am. ^-fourths of 4.
31. If a yard of tape cost 4 cents, how much can yoa
buy for 5 cents ? For 6 cents ? For 7 cents ?
32. Five are how many times 4? Ans. Once 4 and
one-fourth of 4.
33. Six are how many times 4 ? Ans. Once 4 and
two-fourths of 4.
PARTS OF NUMBERS. 53
34. Seven are how many times 4? Ans. Once 4 and
three-fourths of 4.
35. 8 are how many times 4 ? 9 are how many 4's ?
36. 10 are how many times 4 ? 11 are how many 4's ?
37. 12 are how many times 4 ? 13 are how many 4's ?
38. 14 are how many times 4 ? 15 are how many 4's ?
39. 16 are how many times 4 ? 17 are how many 4's f
40. 18 are how many times 4 ? 19 are how many 4's ?
41. 20 are how many times 4 ? 21 are how many 4's ?
42. If a melon is worth 5 cents, what is one-fifth of
it worth? What are two-fifths worth? What are threefifths
worth ? What are four-fifths worth ?
43. What is 1-fifth of 5 ? 2-fifths of 5 ? 3-fifths of
57 4-fifthsof5?
44. If a melon is worth 5 cents, what part of it can
you buy for 1 cent ? For 2 cents ? For 3 ? For 4 ?
45. 1 is what part of 5 ? Ans. I -fifth of 5.
46. 2 is what part of 5 ? Ans. 2-fifths of 5.
47. 3 is what part of 5 ? Ans. ^-fifths of 5.
48. 4 is what part of 5 ? Ans. ^-fifths of 5.
49. If flour is worth $5 a barrel, how many barrels
can you buy for $6 ? For $7 ? For $8 ? For $9 ? For
810? For
50. 6 are how many times 5 ? Once 5 and 1-fifth of 5.
51. 7 are how many tinies-5 ? Once 5 and 2-fifthsof5.
52. 8 are how many times 5 ? Once 5 and 3-fifths of 5.
53. 9 are how many times 5 ? 10 are how many times 5?
54. 11 are how many times 5? 12 are how many times 5?
55. If 1 barrel of apples cost $6, what is 1 -sixth worth?
What 2-sixths ? What 3-sixths ? 4-sixths ? 5-sixths ?
56. What is 1-sixth of 6? What are 2-sixths of 6?
S-sixths of 6? 4-sixths of 6? 5-sixths of 6?
2d Bk. 4
54 KAY'S INTELLECTUAL ARITHMETIC.
57. One is what part of 6 ? 2 is what part of 6 ? 3 is?
what part of 6 ? 4 is what part of 6 ? 5 ?
58. How many yards of cloth at $6 a yard, can you
buy for 7? For 8? For $9? For $10? For 811 ?
For $12 ? For 813 ? For $14 ?
59. Seven are how many times 6 ? Ans. One time 6
&nd I -sixth of 6.
60. 8 are how many times 6 ? 8 are how many 6's ?
61. 9 are how many times 6 ? 10 are how many 6's?
62. 11 are how many times 6? 12 are how many 6's?
63. 13 are how many times 6 ? 14 are how many 6 T
s ?
64. 15 are how many times 6 ? 16 are how many 6's?
65. At 7 cents a yard for ribbon, what is 1 -seventh
worth ? What are 2-sevenths worth ? What are 3-sevenths?
4-sevenths? 5-sevenths? 6-sevenths?
66. What is 1 -seventh of 7 ? What are 2-sevenths of
7 ? What are 3-sevenths ? 4-sevenths ? 5-sevenths ?
6-sevenths ?
67. One is what part of 7 ? 2 is what part of 7? 3
is what part of 7 ? 4 is what part of 7 ? 5 ? 6 ?
68. How many barrels of flour at $7 a barrel, can you
buy for $8? For $9? For $10? For $11? For $12?
For $13? For $14? For $15 ? For $16?
69. 8 are how many times 7 ? 9 are how many 7's ?
70. 10 are how many times 7 ? 11 are how many 7's ?
71. 12 are how many times 7 ? 13 are how many 7's ?
72. 15 are how many times 7 ? 16 are how many 7's ?
73. 17 are how many times 7 ? 18 are how many 7's ?
74. If a melon cost 8 cents, what is 1 -eighth worth?
What are 2-eighths worth ? 3-eighths worth ? 4-eighths
worth? 5-eighths? 6-eighths? 7-eighths?
75. What is 1 -eighth of 8 ? What are 2-eighths of 8T
3-eighths of 8? 4-eighths? 5-eightlw? 6-eighths?
7-eighths ? 8-eighths ? 9-eighths ? 10-eigfcths ?
PARTS OF NUMBERS. 55
76. One is what part of 8 ? 2 is what part of 8 ? 3 is
what part of 8 ? 4 is what part of 8 ? 5 is what part
of 8 ? 6 is what part of 8 ? 7 ?
77. How many gallons of beer, at 8 cents a gallon, can
you buy for 9 cents ? For 10 cents ? For 11 cents ?
For 12 cents ? For 13 cents ? For 14 cents ? Foj 15
cents ? 16 cents ? 17 cents ? 18 cents ?
78. 9 are how many times 8 ? Once 8 and \-eiglitTi of 8.
79. 10 are how man^ times 8 ? 11 are how many 8's ?
80. 12 are how many times 8 ? 13 are how many 8's ?
81. 14 are how many times 8 ? 15 are how many 8's ?
82. 16 are how many times 8 ? 17 are how many 8's ?
83. Eighteen are how many times 8 ? Ans. Two time*
8 and 2-ciglitlis of 8.
84. If an orange cost 9 cents, what is 1 -ninth of it
worth ? What are 2-ninths worth ? 3-ninths ? 4-ninths ?
5-ninths ? 6-ninths ? 7-ninths ? 8-ninths ?
85. What is 1-ninth of 9 ? What are 2-ninths of 9 ?
What are 3-ninths of 9 ? 4-ninths of 9 ? 5-ninths of 9?
6-ninths of 9 ? 7-ninths of 9 ? 8-ninths of 9 ?
86. One is what part of 9 ? 2 is what part of 9 ? 3 is
what part of 9 ? 4 is what part of 9 ? 5 is what part
of 9 ? 6 ? 7 ? 8 ?
87. If cloth is $9 a yard, how much can you buy for
$10? For $11? For $12? For $13 ? For $14? For
$15? For $16? For $17? For $18? For $19?
For $20 ?
88. If apples cost 10 cents a bushel, what is 1 -tenth
of a bushel worth? 2-tenths? 3-tenths ? 4-tenths ?
- 5-tenths ? 6-tenths ? 7-tenths ? 8-tenths ? 9-tenths ?
89. One is what part of 10? 2 is what part of 10?
3 is what part of 10 ? 4 is what part of 10 ? 5 is what
part of 10 ? 6 is what part of 10 ? 7 is what part of
10 ? 8 is what part of 10 ? 9 ? 11 ?
56 RAY'S INTELLECTUAL ARITHMETIC.
90. When cloth is 10 a yard, how much can you buy
for 11 ? For 12 ? For 13 ? For 14 ? For 15 ?
For 16 ? For 17 ? For 18 ? For 19 ?
91. What do you understand by 1 -fifth of any thing?
By 3-fifths? By 4-fifths?
92. What do you understand by 3-sevenths of any
thing? By 2-eighths?
93. If I cut an apple into 6 equal parts, and give you
4, what part of the whole apple would I give you ?
94. If I divide an orange into 8 equal parts, and give
you 5, what part of the orange would I give you ?
95. What is meant by 1 -ninth of any thing ? 2-ninths ?
96. What is meant by 1-tenth of any thing ? 2-tenths ?
LESSON III.
1. If you had 7 cents, how many cakes could you
buy at 2 cents each? Ans. 3 cakes and 1-half.
2. Seven are how many times 2 ? Ans. 3 times 2 and
l-half of 2.
3. If you had 11 cents, how many pears could you
buy at 2 cents each ? At 3 cents ?
4. Eleven are how many times 2 ? How many times
3 ? Ans. Three times 3 and 2-thirds of 3.
5. If you had 15 cents, how many cakes could you
buy at 2 cents each ? At 3 cents ? At 4 cents ? At 5 ?
6. Fifteen are how many times 2? 3? 4? 5?
7. If you had 17 cents, how many oranges could you
buy at 2 cents each ? At 3 cents each ? At 4 cents each ?
At 5 cents ? At 6 cents ? At 7 cents ?
8. 24 are how many times 2 ? 3? 4? 5? 6? 7? 8?
9. 25 are how many times 2 ? 3? 4? 5? 6? 7? 8?
10. 26 are how many times 3 ? 4? 5? 6? 7? 8? 9?
11. . 27 are how many times 3 ? 4? 5? 6? 7? 8? 9?
12. 28 are how many times 3 ? 4 5? 6? 7? 8? 9?
PARTS OF NUMBERS. 5-7
13. 29 are how many times 3 ? 4? 5? 6? 7? 8? 9*
14. 30 are how many times 3? ,4? 5? 6? 7? 8? 9?
15. 31 are how many times 3? 4? 5 ? 6? 7? 8? 9?
16. 32 are how many times 3? 4? 5? 6? 7? 8? 9?
17. 33 are how many times 3 ? 4? 5? 6? 7? 8? 9?
18. 34 are how many times 3? 4? 5? 6? 7? 8? 9?
19. 35 are how many times 3 ? 4? 5? 6? 7? 8? 9?
20. 36 are how many times 3? 4? 5? 6? 7? 8? 9?
21. 37 are how many times 3 ? 4? 5? 6? 7? 8? 9?
22. 38 are how many times 3? 4? 5? 6? 7? 8? 9?
23. 39 are how many times 4? 5? 6? 7? 8? 9?
24. 40 are how many times 4? 5? 6? 7? 8? 9?
25. If you had 41 cents, how many oranges could you
buy at 4 cents each ? At 5 cents eac-h ? At 6 cents ?
At 7 cents ? At 8 cents ? At 9 cents ? At 10 cents ?
26. 42 are how many times 4? 5? 6? 7? 8? 9?
27. 43 are how many times 4 ? 5? 6? 7? 8? 9?
28. 44 are how many times 4 ? 5? 6? 7? 8? 9?
29. 45 are how many times 4? 5? 6? 7? 8? 9?
30. 46 are how many times 4? 5? 6? 7? 8? 9?
31. 47 are how many times 4? 5? 6? 7? 8? 9?
32. 48 are how many times 4? 5? 6? 7? 8? 9?
33. 49 are how many times 4? 5? 6? 7? 8? 9?
34. If you had 56 cents, how many peaches could you
buy at 5 cents each ? At 6 cents each ? At 7 cents ?
At 8 cents ? At 9 cents ? At 10 cents ?
35. 56 are how many times 5? 6? 7? 8? 9? 10?
36. 57 are how many times 5? 6? 7? 8? 9? 10?
37. 58 are how many times 5? 6? 7? 8? 9? 10?
38. 59 are how many times 5? 6? 7? 8? 9? 10?
39. 60 are how many times 5? 6? 7? 8? 9? 10?
58 KAY'S INTELLECTUAL ARITHMETIC.
40. 61 are how many times 6? 7 ? 8? 9? 10?
41. 63 are how many times 6 ? 7 ? *? 9? 10?
42. 65 are how many times 6 ? 7? 8? 9? 10?
43. 66 are how many times 6? 7? 8? 9? 10?
44. 68 are how many times 6? 7? 8? 9? 10?
45. 69 are how many times 6? 7? 8? 9? 10?
46. If 70 hours be required to perform a piece of
work, in how many days can it be done by working 6
hours a day ? By working 7 hours a day ? 8 hours a
day ? 9 hours ? 10 hours ?
47. 71 are how many times 6? 7? 8? 9? 10?
48. 72 are how many times 6? 7? 8? 9? 10?
49. 74 are how many times 6?' 7? 8? 9? 10?
50. 76 are how many times 6? 7? 8? 9? 10?
51. 77 are how. many times 6 ? 7? 8? 9? 10?
52. 79 are how many times 7? 8? 9? 10? 11?
53. 80 are how many times 7 ? 8? 9? 10? 11?
54. 81 are how many times 7 ? 8 ? 9 ? 10 ? 11 ?
55. 82 are how many times 7 ? 8? 9? 10? 11?
56. 83 are how many times 7 ? 8 ? 9 ? 10 ? 11 ?
57. 85 are how many times 7 ? 8? 9?^ 10? 11?
58 . 86 are how many times 7? 8? 9? 10? 11?
59. 87 are how many times 7? 8? 9? 10? 11?
60. 88 are how many times 7 ? 8 ? 9 ? 10 ? 11 ?
61. 89 are how many times 7 ? 8? 9? 10? 11?
62. 90 are how many times 7? 8? 9? 10? 11?
63. 91 are how many times 8 ? 9 ? 10 ? 11 ? 12 ?
64. 92 are how many times 8 ? 9 ? 10 ? 11 ? 12 ?
65. 93 are how many times 8 ? 9? 10? 11? 12?
66. 94 are how many times 8 ? 9? 10? 11? 12?
67. 95 are how many times 8 ? 9 ? 10 ? 11 ? 12 ?
68. 96 are how many times 8 ? 9 ? 10 ? 11 ? 12 ?
PARTS OP NUMBERS. 59
69. 98 are how many times 8 ? 9 ? 10 ? 11 ? 12 ?
70. 99 are how many times 8 ? 9 ? 10 ? 11 ? 121
71. 100 are how many times 8 ? 9 ? 10 ? 11 ? 12 ?
SECTION VIIL
LESSON I.
1. At 2 cents each, what will 2 apples and 1-half of
an apple cost?
ANALYSIS. If 1 apple cost 2 cents, 2 apples will cost
twice 2 cents, which are 4 cents; and 1-half of an apple
will cost 1-half of 2 cents, which is 1 cent; and 4 cents and
1 cent are 5 cents ; therefore, if 1 apple cost 2 cents, 2 apples
and l-half will cost 5 cents.
2. Two times 2 and 1-half of 2 are how many?
3. At 2 cents each, what will 3 pears and 1-half of a
lear cost?
4. Three times 2 and 1-half of 2 are how many?
5. At $3 a yard, what will 3 yards and 1-third of a
yard of cloth cost?
6. Three times 3 and 1-third of 3 are how many?
7. At $3 a barrel, what will 3 barrels and 2-thirds of
a barrel of flour cost?
SUGGESTION. To find 2-thirds of any number, first find 1-third,
then multiply by 2.
8. Three times 3 and 2-thirds of 3 are how many?
9. At S3 a yard, what will 4 yards and 2-thirds of a
yard of cloth cost?
ANALYSIS. If 1 yard cost $3, 4 yards will cost 4 times
$3, which are $12; and 1-third of a yard will cost 1-third
of $3, winch is $1 ; and 2-thirds of a yard will cost twice
as much as 1-third, that is, twice $1, which are $2; and $12
and $2 are $14. Ans. $14.
bU KAY'S INTELLECTUAL ARITHMETIC.
10. If an orange cost 4 cents, what will 3 oranges
and 1 -fourth of an orange cost ?
11. 3 times 4 and 2 -fourths of 4 are how many?
12. 4 times 4 and 3-fourths of 4 are how many ?
, 13. At 85 a barrel, what will 4 barrels and 2-fifths of
a barrel of flour cost?
14. 4 times 5 and 2-fifths of 5 are how many ?
15. 5 times 5 and 3-fifths of 5 are how many ?
16. If a man spend $6 a week, how much will he spend
in 4 weeks and 1 -sixth of a week ?
17. 4 times 6 and 2-sixths of 6 are how many ?
18. 5 times 6 and 3-sixths of 6 are how many ?
19. 6 times 6 and 4-sixths of 6 are how many ?
20. 7 times 6 and 5-sixths of 6 are how many ?
21. At 7 cents a yard, how much will 3 yards and
2-sevenths of a yard of tape cost ?
22. 4 times 7 and 3-sevenths of 7 are how many ?
23. 5 times 7 and 4-sevenths of 7 are how many ?
24. 6 times 7 and 6-sevenths of 7 are how many ?
25. If oranges are worth 8 cents each, how much are
3 oranges and 2-eighths of an orange worth ?
26. 4 times 8 and 3-eighths of 8 are how many ?
27. 5 times 8 and 4-eighths of 8 are how many ?
28. 6 times 8 and 7-eighths of 8 are how many ?
29. If a yard of muslin cost 9 cents, what will 2 yards
and 2-ninths of a yard cost ?
30. 3 times 9 and 5 -ninths of 9 are how many ?
31. 5 times 9 and 6 -ninths of 9 are how many ?
32. 7 times 9 and 8-ninths of 9 are how many ?
33. If a pound of sugar cost 10 cents, what will 2
pounds and 2-tenths of a pound cost ?
PARTS OF NUMBERS. Q\
34. 2 times 10 and 3-tenths of 10 are how many ?
35. 5 times 10 and 6-tenths of 10 are how many ?
36. 6 times 10 and 8-tenths of 10 are how many ?
37. 9 times 10 and 7-tenths of 10 are how many ?
38. 10 times 9 and 8-ninths of 9 are how many ?
39. 12 times 11 and 9-elevenths of 11 are how many ?
SECTION IX.
LESSON I.
1. A boy bought 3 apples at 4 cents each : how much
did they cost? He paid for them with oranges, at 6 cents
each : how many oranges did it take ?
ANALYSIS. First. 3 apples cost 3 times the price of 1 apple,
that is, 3 times 4 cents, or \2 cents. Second. It took as many
oranges as 6 cents are contained times in 12 cents; 6 cents
are contained in 12 cents 2 times, Ans. 2 oranges.
2. A man bought 3 yards of cloth at $4 a yard : how
many dollars did it cost? He paid for it with cider, at
$2 a barrel : how many barrels did it take ?
3. A boy bought 8 apples at 2 cents each : how much
did they cost? He paid for them with pears, at 4 cents
each : how many did it take ?
4. Bought 9 marbles at 2 cents each ; paid for them
with tops, at 6 cents each : how many did it take ?
5. Bought 10 yards of cloth at $2 a yard; paid for
it with flour, at $4 a barrel : how many bar, did it take?
6. Bought 8 pints of cherries at 3 cents a pint j paid
for them with apples, at 6 cents a dozen : how many dozen
did it take ?
7. How many barrels of flour, at $3 a barrel, must be
given for 2 yards of cloth, at $7 a yard ?
S2 RAY'S INTELLECTUAL ARITHMETIC.
8. 5 times 3 are how many times 4 ? 6? 7 ? 8?
9. 4 times 4 are how many times 3? 5? 6? 7 ?
10. 5 times 4 are how many times 3? 6? 7? 8?
11. 2 times 11 are how many times 3? 4? 5? 6?
12. 5 times 5 are how many times 3? 4? 6? 7?
13. 8 times 4 are how many times 3 ? 5? 6? 7?
14. 7 times 5 are how many times 4? 6? 8? 9?
15. 6 times 7 are how many times 4? 5? 8? 9?
16. 6 times 9 are how many times 5? 7? 8? 10?
17. 8 times 7 are how many times 5? 6? 9? 10?
18. 6 times 10 are how many times 5 ? 7 ? 8 ? 10 ?
19. 9 times 7 are how many times 6 ? 8 ? 10 ? 11 ?
20. 8 times 8 are how many times 7 ? 9 ? 10 ? 11 ?
LESSON II.
1. Bought 4 boxes and 3-fifths of a box of raisins, at
85 a box : how much did they cost ? Paid for them with
flour, at 6 a barrel : how many did it take ?
ANALYSIS. It will take as many barrels of flour as $6, the
price of 1 barrel, are contained times in the cost of 4 boxes
and %-fifths of a box of raisins, at $5 a box.
2. Bought 4 gallons and 4-sixths of a gallon of wine,
for 6 a gallon, and paid for it with raisins, at 5 a box :
how many boxes did it take ?
3. Bought 5 kegs and 4-sevenths of a keg of tobacco,
for 7 a keg, and paid for it with paper, at 86 a ream :
how many reams did it take ?
4. Five times 5 and 3-fifths of 5 are how many
times 3? 4? 6? 7? 8? 9? 10?
5. Seven times 4 and 3-fourths of 4 are how many
times 3? 5? 6? 8? 9? 10?
6. Five times 6 and 5-sixths of 6 are how many
times 3? 4? 7? 8? 9? 10?
PARTS SF NUMBERS. 63
7 Seven times 5 and 3-fifths of 5 are how many
times 4? 6? 8? 9? 10?
8. Five times 8 and 1 -eighth of 8 are how many
times 4? 6? 7? 9? 10?
9. Seven times 6 and 2 -sixths of 6 are how many
times 5? 8? 9? 10?
10. Nine times 5 and 2 -fifths of 5 are how many
times 6? 7? 8? 10?
11. Nine times 5 and 4-fifths of 5 are how
times 6? 7? 9? 10?
12. Six times 8 and 3 -eighths of 8 are how many
times 5? 7? 9? 10?
13. Seven times 7 and 4-sevenths of 7 are how many
times 5? 6? 8? 9? 10?
14. Nine times 6 and 3-sixths of 6 are how many
times 5? 7? 8? 10?
15. Eight times 7 and 3-sevenths of 7 are how many
times 5? 6? 9? 10?
LESSON III.
1. Bought 4 apples at 3 cents each: paid for them
lemons at 6 cents each : how many did it take ?
2. Bought 7 yards of tape, at 2 cents a yard : how
many pears at 3 cents each, will it take to pay for it ?
3. If 2 apples cost 4 cents, what cost 3 apples ?
ANALYSIS. Three- apples will cost 3 times as much as 1
apple, and 1 apple will cost l-half as much as 2 apples, that
is, l-half of 4 cents, \-half of 4 cents is 2 cents, and 3 times
2 cents are 6 cents. Ans. 6 cents.
4. If 3 yards of cloth cost $9, what cost 4 yards ?
5. If 3 oranges cost 15 cents, what cost 5 oranges?
6. If 4 barrels of flour cost $24, what cost 7 barrels ?
7. If 2 kegs of lard cost $8, what cost 9 kegs ? 11 kegs ?
G4 RAY'S INTELLECTUAL ARITHMETIC.
8. If 5 dozen eggs cost 35 cents, what will 3 dozen
eost ? What will 8 dozen cost ?
9. If a man get 14 for 7 days' work, how much will
he get for 9 days ? For 3 days ?
10. Bought 4 yards and 2-thirds of a yard of cloth, at
$3 a yard, and paid for it with cheese, at 87 a hundred
weight : how many hundred weight did it take ?
11. Bought 4 pounds and 4-fifths of a pound of nails,
at 5 cents a pound, and paid for them with eggs, at 3
cents a dozen : how many dozen did it take ?
12. Bought 7 pounds and 5-sevenths of a pound of
sugar, at 7 cents a p<?>md, and paid for it with chickens^
at 9 cents each : how any did it take ?
13. Bought 9 pounds and 2-sevenths of a pound of
sugar, at 7 cents a poand, and paid for it with eggs, at 6
eents a dozen : how many dozen did it take ?
14. How many pounds, at 7 cents a pound, must you
give for 8 and 2-ninth yards, at 9 cents a yard ?
15. How many barrels, at $6 a barrel, must be given in
exchange for 4 and 5-seventh yards, at 7 a yard ?
16. Bought 5 pounds and 3-sevenths of a pound of
butter, at 7 cents a pound, and paid for it with raisins, at
6 cents a pound : how many pounds did it take ?
17. How many apples, at 2 cents each, can you buy
for 6 cents? How many for 15 cents?
18. How many pears, at 3 for 7 cents, can you buy
for 21 cents ? For 35 cents ?
19. If 6 pears are worth 2 oranges, how many oranges
ean you buy for 21 pears ?
20. A man bought 15 yards of cloth, at the rate of 3
yards for $5 : how many dollars did it cost ?
21. If a man receive 5 for 4 days' work, how many
dollars will he get for 12 days' work ?
22. How many pears, at 3 for 5 cents, can you buy
for 25 cents ?
PARTS OF NUMBERS. 65
23. If 2 pears are worth 6 cents, how many pears must
be given for 4 oranges, at 6 cents each ?
24. What will be the cost of 3 barrels and 9-elevenths
of a barrel of sugar, at $11 a barrel?
25. What will be the cost of 3 boxes and 4-fifths of
a box of butter, at $5 a box ?
26. Find the cost of 4 and 5-sixth tons of hay, at $6
a ton.
27. How many dozen eggs, at 12 cents a dozen, will
pay for 10 and 10-eleventh pounds of sugar, at 11 cents
a pound ?
SECTION X.
LESSON I.
1. If 1-half of an orange cost 3 cents, what will the
whole orange cost?
ANALYSIS. 1 orange will cost twice as much as \-Jialf of
an orange, that is, twice 8 cents. But twice 3 cents are 6 cents ;
therefore, if 1-half of an orange cost 3 cents, the whole orange
l cost 6 cents.
2. Three is 1-half of what number?
ANALYSIS. 3 is I-half of twice 3; but twice 3 are 6;
Cherefore, 3 is l-half of 6.
3. If 1 -fourth of a barrel of cider cost $2, what will
a barrel cost?
4. Two is 1 -fourth of what number?
5. If a man can walk 2 miles in 1 -third of an hour,
how far can he walk in an hour ?
6. Two is 1 -third of what number?
7. The age of Charles is 1 -third that of Thomas;
Charles is 4 years old : how old is Thomas ?
6o AAY'S INTELLECTUAL ARITHMETIC.
8. Four cents is the 1-third of what number of cents!
9. David has 5 marbles, which is only 1 -fourth as
many as Henry : how many has Henry ?
ANALYSIS. 5 is I -fourth of four times 5; but 4 times 5
are 20: therefore, 5 is 1-fourth of 20.
10. Six is 1 -fourth of what number ? 8 is 1 -fourth of
?rhat number ? 9 is 1 -fourth of what?
11. Six is 1-third of what number? 7 is 1-third of
what number? 9 is 1-third of what?
12. Three is 1-fifth of what number? 5 is 1-fifth of
what number? 9 is 1-fifth of what?
13. Five is 1-sixth of what number? 7 is 1-sixth of
what number?. 9 is 1-sixth of what?
14. Two is 1 -seventh of what number? 5 is 1 -seventh
of what number ? 8 is 1 -seventh of what?
15. Five is 1-eighth of what number? 7 is 1-eighth
of what number? 9 is 1-eighth of what?
16. Seven is 1-ninth of what number? 8 is 1-ninth
of what number? 9 is 1-ninth of what?
LESSON II.
1. James had 4 apples, and gave his brother 1-half
of them : how many did he give him ?
2. If you divide 6 apples equally between 2 boys,
what part of them must each have? Ans. \-lialf.
3. If you divide 6 apples equally between 3 boys,
what part of them must each have? Ans. 1-third.
4. If 3 yards of cloth cost $9, what part of 9 will 1
yard cost ? What part of 89 will 2 yards cost ?
5. If 4 oranges cost 12 cents, what part of 12 cents
will 1 orange cost? What part of 12 cents will 2 cost?
What part will 3 cost?
6. What is 1 -fourth of 12? What are 2-fourths
c<f 12? 3-fourthsof 12?
PAETS OF LUMBERS. 67
7. If 5 barrels of flour cost $30, what part of $30
will 1 barrel cost? What part of $30 will 2 barrels cost?
What part will 3 bar. cost ? What part will 4 cost ?
8. WT
hat is 1-fifth of 30? What are 2-fifths of 30?
3-fifths of 30? 4-fifths of 30 ?
9. If 1 apple cost 4 cents, what will 1-half of an
apple cost ? What will 3-halves cost ?
10. If 2 oranges cost 8 cents, what part of 8 cents
will 1 orange cost ? 3 oranges ? 5 ?
11. If 3 barrels of cider cost $12, what part of $12 will
1 barrel cost? What part of $12 will 2 barrels cost? 4
barrels ? 5 barrels ?
12. What is 1 -third of 12 ? What are 2-thirds of 12?
4-thirds of 12? 5-thirds of 12?
13. What is 1 -fourth of 24? 2-fourths? 3-fourths?
5-fourths ? 6-fourths ? 7-fourths ? 9-fourths ?
14. What is 1-fifth of 25? 2-fifths? 3-fifths?
4-fifths? 6-fifths? 7-fifths? 8-fifths? 9- fifths ?
15. What is 1-sixth of 24? 2-sixths ? 3-sixths ?
4-sixths ? 5-sixths ? 6-sixths ? 7-sixths ?
16. What is 1-seventh of 56? 2-sevenths? 3-sevenths?
4-sevenths? 5 -sevenths? 7-sevenths?
17. What is 1 -eighth of 72 ? 2-eighths ? 5-eighths?
8-eighths ? 10-eighths ? 11-eighths ?
18. What is 1-ninth of 54? 3-ninths ? 5-ninths?
7-ninths ? 9-ninths ? 11-ninths ? 12-ninths ?
19. What is 1-tenth of 60? 3-tenths ? 7-tenths?
9-tenths? 11-tenths?
20. What are 2-thirds of 6 ?
ANALYSIS. Two-thirds of 6 are twice l-third. One-third
vf 6 is 2, and twice 2 are 4 ; therefore, 2-thirds of 6 are 4.
21. What are 2-thirds of 12? 3-fourths of 12?
22. What are 4-fifths of 20 ? 5-sixths of 30 ?
23. What is 1-ninth of 27 ? 1 -fourth of 36 ?
RAY'S INTELLECTUAL ARITHMETIC.
24.
PARTS OF NUMBERS. 69
16. 11 -sevenths of 49 are how many times 8 ?
17. 3-ninths of 54 are how many times 7 ?
18. 10-ninths of 63 are how many times 8 ?
19. 8-ninths of 54 are how many times 5 ?
20. 9-sevenths of 42 are how many times 8 ?
LESSON IV.
1. James gave his brother 2 apples, which were
Vthird of all he had: how many had he left?
2. Thomas gave his brother 5 cents, which were
1 -fourth of all he had : how many had he ?
3. If, in traveling, I walk 1 -fifth of my journey in 2
hours, at that rate, in what time can I complete the remaining
4-fifths ?
4. One pint is 1 -eighth of a gallon : if a pint of wine
cost 7 cents, what will a gallon cost ?
5. A boy found a purse containing $12, and received
1 -sixth of the money for returning it to the owner : how
much did he receive?
6. If $42 be equally divided among a number of
men, giving each man 1 -sixth of the money, how many
men would there be, and what would each receive ?
7. Thomas had 28 marbles: he gave 1 -fourth of them
to James, and twice as many to William as to James :
how many did each receive ?
8. William had 24 apples: he gave 1-half to Thomas,
and 1 -third to James: how many did he give to both?
How many had he left?
9. A boy had 12 cents: he spent 1 -third of them fcaf
apples, and 1 -fourth for cakes : how many had he left?
10. A little girl received 20 cents from her mother;
her brother gave her 1 -fifth as many as her mother, and
her sister 1-half as many as her brother : how many did
she then have ?
2d Bk.
'
5
70 RAT'S INTELLECTUAL ARITHMETIC.
11. A boy having 40 cents gave 3-fifths of them foi
2 arithmetics : what was the price of 1 arithmetic ?
ANALYSIS. 3-Jifths are 3 times \-fifih: \-fifih of 40 cents
is 8 cents, and 3fifths are 3 times 8 cents, ivhich are 24 cents.
If 2 books cost 24 cents, 1 book will cost \-Jialf of 24 cents>
which is 12 cents. Ans. 12 cents.
12. James had 14 cents, and gave 4-sevenths of them
to his sister : how many cents had he left ?
13. John had 15 pears: he gave 1-third to Frank,
and 3-fifths to Harry : how many had he left ?
14. A man had 30 yards of cloth, and sold 2-fifths of
it for 48 : how much was that a yard?
15. John had 25 cents, and gave 3-fifths for peaches,
at 2 cents each : how many did he buy ?
16. A boy having 54 chestnuts, divided 5 -ninths of
them among 3 girls : how many did each receive ?
17. A man had 28 barrels of flour, and sold 2-sevenths
of them for 24 : what was that a barrel ?
18. Bought 7 yards of cloth for 42 : I gave 3 yards
for 9 barrels of cider : how much was it a barrel ?
19. A man had 40, and lost 3-fifths of them: he expended
the remainder in flour, at 84 a barrel : how many
barrels did he buy ?
20. I had 10 oents, and lost 1 -fifth : spent the rest for
apples, at 2 cents each : how many did I buy ?
21. James had 48 cents : he gave 3-eighths to his
brother, and spent the rest in chestnuts, at 5 cents a
quart : how many quarts of chestnuts did he buy ?
22. Thomas had 28 cents: he gave 1-fourth to his
sister, and 3-sevenths to his brother, and with the remain
der he bought 3 books : what did each cost ?
PARTS OF NUMBERS. 71
SECTION XI.
LESSON I.
1. If 2-thirds of a melon cost 4 cents, what will
1 -third cost?
ANALYSIS. l-third is I -half of 2-thirds: if 2-thirds cost
4 cents, l-third will cost l-half of 4 cents, or 2 cents, Ans.
2. Four is 2 times what number? Ans. Four is 2
times l-half of 4, which is 2.
3. If 3-fourths of a yard of cloth cost $6, what will
1 -fourth cost?
4. Six is 3 times what number?
5. If 2-thirds of a barrel of flour cost $8, what will
l-third cost?
6. Eight is 2 times what number?
7. If 3-fifths of a pound of butter cost 9 cents, what
will 1 -fifth cost?
8. Nine is 3 times what number ?
9. If 4-fifths of a pound of coffee cost 16 cents, what
will 1 -fifth cost?
10. Sixteen is 4 times what number?
11. If 5 -sixths of a gallon of wine cost 35 cents, whatwill
1 -sixth cost?
12. If 6-tenths of a yard of '4oth cost 30 cents, what
will 1 -tenth cost?
13. If 4-sevenths of a yard of muslin cost 28 cents,
what will 1 -seventh cost?
14. If 2-thirds of an orange cost 4 cents, what cost
l-third? If l-third cost 2 cents, what cost the whole?
15. Four is 2-thirds of some number: what is l-third
of that number? 2 is l-third of what number? Then
4 is 2-thirds of what number ?
16. If 2-thirds of a yard of cloth cost $6, what cost
1 -third of a yard ? If l-third cost $3, what eo3*-a yard ?
72 KAY'S INTELLECTUAL ARITHMETIC.
17. Six is 2-thirds of some number: what is 1 -third
of that number?' 3 is 1 -third of what number? Then
6 is 2-thirds of what number ?
18. If 3-fourths of a barrel of flour cost $6, what will
1-fourth cost? If 1-fourth cost $2, what cost a barrel?
19. Six is 3-fourths of some number : what is 1-fourth
of the same number? 2 is 1-fourth of what number?
Then 6 is 3-fourths of what number ?
20. If 2-fifths of a melon cost 8 cents, what cost 1-fifth?
If 1-fifth cost 4 cents, what cost the whole ?
21. Eight is 2-fifths of some number : what is 1-fifth
of the same number? 4 is 1-fifth of what number?
Then 8 is 2-fifths of what number?
22. If 3-fifths of a pound cost 9 cents, what cost 1-fifth?
If 1 -fifth cost 3 cents, what cost a pound ?
23. Nine is 3-fifths of some number : what is 1-fifth
of that number? 3 is 1-fifth of what number? Then 9
is 3-fifths of what number ?
24. If 4-fifths of a pound cost 8 cents, what cost 1-fifth?
If 1-fifth cost 2 cents, what cost a pound?
25. Eight is 4-fifths of some number: what is 1-fifth
of that number? 2 is 1-fifth of what number? Then
8 is 4-fifths of what number ?
26. If 3-fourths of a dozen eggs cost 9 cents, what
fcost 1-fourth ? If 1-fourth cost 3 cents, what cost a dozen ?
27. Nine is 3-fourths of some number, what is 1-fourth
of that number? 3 is 1-fourth of what number? Then
9 is 3-fourths of what number ?
28. If 2-thirds of a yard of tape cost 20 cents, what
cost 1-third? If 1 -third cost 10 cents, what cost a yard?
29. Twenty is 2-thirds of some number : what is
1 -third of that number? 10 is 1 -third of what number?
Then 20 is 2-thirds of what number ?
30. Six is 3-fourths of what number?
PARTS OF NUMBERS. 73
ANALYSIS. 1-fourth is l-third of 3-fourths : if 3-fourths
are 6, 1-fourth is l-third of 6, which is 2, and 4-fourths3 (or
the whole number,) are 4 times 2, which are 8, Am.
31. 9 is 3-sovenths of what number ?
32. 10 is 2-S3venths of what number ?
33. 14 is 7 -eighths of what number ?
34. 15 is 3-eighths of what number ?
35. 16 IB 4-ninths of what number ?
36. 18 is 6 -tenths of what number ?
37. 20 is 5-fourths of what number?
38. 15 is 5-fourths of what number?
39 . 33 is 11 -sixths of what number?
40. 22 is 2-elevenths of what number?
41. 81 is 9-thirds of what number?
LESSON II.
1. If 3-fourths of a pound of raisins cost 9 cents,
how much will a pound cost ? How many lemons, at 2
cents each, will pay for 1 pound of raisins ?
ANALYSIS. First. 1-fourth will cost l-third as much as
3-fourths, and 4-fourths, or a pound, will cost 4 times as
much as 1-fourth ; if 3-fourths cost 9 cents, \-fourth will cost
l-third of 9 cents, or 3 cents, and 4-fourths will cost 4 times
3 cents, or 12 cents.
Second. At 2 cents each, it will require as many lemons as
2 cents are contained times in 12 cents; 2 cents are contained
in 12 cents 6 times. Ans. 6 lemons.
2. If 2-thirds of a pound of sugar cost 16 cents,
much will a pound cost ? How many oranges, at 4 cents
each, will pay for 1 pound of sugar ?
3. If 7-eighths of a barrel of wine cost $42, how
many bar. of cider, at $6 a bar., will pay for 1 bar. of wine?
74 KAY'S INTELLECTUAL ARITHMETIC.
y
4. If 3-fifths of a hogshead of sugar cost $24, how
many barrels of flour, at $4 a barrel, will pay for 1 hogshead
of sugar?
5. Sold a horse for 25, which was 5-eighths of his
cost : how much did he cost me ? I paid for him with
sloth, at $6 a yard : how many yards did I give ?
6. Thirty is 5-eighths of how many times 5 ?
Ans.^Of as many times 5, as 5 is contained times in the
number of which l-fifth of 30 is \-eiglitli.
ANALYSIS. If 30 is 5-eighths, \-fiftli of 30, which is 6, is
i-eighth, and the number is 8 times 6, which are 48; 48 is 9
times 5 and 3-fifths of 5.
7. 12 is 4-sevenths of how many times 5?
8. 18 is 3-eighths of how many times 9 ?
9. 16 is 2 -sevenths of how many times 9 ?
10. 36 is 4-sevenths of how many times 8 ?
11. 45 is 5 -ninths of how many times 7?
12. 24 is 4-thirds of how many times 5 ?
13. 72 is 8-fifths of how many times 7 ?
14. 81 is 9-fourths of how many times 3 ?
15. 50 is 10-sevenths of how many times 4?
16. 63 is 7-sixths of how many times 5 ?
LESSON III.
1. James gave his brother 4 marbles, which were
2-thirds of all he had : how many marbles had he ?
2. Thomas sold a knife for 15 cents, which were
3 fifths of its cost : how much did it cost ?
3. William lost 6 marbles, which were 3-eighths of
all he had : how many had he ?
4. I sold a horse for S42, which were 6-fifths of hi?
cost: how many dollars did I gain?
PARTS OF NUMBERS. 75
5. A grocer sold a lot of flour for $40, which were
5-fourths of the cost : what was the cost ?
6. Sold a horse for $56, which were 8-fifths of the
cost : paid with flour, at $4 a barrel : how many barrels
did I give?
7. A man sold a watch for $28, which were 4-thirds
of its cost : how much did it cost ?
8. A man purchased a horse : after paying 3-fifths of
the price, he owed $20 : what was. the cost?
9. Alexander sold a book for 25 cents, and lost
2-sevenths of the cost : what was the cost ?
10. If a boy can earn 32 cents in a whole day, how
\nuch can he earn in 5-eighths of a day?
11. If 7-ninths of a cask of wine cost $42, how much
flour, at $8 a barrel, will pay for 1 cask of wine ?
12. If there are 10 links in 5 -ninths of a chain, how
many links are there in the whole chain?
13. In an orchard there are 12 cherry-trees : the remaining
5-sevenths of the orchard are apple-trees : how
many apple-trees are there?
14. There is a pole, 4-fifths of which is under water,
and 6 feet out of water : how long is the pole ?
15. There is a pole, 3-fifths of which is in the earth,
and 12 feet in the air : how long is the pole ?
16. One-fifth of a pole is in the mud, 2 -fifths in the
water, and 14 feet in the air : how long is the pole ?
17. The age of Joseph is 25 years, which is 5-eighths
of the age of his father : the father's age is 10 times that
of his youngest son : what is the age of the father? whal
is the age of his youngest son?
18. A man sold a horse for $45, which were 5 -thirds
of the cost : how much did he gain ?
19. A man paid $24 for a watch, and sold it for
7-fourths of the cost : he was paid in cloth, at $5 a yard :
how many yards did he receive?
76 RAY'S INTELLECTUAL ARITHMETIC.
20. A watchmaker sold a watch for 18, and lost 2-fifths
of its value : how much did he lose ?
ANALYSIS. Since he lost 2-fifths, he sold it for ^-fifths of
its value; then ^-fifths are $18, and \-fifth is l-third of
$18, or $6; 2-fifths are 2 times $6, or $12, Ans.
21. A watchmaker sold a watch for 8-15, and gained
2-sevenths of the cost : what was the cost ?
22. A boy spent 3-sevenths of his money, and had
12 cents left : how much had he ?
SECTION XII.
LESSON I.
1. If you have 12 cents, and 3-fourths of your money
equal l-third of mine, how many cents have I ?
ANALYSIS. \-fourth of 12 is 3, and 3-fourths are 3 times
3, or 9 ; 9 is l-third of 3 times 9 : 3 times 9 are 27, Ans.
2. 2-thirds of 9 are 1 -fifth of what number?
3. 4-fifths of 10 are 1-half of what number?
4. 3-sevenths of 14 are 1 -sixth -of what number?
5. 5-sixths of 12 are 1-fourth of what number?
6. 3-eighths of 16 are 1 -fifth of what number?
7. 2-fifths of 20 are how many thirds of 15 ?
8. 4-sevenths of 21 are how many fifths of 25 ?
9. 5-sixths of 30 are how many fourths of 32 ? ,
10. 7-ninths of 45 are how many sevenths of 21 ?
11. 3-fourths of 36 are how many thirds of 18 ?
12. 5-eigl^ths of 40 are how many ninths of 36 ?
13. Divide 1-fourth of 32 by 2-thirds of 9. Threefourths
of 28 by 2-thirds of 12.
PARTS OF NUMBERS. 77
14. Divide 3-fifths of 40 by 3-sevenths of 14. Five*
sevenths of 42 by 3-eighths of 24.
15. Divide the 7-ninths of 54 by 5-sixths of 12.
16. Nine-elevenths of 88 are how many times 5-ninths
of 18?
LESSON II.
1. William says to Frank,
" Your age is 15 years,
and 4-fifths of your age are 6-sevenths of mine : what
is my age ?"
ANALYSIS. l-fifth of 15 is 3, and 4-fifths are 4 times 3,
which are 12; if 12 is Q-sevenths, l-sixth of 12, which is 2,
is l-seventh, and if 2 is l-seventh, the number is 7 times 2,
which are 14. Ans. 14 years.
2.
78 RAY'S INTELLECTUAL ARITHMETIC.
14. Four-fifths of 20 are 8-sevenths of how many
times 1 -third of 15 ?
15. Six-sevenths of 21 are 2-fifths of how many
times 1 -fifth of 30?
16. Four-ninths of 27 are 2-thirds of how many
times 1 -seventh of 35 ?
17. Five-sevenths of 42 are 5 -ninths of how many
times 1 -eighth of 56 ?
18. Three-fourths of 24 are 9-tenths of how many
times 2-fifths of 10 ?
19. Five-sixths of 12 are 2-sevenths of how many
times 3-fourths of 8 ?
20. Five-eighths of 32 are 4-ninths of how many
times that number of which 2 is 1 -third?
21. Six-sevenths of 28 are 8-elevenths of how many
times that number of which 4 is 2-fifths ?
22. Eight-ninths of 45 are 10-thirds of how many
times that number of which 14 is 7-fourths?
23. Seven-tenths of 50 are 5-ninths of how many
times that number of which 8 is 4-fifths ?
24. Nine-sevenths of 56 are 12-fifths of how many
times the number of which 6 is 6 -sevenths ?
25. One-third of a certain number is 2 more than
1-half of 12 : what is the number ?
26. One-fourth of a certain number is 3 less than
1 -fifth of 30: what is the number?
27. Two-fifths of 20 is 6 less than how many thirds
of 21 ?
28. Three-fourths of 24 is 6 more than 2-thirds of
what number ?
29. Five-sixths of 30, increased by 4, is 1 less than
3-fourths of some number : what is that number ?
30. Three-fifths of 40 is just 3 less than 9 -sevenths of
how many times 7 ?
MISCELLANEOUS REVIEW. 79
SECTION XIII. REVIEW.
LESSON I.
1 William had 23 cents : Thomas gave him 8 cents
more, treorge 6, James 5, and David 7: he gave 15 cents
for a book: how many cents had he left?
2. A grocer paid $12 for sugar, $9 for coffee, $5 for
tea, $7 for flour, and had 10 left : how many dollars had
he ai first?
3. A boy had 11 cents : his father gives him 9 cents,
his mother 6, and his sister enough to make 34 : how
many cents did his sister give him ?
4. Five men bought a horse for $42 : the first gave
$13, the second 7, the third 5, and the fourth 9: how
many dollars did the fifth give ?
5. A man purchased 8 sheep at $4 a head, 5 barrels
of flour at $3 a barrel, 4 yards of cloth at $3 a yard,
and 5 ounces of opium at $1 an ounce : how much did
he spend?
6. A boy lost 25 cents : after finding 15 cents, he had
25 cents : how many cents had he at first ?
7. A man owed a debt of $28, and paid all but $9 :
how much did he pay?
8. Borrowed $56 : at one time paid $23 ; at another,
all but $7 : how much did I pay the last time ?
9. James borrowed 37 cents : at one time he paid 5,
at another 8, and the third time, all but 15.: how many
cents did he pay the third time ?
10. A farmer sold 2 cows at $9 each, and 5 hogs at
$3 each, and received in payment 3 sheep at $3 each, and
the rest in money : how much money did he receive ?
11. A farmer sold 12 barrels of cider at $3 a barrel :
he then purchased 5 barrels of salt at $3 a barrel, and
some sugar, for $8 : how many $'s had he left?
80 BAY'S INTELLECTUAL ARITHMETIC.
12. A merchant purchased 13 hats at 4 each, 5 pairs
of shoes at 2 a pair, and an umbrella, for 7 : what
must he sell the whole for, to gain $9 ?
13. If 1 barrel of flour cost S3, what cost 7 barrels?
14. If 2 pounds cost 16 cents, what cost 5 pounds?
15. If 3 barrels of cider cost 12, what cost 4 barrels ?
16. If 4 yards of cloth cost $28, what cost 7 yards ?
17. If 3 pounds cost 27 cents, what cost 8 pounds ?
18. If 5 barrels of flour cost 35, what cost 8 barrels ?
19. If 7 apples cost 28 cents, what cost 3 apples?
20. If 8 oranges are worth 24 apples, how many apples
are 3 oranges worth ?
21. If 4 pounds of cheese cost 36 cents, what will 3
pounds cost?
22. If 8 yards of cloth cost $56, what cost 7 yards ?
23. What sum of money must be divided among 11 men
that each shall receive 9 ?
24. A walks 5 miles, while B walks 3 : when A ha\
gone 35 miles, how far has B gone ?
ANALYSIS. In 35 there are 7 fives, and B walks as many
threes as A walks fives; that is, 7 times 3, or 21 miles.
25. Joseph and his father are husking corn : the father
can husk 7 rows while Joseph husks 3 : how many rows
will Joseph husk while his father husko 42 ?
26. Charles can earn 9 while Mary earns 4 : how
many S's will Charles earn while Mary earns 28 ?
27. If 6 horses eat 12 bushels of oats in a week, how
many will 10 horses eat in the same time ?
28. If 5 horses eat 16 bushels in 2 weeks, how long
would it take them to eat 56 bushels ?
29. If 6 apples are worth 18 cents, how many apples
must be given for 5 oranges worth 6 cents each ?
30. How many horses can eat in 9 days, the same
amount of hay that 12 horses eat in 6 days?
MISCELLANEOUS REVIEW. 81
31. If 5 men earn $30 in 3 days, how much will 2
men earn in the same time ? How much will 2 men earn
in I day?
LESSON II.
1. How many times 6 in 3-fifths of 40?
2. How many times 3-sevenths of 14 in 54 ?
3. How many times 5-sixths of 12, in 4-ninths of 72?
4. How many times 3-fifths of 20, in twice that number
of which 14 is 7-ninths ?
5. If 3-eighths of a tun of hay cost $9, what will
5-sixths of a tun cost ?
6. If a man can earn $7 in 1 -fifth of a month, how
many dollars can he earn in 1 month ?
7. If a man can earn $42 in 1 month, how many
dollars can he earn in 1 -sixth of a month?
8. If a hogshead of sugar is worth $96, what are
7-eighths worth?
9. Five men paid $20 for a horse : what part of $20
did 2 men pay ?
10. If $7 will buy 56 yards of muslin, how many
yards will $4 buy ?
11. If 3 men can do a job of work in 16 days, in how
many days can 4 men do it ?
12. If 3 men spend $12 in 1 week, at the same, rate,
how many would 2 men spend in 6 weeks ?
13. If 6 men do a piece of work in 7 days, in how
many days can 3 men do it ?
14. If 5 men do a piece of work in 8 days, in how
many days can 4 men do a job twice as large ?
15. If 6 men do a job in 5 days, in how many days
can 2 men do a job half as large?
16. James had 16 apples : he kept 1-fourth himself
and divided the remainder equally among 3 of his companions
: how many did each receive ?
82 RAY'S INTELLECTUAL ARITHMETIC.
-**;
17. Three-fourths of 24, increased by 2-thirds of 12,
are equal to how many ?
18. Five-sixths of 24, diminished by 3-fourths of 20,
equal how many ?
19. One-half, and 2-thirds, and 3-fourths of 12 are
how many ?
20. Two-thirds of 12, less 1-half of 12, are 2-fifths of
what number?
21. From 10 take 3-fifths of itself; add to the remainder
its 1-half: what is the result?
22. Thomas had 28 cents : he gave* 2-sevenths to his
sister, and 2-fifths of the remainder to his brother : how
many more did he give, than he had left?
23. James had 35 marbles : he gave to Thomas
3-sevenths, to Charles 2-fifths : to which did he give the
most, and how many ? What number had he left ?
24. Thomas had 28 : he kept 2-sevenths, and divided
the remainder equally among his 4 brothers: how many
dollars did each receive ?
25. A grocer had 14 barrels of flour : he sold 4-sevenths
at S3 a barrel, and the remainder at 5 a barrel :
what did it amount to ?
26. Bought 15 yards of cloth, at 82 a yard : I sold
1 -third at 4 a yard, 2-fifths at 83 a yard, and the re,
mainder at 85 a yard : how much did I gain ?
27. Bought 10 yards of cloth for 90, and sold 2-fifths
of it for 840 : how much a yard did I gain ?
28. Two men travel the same direction : A is 40 miles
ahead of B ; but B travels 23 miles a day, and A 18 :
in how many days will B overtake A ?
29. A hare is 90 yards in advance of a hound : the
hound goes 10 yards in a minute, and the hare 7 : in how
many minutes will the hound overtake the hare? How
far will each run ?
30 .If a hound runs 7 rods while a hare runs 4,
far will the hare run while the hound runs 35 rods ?
MISCELLANEOUS KEVIEW. 83
31. One man is pursuing another, who is 4 miles in
advance : he travels 3 miles in pursuit, while the other
advances 1 : how much aoes he gain in going 3 miles ?
How far must he travel to overtake him ?
32. C and D travel in tne same direction : C is 15 miles
ahead of D ; but D travels 5 miles an hour, and C only
2 : in how many hours wnl L overtake C ? How far will
D have traveled ?
33. A cistern containing 2 gallons, is filled by a pipe
at the rate of 8 gallons an hour, and emptied by a pipe
at the rate of 5 gallons an hour : if both pipes are open,
how many gallons will remain in the cistern each hour?
How long will the cistern be in filling ?
34 .A cistern containing 36 gallons has 2 pipes ; by
the first it receives 6 gallons an hour, and by the second
it discharges 9 gallons an hour : if both pipes are left
open, how long will it take to empty the cistern ?
35. My pants cost $8, which were 2-fifths the cost of
my coat ; my vest cost 1-haif as much as my pants : what
was the cost of the whole ?
LESSON III.
1. What are the divlbors of a number called ? Ans.
Its factors.
2. What is the divisor of a number ? Ans. Any
number which is contained in it an exact number of times;
that is, without a remainder.
3. What is a multiple of a number ? Ans. Any
product of which that number is a factor.
4. Twelve can be divided by 2, 3, 4, and 6 : what are
the factors of 12 ? of 24 ?
5. What 2 numbers multiplied together, will make
28 ? What are the factors of 28 ?
6. What numbers w'll divide 22? What are th*
factors of 22 ?
84 RAY'S INTELLECTUAL ARITHMETIC.
7. Six is a factor of 24; what number is a multipk
of 6 ? of 12 ?
8. Seven is a factor of 21 ? What number is a multiple
of 7 ? of 3 ?
9. What three numbers are multiples of 7 ? of 8 ?
of 9? of 10?
10. What numbers will divide 18? 27? 33? 36?
39? 48? 49? 63? 64? 65? 72? 75?
11. What number will divide both 10 and 15? 12
and 16 ? 20 and 32 ? 64 and 80 ? 72 and 96 ?
12. What is the greatest number less than 12, that will
divide 12 ? Less than 16 that will divide 16 ? Less than
20 that will divide 20 ?
13. What i? the greatest number that will divide both
6 and 10 ? 12 and 18 ? 20 and 32 ? 24 and 36 ? 42
and 48 ? 56 and 72 ?
14. What numbers can be divided by 9 ? 12 ? 8 ?
7? 16? 14? 15?
15. What is the least number that can be divided by
2 and 3? 4 and 7 ? 2 and 9 ? 7 and 21 ? 10 and 12?
12 and 20? *
The answers to the 13th question are called the Greatest
Common Divisors ; to the 15th question, the Least Common
Multiples or Dividends.
SECTION XIV. FRACTIONS.
LESSON I.
A single or whole thing of any kind, is called a unit
one ; as 1 yard, 1 dollar, 1 mile, 1, &c.
1. The above line represents a yard of tape : it is
called 1 yard.
FRACTIONS. 85
2. If you divide it into two equal parts, one of the
$arts is called one-half of a yard.
one-half. one-half.
One part is represented thus, |,
and is read one-half.
3. If the yard of tape is divided into three equal
parts, one of the parts is called one-third; two parts are
called two-thirds.
one-third. one-third. one-third.
One part is represented thus, ^, and is read one-third;
2 parts are represented thus, |, read two-thirds.
4. If the yard of tape is divided into four equal parts,
one of the parts is called one-fourth; two parts two-fourths;
3 parts three-fourths.
one-fourth. one-fourth. one-fourth. one-fourth.
One part is represented thus, |, and is read one-fourth ;
2 parts are represented thus, |, read two-fourths; 3 parts
thus, |, read three-fourths.
5. If the yard of tape is divided into five equal parts,
one of the parts is called one-fifth; two parts two-fifths;
three parts three-fifths.
A FRACTION is a part of a unit, or one. In every
fraction the unit is divided into equal parts. The lower
number shows into how many equal parts the unit is
divided : the upper number, how many parts are tak^n,
2d Bk. 6
86 RATS INTELLECTUAL ARITHMETIC.
The lower number is called the DENOMINATOR, because
it denominates or names the parts : the upper number the
NUMERATOR, because it numbers the parts.
In reading fractions, that is, in expressing them bj
'
words, read the Numerator first j the Denominator last.
6. If you divide a yard into 6 equal parts, whai
figures will express 2 parts ? What 5 parts ?
7. If you take away 2 of the 6 parts, what fraction
will express the remainder ?
8. How can you express in figures 3-sevenths of 1 ?
9. In the fraction f, into how many parts is the unit
(one) divided, and how many are represented ?
10. Read the following Fractions: 11111111 i i i i i i i
~g> 3> o? 6> 7) 6' ? To> TT> 72> 7.3? 74> To? 76? T7>
i I, I, I. 4, I i, T
3
j> T
9
T, IS. T
9
a> if, M> & IS-
11. What does the fraction f represent? Into how
many equal parts is the unit (one) divided?
12. What do you understand by T
9
T ?
j-f
? |f ?
13. Divide an apple into 10 equal parts, and take away
3 parts : what fraction will express the remainder ?
14. What fraction will represent 1-half of 1 ? 1-half
of 3 ? 1-third of 5 ?
15. Which is the greater fraction, 4 or 4 ? - or j ?
ior? |or|?
16. What is a fraction ? What does the denominator
of a fraction show ? What does the numerator show ?
17. How many apples in 2-halves of an apple ? In
3-thirds ? In 4-fourths ? In 5 -fifths ?
18. When the numerator of a fraction is equal to the
denominator, as |, |, |, &c., what is its value ? Ans. 1.
19. Are 3-halves of an apple more or less than a whole
apple ? When the numerator is greater than the denominator,
is the value of the fraction greater or less than 1 ?
FKACTIONS, 7
20. If jker denominators of fractions be increased, do
the fractions become greater or less ? Why ?
21. If the numerators be increased, do the fractions
become greater or less ? Why ?
22. Such fractions as , |, |, &c., being either equal
to, or greater than 1, are called improper fractions; those
less than 1, as ^, |, &c., proper fractions.
23. Whole numbers and fractions joined together, as
14, 2|, 5j, 6|, 10|, &c., are called mixed numbers.
SECTION XV.
LESSON I.
1. How do you find the number of halves in any
number ? Why ?
2. In 1, how many halves ? In U ? In 2 ? In 2^ ?
In 3 ? In 3J ? In 4 ? 4 ' ? 5 ? 6 ? 7 ? 7i ?
3. How do you find how many thirds there are in
any number ? Why ?
4. How many thirds in 1 ? In 1| ? In If ? In 2 ?
In 21? In 23? 3? 31? 3 ? 4? 44 ? 5? 6? O O DO O
5. How do you find how many fourths there are in
any number ? Why ?
6. How many fourths in 1 apple? In 1 and 1-fourth?
In 1 and 2-fourths ? In 1 and 3-fourths ? In 2 ? In 2
and 3-fourths ? In 3 ? In 4 and 3-fourths ?
7. How find how many fifths are in any number ?
8. How many fifths in 1 ? In H ? In If? In If ?
In 11 ? In 2 ? In 2i ? In 2| ? In 2| ? In 2| ?
'
IB
B? "inSf? In3?In4f? In 54?
9. In 4 and 2-sevenths how many sevenths ?
10. In 5 and 3-eighths how many eighths ?
Jl. In 4 and 7-ninths how many ninths?
88 RAY'S INTELLECTUAL ARITHMETIC
12. In 5 and 3-tenths how many tenths ?
13- In 12 and 1-third how many thirds?
14 In 11 and 1-half how many halves ?
15- In 6 and 3-fifths how many fifths ?
16* In 7 and 3-fourths how many fourths?
17. In 6 and 1-sixth
^
how many sixths ?
18. In 5 and 7-eighths how many eighths ?
19. In 11 and 4-sevenths how many sevenths ?
J
20. In 7 and 9-tenths how many tenths?
LESSON II.
1. James's brother gave him 3-halves of an apple:
how many apples did he give him ? Ans. One apple
and 1-half.
2. In 3 halves how many ones ?
ANAL. Since 2-halves make 1, in 3-halves are as many ones
as 2-halves are contained times in 3-halves; 2-halves in
3-halves, one and one-half times. Ans. 1 and 1-half.
3. John's father gave him 4 half- dollars : how many
dollars did he give him ?
4. How many ones in 4-halves? in 5-halves? in
6-halves ? in 7-halves ? in 8-halves ? in 9-halves ?
5. How do you find how many ones there are in any
number of halves?
6. If you divide an orange into 3 equal parts, what is
1 part called ? How many thirds in 1 orange ?
7. If 1 orange make 3-thirds, how many oranges would
there be in 3-thirds of an orange? in 4-thirds of an
orange? in 5 -thirds? in 6 -thirds?
8. How many ones in 3-thirds? in 4-thirds? in
5-thirds ? in 6-thirds ? in 7- thirds ? in 8-thirds ?
9. How do you find how many ones there are in any
number of thirds ?
FRACTIONS. 89
10. In 4-fourths of a dollar how many dollars ?
In 5-fourths? In 6-fourths ?
In 7-fourths? In 8-fourths ?
In 9-fourths? In 11-fourths?
In 13-fourths ? In 15-fourths ?
In 18-fourths ? In 19-fourths ?
11. How many ones in 4-fourths ? in 5-fourths ? in
6-fourths? in 7-fourths? in 8 fourths? in 9-fourths?
11-fourths? 13-fourths? 15-fourths? 19-fourths?
12. How do you find how many ones there are in any
number of fourths ?
13. In 5 -fifths of an orange, how many oranges ?
In 6-fifths? In 7-fifths?
In 8-fifths? In 9-fifths?
In 10-fifths ? In 11-fifths ?
Inl3-fifths? Inl5-fifths?
Inl7-fifths? In 18-fifths?
14. How many ones in 5-fifths ? 6-fifths ? 7-fifths ?
8-fifths? 9-fifths? 10-fifths? 11-fifths? 13-fifths?
15. How do you find how many times 1 there are in
any number of fifths ?
16. How many times 1, that is, how many whole ones,
in 23-sixths ? Ans. Three and 5 -sixths.
17. In 30-sevenths are how many times 1 ?
18. In 35-sevenths ? 22. In 23-halves ?
19. In46-ninths? 23. In 33-fifth s ?
20. In 53-tenths ? 24. In 31-fourths ?
21. In37-thirds? 25. In37-sixths?
26. In 47-eighths are how many whole ones ?
27. In 81-sevenths ? 30. In75-tenths?
28. In79-tenths? 31. In 89-elevenths?
29. In 53-ninths ? 32. In 93-twelfths ?
90 KAY'S INTELLECTUAL ARITHMETIC.
SECTION XVI.
LESSON I.
ILLUSTRATION. The first line represents a yard of
tape divided into two equal parts ; the second, a yard of
tape divided into four equal parts.
one-half. one-lialf.
one-fourth. one-fourth. one-fourth. one-fourth.
One-half Q), is equal to two-fourths (f).
1. If I give to Mary 1-half of an orange, and to Jane
1 -fourth, how much more will Mary have than Jane?
what part will be left ?
2. How much greater is 1-half than 1 -fourth ? How
much are 1-half and 1 -fourth?
3. James divided a melon, giving to his sister 1-half,
and to his brother 1 -fourth : what part did he give away ?
4. Thomas gave 3-fourths of a dollar for a geography,
and 1-half of a dollar for an arithmetic and slate : how
much did he give for both ?
5. How much are 1-half and 3-fourths?
ILLUSTRATION. The first line represents a yard of
ribbon divided into 2 equal parts; the second, a yard
divided into six equal parts.
one-half. one-half.
I -sixth. 1 -sixth. 1- sixth. I-sixth. 1-sixth. 1-sixth.
6. One-half is how many sixths? 1-third is how
many sixths ? 2-thirds are how many sixths ?
FRACTIONS. 91
7. James received 1-half of an orange, and Charles
I-third : how much more had James than Charles ?
J3. How much are 1-half and 1 -third?
9. If 1 -third is 2-sixths, how many sixths are there
in 2-thirds ?
10. A yard of flannel costs half a dollar : a yard of
sloth 2-thirds of a dollar : how much do both cost ?
11. James bought 2 melons; he gave to Lucy half of
the first ; to Jane 2-thirds of the second : what part of a
melon had Jane more than Lucy ?
12. How much greater are 2-thirds than 1-half?
13. After taking away 1-half and 1 -third of an apple,
what part will be left ?
14. I wish to divide an orange, and give to Mary 1-half,
to Jane 1 -fourth, and to William 1-eighth: how must 1
divide it ? how many eighths will each have ?
15. Two-fourths are how many eighths ? 3-fourths are
liow many eighths ?
16. One-fifth is how many tenths? 2-fifths are how
many tenths ? 3-fifths ? 4-fifths ?
17. Thomas wishes to divide an orange, and give Ann
1-half, and Lucy 2-fifths: how must he divide it? what
part will he have left ?
18. How much are 1-half and 2-fifths?
19. One-third is how many twelfths? 1-fourth is how
many twelfths? How many twelfths in 2-thirds? in
2-fourths ? in 3-fourths ?
20. A farmer sows 1-half of a field in wheat, 1-third
in rye ; the rest in barley : how many twelfths are in
wheat ? how many in rye ? how many in barley ?
21. How many twelfths in 1-fourth? in 1-fourth and
1-third?
22. David bought a pound of figs: he gave 1-third of
them to his mother, 1-fourth to his sister, and 1 -sixth to
his brother : what part had he left ?
2 RAY'S INTELLECTUAL ARITHMETIC.
23. How many eighths are in 1-half? in 3-fourths?
24. How many tenths are in 1-half? in 1 -fifth?
in2-fifths? in 3-fifths ?
25. How many fifteenths in 1-third? in 1-fifth?
in 3-fifths ? in 4-fifths ?
26. How many twentieths in 1-half? in 1 -fourth ?
in 1-fifth ? in 3-fourths ? in 3-fifths ?
27. Reduce -f and f to twelfths : ^ and f to fifteenths :
also,
i and | to tenths.
28. Reduce f and f to twentieths : ^, |, and 1- to twentieths.
2& Reduce | and ^ to fourteenths: ^ and f to eighteenths.
30. Reduce f, f, and | to twenty-fourths : | and T
3 to
fortieths.
REMARK. When two or more fractions have the same denominator,
they are said to have a common denominator;
thus, | and | have 6 for a common denominator.
When the denominators of two fractions are not the same, a
common denominator may be found by multiplying the denominators
together.
Find the Common Denominator
31. Of and f .
FRACTIONS. 93
41.Reduce |, |,
42.Reduce -J, ^,
j, T
7
3 , and ^ to forty-eighths.
J, ^, , TV, and y^ to seventy-seconds,
LESSON II.
ILLUSTRATION. The first line represents a yard of
ribbon divided into 2 equal parts ; the second line, a
yard divided into 4 equal parts.
one-half. one-half.
one-fourth. one-fourth. one-fourth. one-fourth.
The first of the lines below represents a yard divided
into 3 equal parts ; the second, a yard divided into 6 equal
parts. From this it is seen that ^ is equal to f ; and
that | are equal to |.
one-third. one-third. one-third.
1-sixth. I-sixth. \-sixth. 1-sixth. 1-sixth. 1-sixth.
REMARK. The Numerator and Denominator are called the
terms of the fraction; when these are the smallest numbers of
-which the fraction admits, it is said to be in its lowest terms.
1. Reduce to its lowest terms. Divide loth terms ly
the greatest number (3), that will exactly divide loth.
2. Reduce | to its lowest terms. to its lowest
terms. | to its lowest terms.
3. Reduce to their lowest terms, |, .
4. Reduce to their lowest terms, j
2
^, j
5
^.
5. Reduce to their lowest terms, y
4
^, T% -j%, -,.
6. Reduce to their lowest terms, y^, y%, y%, -f%, T
9
2, yf
7. Reduce to their lowest terms. ^, -f~, y%, y^, |f, yf .
94 RAY'S INTELLECTUAL ARITHMETIC.
8. Reduce to their lowest terms, T%, T
6
6 , T%, g, ft, ft.
9. Reduce to their lowest terms, T
2
g, T
4
g , 7
*
g , T
8
g , ig, if.
10. Reduce to their lowest terms, 3
6
5 , o
3
T, ^, g, i, if.
11. Reduce to their lowest terms, ||, if, if, j, f{j, f|,
SECTION XVII.
LESSON I.
1. David divided an orange, giving to William 1 -fifth,
and to Sarah 2-fifths : how many fifths did he give away ?
how many fifths had he left ?
2. After taking from any thing 3-fifths of itself, what
part will be left?
3. John bought a quart of chestnuts : he gave 2-sixths
to Mary, and 3-sixths to Eliza : how many sixths did he
give away ? what part had he left ?
4. One-fifth, 2-fifths, and 7 -fifths of an orange, are
how many fifths ? how many oranges ?
5. How much are i, f, and J?
6. One-eighth, 3-eighths, and 7-eighths of a dollar
are how many eighths? how many dollars?
7. What is the sum of f, |, and | ?
8. Daniel's mother gave him $J, and his uncle $J:
how many sixths of a dollar did he receive from both ?
ANALYSIS. One-half is 3-sixths, and \-third is 2-sixths :
3-sixths and 2-sixths are 5-sixths. Ans. 5-sixths of a dollar.
The fractions are reduced to a common denominator before
adding, because you can not add things of different kinds.
This is the first step in adding or subtracting fractions.
9. Lucy divided an orange, giving to her sister 1 -third,
and to her brother 1 -sixth : how many sixths did she
give ? how many sixths did she have left ?
FRACTIONS. 95
10. James bought a lemon, and gave to Lucy 1 -eighth,
and to Susan 1 -fourth: how many eighths did he give?
what part did he retain ?
11. William cut a pine-apple, and gave to Mary
1 -third, and to Eliza 1-ninth: how many ninths did he
give to both ? what part did he retain ?
12. Thomas bought a copy-book for fty
1
^, and a reader
for $J : how many tenths of a dollar did they both cost?
13. David gave 1 -fourth of a melon to Eliza, 1 -third
to his mother, and kept the remainder : how many twelfths
did he give ? what part did he retain ?
14. I bought 1^- yards at one store, and 2J yards at
another : how many yards did I purchase ?
15. I planted 2J acres of ground in corn ; 8| acres in
oats : how many acres were in both pieces ?
16. John bought a knife for $J, a slate for $J, and a
book for $f : how much did the whole cost ?
17. Add A and f. and f . f and |.
18. Add -| and f. and i. \ and \.
19. Add i and 4. f and If. | and f.
20. Add | and f . f and f . f and f .
21. Add 1| and 2i. 1, 1, |, and f
22. Add 3f and 4|. f , |, |, and .
23. Add 4f and 5f 1', 2|, |, and6T%.
24. Add 5| and 4|. |, 3j, 4, and 5 f .
LESSON II.
1. Mary had 3-fourths of an orange ; she gave her
uister 1 -fourth : how many fourths had she left?
2. If you take 1-fifth from 3-fifths, what will be left ?
3. I bought 5-sixths of a quart of nuts, and gave
3-sixths to my sister : how many sixths had I left ?
4. One is how many sixths ? If you take 5-sixths
from 1, what will be left?
96 RAY'S INTELLECTUAL ARITHMETIC.
5. Two is how many fifths? If you take 3-fifths
from 2, what will be left ?
6. If you take i from f, what will be left ?
7. | from f?
'
10. T% from Jfc?
8. | from I? 11.
-| from 1?
9. from I ? 12. f from 2 ?
13. Thomas divided an orange, giving his brother
1-half, and his sister 1 -third : how many sixths did each
receive? how much more the brother than the sister?
14. If a bushel of wheat cost $J, and of corn $J, how
much will the wheat cost more than the corn ?
15. If you take \ from f, what will remain ?
16. Joseph bought a quart of chestnuts, and gave 1-half
of them to his mother, and 1-sixth to his sister : how
many sixths did he give his mother more than his sister ?
17. Take \ and \ from 1, what will be left?
18. Jane divided an orange, giving to her sister
3-eighths, and to her brother 1 -fourth : to which did she
give the most ? what part of the orange was left ?
19. A man having 72 miles to travel, went 1 -third the
distance the first day, 2-ninths the second, the remainder
the third day : how many ninths of the distance did he
travel further on the first day, than on the second ? what
part did he travel the last day ?
20. If you take f from \, what will be left? If you
take | and -J from 1, what will be left?
21. If you take 1 from ^, what will be left?
ANALYSIS. \-fiftli is 2-tenths, and l-half is 5-tenths;
2-tenths from 5-tenths leave 3-tenths. Ans. 3-tenths.
What will be left if you take
om J ? I from | ? 1 fr
om ? f from ? fr
24. * from | ? 4 from f ? I from f ?
o 4 o 55 O
22. | from from \ ?
23. 1 from from f ?
FRACTIONS. 97
25. I from f ? J> from f ? ^ from -|
?
26. from f ? f from | ? f from | ?
27. J from f ? | from f ? * from 2 ?
LESSON III.
1. Mary divided a quart of pecans, giving Ann 1 -third,
and Jane 1-fourth of them : what part had she left?
2. A farmer had 1^ bushels of wheat : he gave to 1
poor man J of a bushel, and to another | of a bushel :
how much wheat was left ?
3. James had ^ of a pound of raisins : he gave to
his brother i of a pound, and to his sister \ of a pound :
how much had he left ?
4. If from 3^ bushels of corn, 1^ bushels be taken, how
much will there be left?
5. A lady bought 3| yards of muslin at one store,
and 2\ yards at another : after using 14 yards, how much
had she left ?
6. William's father gave him $| : he gave to a poor
person $|, for apples S-Jg, and for a book $| : what part
of a dollar had he left ?
7. James's mother gave him a book : he read the first
day|, ^he second | , the third ^, and the fourth the remainder
: what part did he read the fourth day ?
8. A farmer has a flock of 84 sheep in 4 fields : the
first contains i, the second J, and the third \ of them :
what part does the fourth field contain ?
9. Daniel spends -A of his time in sleep, \ of it at
school, y
1
^ in reading, and ^ in learning music: what
part of his time is not employed?
10. A pole is standing in a pond; ^ of it is in the air,
pud | in the water : what part is in the earth ?
11. A student devotes | of his time to sleep, -|
% to
Btudy, 75^ to reading, ^ to exercise, and TV to deeds of
charity 7 what part of his time is unemployed ?
98 RAY'S INTELLECTUAL ARITHMETIC.
12. One-third of an orchard is apple trees, ^ pear trees,
| plum trees, 7V quince trees, and the remainder peach
trees : if the orchard contain 96 trees, how many are
there of each kind ?
13. After spending ^ and
-Jof
my money, and losing
jV, I had 8 remaining : how much had I at first?
14 .The difference between | and i of my money
is $3 : how much have I ?
15 .1 ate f of my peaches, gave away 10, and have 10
left : how many had I at first ?
SECTION XVIII.
LESSON I.
1. A father gave each of his two sons half a dollar:
how much did he give them both ?
2. A mother gave each of her 3 children half an
orange : how many half oranges did it take ? how many
oranges ? why ?
3. What are 3 times 1-half? 4 times 1-half?
4. John fed 5 horses, giving to each half a peck of
oats : how many half pecks did it take ? how many pecks ?
why?
5. What are 5 times 1-half? 6 times 1-half?
6. What are 8 times 1-half? 9 times 1-half?
7. James gave 1 -third of an orange to each of his
sisters : how much did he give to both ? why ?
8. If a man can eat 1 -third of a pound of meat in 1
day, how much can he eat in 3 days ? why ?
9. What are 4 times 1 -third ? 5-times 1 -third ?
10. John gave 2-thirds of a pine-apple to each of his 2
brothers : how many thirds did he give to both ? How
many pine-apples did it take ? Ans. 1 and 1-third.
FRACTIONS. 99
11. What are 4 times 2-thirds? 5 times 2-thirds?
12. Thomas gave 1 -fourth of an apple to each of his 3
playmates : how many fourths did it take ? why ?
13. If 1 bushel of oats cost 1 -fourth of a dollar, how
much will 4 bushels cost ? why ?
14. Charles gave 3-fourths of a pint of chestnuts to
each of his 2 brothers : how many fourths of a pint did
it take ? how many pints ? why ?
15. Mary gave 3-fourths of an orange to each of her
three brothers : how many fourths of an orange did it
take ? how many oranges ? why ?
16. What are 5 times 3-fourths ?
ANALYSIS. 5 times 3-fourths are I5-fourths ; 15-fourths
are as many ones as 4-fourths are contained times in
15-fourths ; 4-fourths in 15-fourths, 3 and 3-fourths times.
Ans. 3 and 3-fourths.
17. What are 6 times | ?
18. What are 3 times | ?
19. What are 3 times 4 ?
20. What are 5 times ?
21. What are 2 times ^ ?
What are 8 times f ?
What are 5 times X ?
22.
23.
24.
25.
26.
What are 7 times f ?
What are 3 times
What are 7 times
3?
I?
7
100 BAY'S INTELLECTUAL ARITHMETIC.
AXAL. 3 times I are 3, and 3 times l-half are 3-halves,
equal 1 and l-half; this added to 3 makes 4 and l-half.
4. If 1 bushel of wheat cost $1 and 1 -third of a dollar,
what will 2 bushels cost ?
5. How many are 3 times lJ ? 2 times 2f ?
6. How many are 3 times 3| ? 4 times 4^ ?
7. How many are 5 times 2| ? 6 times 3| ?
8. How many are 8 times 3| ? 9 times 4| ?
9. If 1 bushel of barley cost $1 and 1-fourth of a
dollar, what will 3 bushels cost ? 4 bushels ?
10. How many are 5 times 1| ? 6 times 1| ?
11. How many are 2 times 1| ? 3 times 2| ?
12. How many are 4 times 3i ? 5 times 3| ?
13. How many are 6 times 3| ? 8 times 3| ?
14. How many are 7 times 2| ? 9 times 2| ?
15. How many are 10 times 1| ? 10 times 3^ ?
16. How many are 11 times 2| ? 12 times 3| ?
17. If a family consume 3 and 1 -fifth barrels of flour
io one month, how much will they require for 3 months ?
18. How many are 4 times 3| ? 5 times 3| ?
19. How many are 2 times 6f ? 3 times 2| ?
20. How many are 6 times 4i ? 6 times 3| ?
21. How many are 7 times 4| ? 8 times 3| ?
22. How many are 9 times If ? 9 times 3\ ?
SECTION XIX.
LESSON I.
1. If you divide an apple into two equal parts, >yhat
I part called? what is l-half of 1 ?
FRACTIONS. 101
2. If you divide 3 apples between 2 boys, how many
apples will each have ? how will you divide them ? what
is 1-half of 3?
ANALYSIS. l-half of I is l-half, and l-half of L is 3
times l-half of 1, which are 3-halves ; 3-halves are as many
ones as 2-halves are contained times in %-halves; 2-halves in
Sj 1 and l-half times. Ans. 1 and l-half
3. John bought 5 oranges, and divided them between
his 2 sisters : how many oranges did each have ?
4. What is l-half of 6? f of 7? Jof8? Jof9?
of 11? of 13?
5. If cloth is SI a yard, what must be paid for 1 -third
of a yard ? what is 1 -third of 1 ?
6. James has 2 oranges to divide among his 3 sisters :
how must he divide them, and what part of an orange
will each have? what is 1 -third of 2?
7. If 3 bushels of pears cost $4, how much is that a
bushel ? what is 1 -third of 4?
ANALYSIS. l-third of 4 is the same as 4 times l-third of
1, which are 4-thirds ; 4-thirds are as many ones as 3-thirds
are contained times in 4-thirds: 3-thirds w 4-thirds 1 and
l-third times. Ans. 1 and l-lhird.
8. A carpenter receives ?5 for 3 days' work: how
much. is that a day? what is l-third of 5?
9. What is l-third of 6? J of 7 ? J of 8 ? Jof9?
of 10? of 11? of 13?
10. What is l-third of 16? of 17? of 18? of 19?
of 20 ? of 21 ? of 22 ?
11. What is l-third of 23? of 24? of 25 ? of 26?
of 27? of 28? of 31?
12. If 4 bushels of oats cost $1, what part of a doL
lar will 1 bushel cost? what is 1 -fourth of 1 ?
13. A mother divided 3 pears equally among her 4
children, what part of a pear did each receive ?
2d Bk. 7
102 BAY;S INTELLECTUAL ARITHMETIC.
14. What isl-fourthof 4? Jof5? Jof6? of 7?
of 8? of 9? of 10? of 11?
15. What is 1 -fourth of 12 ? of 13 ? of 15 ? of 17 ?
of 19? of 23?
16. What is 1 -fourth of 25 ? of 26 ? of 27 ? of 28 ?
of 29 ? of 30 ? of 31 ?
17. If a melon cost 5 cents, what part of a melon can
you buy for 1 cent? what is 1 -fifth of 1 ?
18. James had 2 pine-apples, and divided them among
5 of his companions : what part of a pine-apple did each
have ? what is 1 -fifth of 2 ?
19. What is 1-fifth of 3? of4? i of 5 ? of 6?
of 7? of 8.? of 9? of 10? of 11? of 12?
20. What is 1-sixth of 1 ? J of 2 ? I of 3 ? of 4 ?
of 5? of 6? of 7? of 8? of 10? of 15?
21. What is 1-ninth of 2? ' of 4? ^of7? of 8?
of 9? of 12? of 15? of 17? of 19?
22. How do you find one-half of any thing? onethird?
one-fourth? one-fifth? one-eighth?
LESSEN II.
1. If 3 bushels of wheat cost $1, what part of $1
will 1 bushel cost ? what part will 2 bushels cost ?
2. What is 1 -third of 1 ? what are 2-thirds of 1 ?
3. If 3 bushels of wheat cost $2, what part of 81
will 1 bushel cost ? what part will 2 bushels cost ?
ANAL. 1 bushel will cost l-third of $2, which is 2-thirds;
and 2 bushels will cost twice as much as 1 bushel; that is>
2 times 2-thirds, which are 4-thirds of $1, equal $1 and
l-third of a dollar. Ans. $1 and l-third.
4. If 3 barrels of cider cost $4, what part of a dollar
will 1 barrel cost ? what part will 2 barrels cost ?
5. What is l-third of 4 ? 2-thirds of 4 ?
6. What is l-third of 5 ? 2-thirds of 5 ?
FRACTIONS. 103
7. What is 1-third of 6? 2-thirds of 6?
8. What is 1-third of 7 ? 2-thirds of 7 ?
9. What is 1-third of 8 ? 2-thirds of 8?
10. What is 1-third of 9 ? 2-thirds of 9 ?
11. What is 1-third of 10 ?
' 2-thirds of 10 ?
12. If 4 barrels of apples cost $3, what part of a dollar
will 1 barrel cost ? 2 barrels ? 3 barrels ?
, 13. If 5 apples cost 2 cents, what part of a cent will
1 apple cost ? 2 apples ? 3 apples ? 4 apples ?
14. What are 2-fifths of 13 ? 3-fifths of 13 k
15. What is 1-sixth of 7 ? 5-sixths of 7 ?
16. What are 3-sixths of 10 ? 4-sixths of 10?
17. What is 1-seventh of 9 ? 3-sevenths of 9 ?
18. What are 2-sevenths of 11 ? 3-sevenths of 12 ?
19. What is 1-eighth of 13 ? 3-eighths of 13?
20. What are 5-thirds of 4 ? 7-thirds of 5 ?
21. What are 7-eighths of 10 ? 9-eighths of 3 ?
22. What are 3-tenths of 7 ? 7-tenths of 3 ?
23. What are 5-ninths of 11 ? 9-fifths of 7 ?
24. Which is the greater 4 of 2. or * of 1 ? i of 3.
or f of 1 ?
LESSON III.
1. James bought 3 lemons for 7 cents: how much
was that apiece?
2. William bought 5 quarts of chestnuts for 18 cents :
at that rate, what was the cost of 2 quarts ?
3. If 7 pounds of cheese sell for 40 cents, how much
should 5 pounds sell for?
4. Bought 6 yards of muslin ; gave my mother 3-fifths,
and kept the remainder : how many yards had each ?
5. If a quantity of provisions serve 2 men 7 days,
how long will it last 1 man ? how long 3 men ?
104 RAY'S INTELLECTUAL ARITHMETIC.
6. If 4 men perform a job of work in 8 days, how
long will it require 5 men ?
ANALYSIS. It will take 1 man 4 times as long as 4 men ;
*,nd 5 men \-fifth as long as 1 man.
7. If a barrel of cider will last 5 men 8 days, how
long will it last 3 men ?
8. If 1 barrel of flour serve 8 persons 10 days, how
long will it last 11 persons?
9. If 7 men can do a piece of work in 5 days, how
long will it require 8 men ?
10. If 2 men build a wall in 12 days, how long will
it take 7 men ?
11. If it requires 11 days, of 8 hours each, to do a
job, how many will be required of 10 hours each?
12. A man paid 37 cents for riding 8 miles : at the
same rate, what will it cost to ride 11 miles?
13. Two pipes of a certain size will empty a cistern
in 17 minutes : in what time will 3 pipes empty it ?
14. If 18 bushels of oats last 5 horses 1 week, how
many bushels will 7 horses require ?
15. If a laborer receive 5 bushels of wheat for 7 days'
work, how much should he receive for 11 days?
16. If a carpenter earn $8 in 5 days, how much will
he earn in 9 days ?
17. If 3 yards of cloth cost $13, what cost 8 yards?
18. A pole, 18 feet long, is 2-sevenths in the earth,
the rest in the air : what is the length of each part ?
19. Three men, A, B and C, found a bag containing
815: A got 2-ninths, B 1-third, and C the remainder:
what was the share of each ?
20. If J a yard of cloth cost 4, what cost 6 yards ?
21. If 3 ounces cost 36 cents, what should be charged
for of an ounce ?
22. If | of a number equal 18, what is the | of it?
FRACTIONS. ,05
23. How much must a man earn a day to receive $72
for 8 weeks, 6 days to a week ?
SECTION XX.
LESSON I.
ILLUSTRATION. If 1-half of a yard of tape be divided
into 2 equal parts, one of the parts is 1-half of 1-half;
that is, 1 -fourth of the whole yard.
one-half. one-half.
one-fourth. one-fourth. one-fourth. one-fourth.
If 1 -third of a yard be divided into two equal parts,
one of the parts is 1 -sixth of the whole yard; that is,
1-half of 1-third, is 1-sixth.
one-third. one-third. one-third.
l-sixth. 1-sixth. 1-sixth. 1-sixth. 1-sixth. 1-sixth.
1. Mary having 1-half of an orange, gave her brother
1 -half of what she had : what part of a whole orange did
she give him ?
2. James divided 1-third of an apple equally between
his 2 brothers ; what part did each receive ?
3. What is J of |? i of ?
ANALYSIS. 1-half of 1-third is the fraction obtained by
dividing a unit into 3 equal parts, and one of these parts
into 2 equal parts ; but if a unit (one) be thus divided, there
will be 6 parts, and each part will be 1-sixth of the whole;
therefore, 1-half of 1-third is 1-sixth. <
106 RAY'S INTELLECTUAL ARITHMETIC.
4. Thomas divided 1-half of a lemon equally between
his 3 sisters : what part did each receive ?
5. If 1 -fourth of an orange be divided into 2 equal
parts, what is 1 of the parts called ?
6. What is J of J ? J of J ?
7. If 1- third of an apple be cut into 3 equal parts,
what part of the apple will each piece be ?
8. If each half of an apple be divided into 5 equal
parts, how many parts will there be ? what is 1 part
called ? what is 4 of ?
9. If you divide an orange into 4 equal parts, and cut
each part into 3 pieces, what is 1 piece called ?
What single Fraction equals
10. I of
11. i of
12. 4 of
of |?
on?
? 4 of 4? 4 of 4? 4 of A? 4 * 5 ' 3
A g ' 5 U1 5 ' 3 UI 9
is. ion? i f i? iu? i f 4?
14. Thomas has 2-thirds of an apple, and wishes t<?
give his brother 1-half of what he has: what part of the
whole apple must he give him ?
15. What is J of J ? J of ?
ANALYSIS. I-half of 2-thirds is 2 times l-half of 1-third;
\-half of l-third is l-sixth; 2 times l-sixth are 2-sixths.
16. Daniel has 3-fifths of a melon to divide equally
between his brother and sister : how must he divide it
and what part of the whole will each receive ?
17. What is i of | ? i of | ? i of | ?
18. What is | of | ? 1 of ? i of | ?
19. What is I of 4 ? 4 of g ?
-i of ?
20. What is 4 of | ? of 3 ? of f ?
21. Edward has 4-fifths of a melon, and wishes to give
his sister 2-thirds of what he has : what part of the whole
melon will she receive ?
FRACTIONS. 107
22. What'is of T% ? | of f ? f of f ?
ANALYSIS. 2-thirds of 4-jfifths are 2 times 1 -third of
trfifths, and l-third of 4-ffths is 4 times l-third of l-Jifth.
23. What are f of f ? | of f ? | of f ?
24. What are | of
jj
? | of f ? | of | ?
25. What are f of f ? | of 4 ? | of Ty
26. What are f of f ? f of _5_? f of f ?
27. If 1 yard of cloth is worth 2J bushels of wheat,
what is l-half of a yard worth ?
ANALYSIS. 2 and l-half are 5-halves, and l-half of 5-halves
is 5 times l-half of l-half, or 5-fourths, equal 1 and l-fourth.
Ans. 1 and l-fourth bushels.
What single Fraction will represent
28. i of 2-i ? i of 11 ? J of If ? | of 2?
29. A of 2| ? i of 3i ? i of 4| ? i of 51 ?
30. f of U ? | of 1| ? f of LI ? | of 1| ?
31. | of 21? | of 4| ? | of 2|? I of 3?
LESSON II.
1. A person owning | of a ship, sold | of his share :
what part of the whole ship did he sell ?
2. If 3 yards of cloth cost $2|, what cost 2 yards?
ANALYSIS. 1 yard will cost l-third as much as 3 yards,
and 2 yards will cost 2 times as much as 1 yard; that is,
2-thirds as much as 3 yards.
3. If 2 yards of cloth cost $1|, what cost 3 yards?
4. If 3 yards of cloth cost $5^, what part of that
sum will 2 yards cost?
5. If 5 gallons of molasses cost $3, what part of
that sum will 3 gallons cost ?
108 KAY'S INTELLECTUAL ARITHMETIC.
6. If 7 pounds of sugar cost l-i, what cost 4 pounds ?
7. If 8 pounds of butter cost $1J , what cost 7 pounds ?
8. If 7 yards of cloth cost $5|, what will be the cost
of 3 yards ? of 4 yards ?
9. If 3 barrels of cider cost $4|, what part of that
sum will 5 barrels cost?
10. If 5 gallons of oil cost $2|, what part of that sum
will 7 gallons cost ?
11. If 2 gallons of molasses cost $1|, what will be the
cost of 3 gallons ?
12. If 3 bottles of wine cost $2|, what will be the cost
of 8 bottles? of 10 bottles?
13. If 2 men do a job of work in 3| days, how many
days will it take 1 man ? 3 men ?
14. A man can do a job in 3| days, of 10 hours each:
how many days will it take of 7 hours each ?
15. If a man can do a piece of work in 15| days,
working 5 hours a day, how many days will it take,
working 8 hours a day ?
SECTION XXI.
LESSON I.
1. A boy had 2-thirds of an orange, which he divided
equally between his two sisters : what part of an orange
did each receive ?
ANALYSIS. Since 2 apples divided by 2 give 1 apple, and
$2 divided by 2 give $1, it follows that 2-thirds divided by 2
must give l-third. Each, therefore, must have received l-third
of an orange.
2. How often is 3 contained in 3-fourths ?
3. If 2 and 2-thirds dollars be divided among 4 men,
what is each man's share ?
FRACTIONS. 109
4. If 2f be divided by 7, what will be the result ?
5. How many times 6, in 3 and 3-fifths ?
6. How many thirds of 9, are contained in 3| ?
7. Divide 6 and 3-fourths cents among 9 boys.
8. A man having lOf acres of land, divided it equally
between his 5 children : how much did each receive ?
9. If 4f be divided by 11, what will be the quotient?
10. If $| be equally divided among 3 men, what part of
a dollar will each get ?
11. How divide a fraction by any whole number ?
LESSON II.
1. If 1 apple cost i a cent, how many will 1 cent buy?
why ? Ans. Because there are 2 half-cents in 1 cent.
2. If 1 pear cost 1-third of a cent, how many pears
can you buy for 1 cent? why?
3. When 1 bushel of wheat costs &|, how many bushels
can you buy for $H ?
4. How often is ^ contained in 2i or f ?
ANALYSIS. One-half is contained in Fiw-halves, as often
as 1 is contained times in 5 ; that is, 5 times : for, if 1 apple
is contained in 5 apples 5 times ; 1 cent in 5 cents, 5 times;
\-half must be contained in 5-halves, 5 times.
5. If 1 lemon cost half a cent, how many lemons can
be bought for 5 cents ?
6. If 1 apple cost 1 -third of a cent, how many apples
can John buy with 4 and 1 -third cents?
7. If 1 peach cost 2-thirds of a cent, how many
peaches can you purchase for 4 and 1 -third cents?
ANALYSIS. 4 and l-third are 13-thirds ; and Two-thirds are
contained in THiRTEEN-rffaVds as often as 2 is contained in 13,
that is, 6 and \-half times. Ans. 6^ peachesHO
RAY'S INTELLECTUAL ARITHMETIC.
8. I distributed 2f bushels of wheat among a number
of poor persons, giving to each, 2-thirds of a bushel :
now many persons were there ?
9. At | a yard, how many yards of calico can be
purchased for &3f ?
10. At $| a yard, how many yards of , cloth can be
purchased for S3]
?
11. If a lemon cost 3-fourths of a cent, how many
can be purchased for 3| cents ? for 5 cents ?
12. One bushel of rye is worth 3-fourths of a bushel
of wheat: how many bushels of rye can be bought
with 4f bushels of wheat ? with 8| bushels ?
13. At 1 -fifth of a cent each, how many cherries can
Mary purchase with 3 cents ? why ?
14. At $? a gallon, how many gallons of vinegar can
you buy for ?2f ? for
15. How often is f contained in 2| ? in 4f ? in 61 ?
16. How often is J contained in 3| ? in 51 ? in 4| ?
17. How often is f contained in 1 ? in 3f ? in
18. How often is f contained in 3 ? in 4| ? in
19. At $| a yard, how much cloth can be had for
ANALYSIS. As many yards as l-third of a dollar, the price
of 1 yard, is contained times in I -half of a dollar; l-half is
equal to 3-sixths, and l-third to 2-sixths : 2-sixth.s in 3-sixths,
1 and l-half times. Ans. 1 and l-half yards.
20. At $^ a yard, how much gingham can be purchased
for $| ?
21. How often is i contained in | ?
22. If 1 yard of calico cost $A, how much can be purchased
for $| ?
23. How often is i contained in |?
NOTE. Reduce the fractions to a common denominator, then
divide their numerators.
FRACTIONS. 1 1 1
24. How often is ^ contained in | ? in f ? in { ?
25. How often is
-|
contained in f ? in f ? in }| ?
26. How often is f contained in f ? in | ? in i| ?
27. How often is | contained in f ? in | ? in T
9 ?
28. How often is U contained in | ? in | ? in 2| ?
29. How often is 2\ contained in ? in | ?. in 3| ?
30. How often is 3| contained in | ? in f ? in 5| ?
SECTION XXII.
LESSON I.
1. James gave his brother 1 apple and 1-half, which
Was 1-half of what he had : how many had he ?
2. If 1-third of a yard cost $l, what cost a yard?
3. If a man travel 1 and 1-third miles in 1- ourth of
an hour, how far can he travel in 1 hour ?
4. Daniel bought 1 -fifth of an orange for 3J cents :
at that rate, what will a whole one cost ?
5. If 1 -fourth of a yard of tape cost 3 and 2-thirds
cents, what will 1 yard cost ?
ANALYSTS. 3 and 2-thirds are l-fourth of 4 times 3 and
2-thirds ; 4 times 2-thirds are S-thirds, equal 2 and 2-thirds ;
4 times 3 are 12, which added to 2 and 2-thirds^ make 14
und 2-thirds. Ans. 14 and 2-thirds.
6. 5 and 1-third are 1-half of what number?
7. 7 and 3-fourths are 1-third of what number ?
8. 5 and 3-eighths are 1 -fifth of what number?
9. 3 and 4-sevenths are 1-seventh of what number?
10. 4 and 4-fifths are 1-six.th of what number?
11. 9 and 2-thirds are 1 -eighth of what number?
112 RAY'S INTELLECTUAL ARITHMETIC.
12. If 2-thirds of a yard of ribbon cost 3 cents, what
will 1-third cost? If 1 -third of a yard cost 1 and 1-half
cents, what will a yard cost ?
13. One and 1-half is 1-third of what number ? Three
is 2-thirds of what number ?
14. If 2-thirds of a barrel of flour cost $5, what cost
1-third of a barrel ? If 1-third of a barrel cost $2J-, what
will a whole barrel cost ?
15. Five is 2 times what number ?
ANALYSIS. 5 is 2 times l-half of 5; 1-half of 5 is 2 and
1-half; therefore, 5 is 2 times 2 and 1-half
16. Five is 2-thirds of what number?
ANALYSIS. 5 is 2-thirds of that number of which 1-halJof
5 is 1-third: 1-half of 5 is 2 and 1-half and 2 and 1-half
are 1-third of 3 times 2 and 1-half which are 7 and 1-half;
therefore, 5 is 2-thirds 0/7 and 1-half
17. If 3-fourths of a yard of velvet cost $4, what will
1 -fourth of a yard cost? If 1 -fourth cost $1|, what will
a whole yard cost ?
18. Four is 3-fourths of what number ?
19. If 2-fifths of a quart of chestnuts cost 3 cents,
what will 1 -fifth cost? If 1 -fifth of a quart cost 1 and
1-half cents, what will a whole quart cost?
20. Three is 2-fifths of what number ?
21. If 4-fifths of a yard cost $7, what will 1-fifth cost?
If 1-fifth of a yard cost $1|, what cost a yard?
22. Seven is 4-fifths of what number ?
23. If 3-eighths of a melon cost 4 cenfs, what will
1-eighth cost? what will a whole melon cost ?
24. If 3-fifths of a pound of sugar cost 10 cents, what
will 1-fifth cost ? what will 1 pound cost?
25. If 5-sixths of a barrel of beef cost $6, what will
1 -sixth cost? what will a barrel cost?
FRACTIONS. 113
26. Fifteen is 2-ninths of what number ?
27. Fourteen is 3-eighths of what number ?
28. Thirteen is 3-fourths of what number ?
29. Seventeen is 8-ninths of what number ?
30. If 1 yard and a half, that is, 3 half yards of ribbon
cost 6 cents, what will a yard cost ?
31. Six is 3-halves of what number ?
ANALYSIS. Of that number of which l-third of 6 is
l-half; l-third of 6 is 2 ; 2 is \-lialf of 2 times 2, which are
4; therefore, 6 is 3-halves of 4.
32. If 1 yard and l-third of a yard, that is, 4-thirds
of a yard of cloth, cost $3, what will a yard cost ?
33. If a man travel 9 miles in 1 hour and 2-sevenths
of an hour, how far will he travel in 1 hour ?
34. By selling a watch for $18, I gained 1 -fourth of
what it cost me : how much did it cost ?
35. A grocer, by selling a lot of flour for $25, gained
1 -fifth of what it cost him: what was the cost? how
much did he gain ?
36. If a man pay $6 for 1 and l-third yards of cloth,
what is the cost of 1 yard ?
37. If a man receive $10 for 2 and 2-thirds days'
work, how much is that per day ?
38. If 2-thirds of a yard cost $4|, what cost a yard ?
39. Four and 6-sevenths, are 2-thirds of what ?
ANALYSIS Of that number of which l-half of 4 and 6-se&
#nths is l-third.
40. Four and 2-thirds .are 2-fifths of what ?
41. Three and 3-fourths are 5-sixths of what?
42. One and 3-fifths are 3-fifths of what?
43. Three and 2-thirds are 3-fourths of what ?
44. Four and 4-fifths are 6-halves of what ?
114 RAY'S INTELLECTUAL ARITHMETIC.
LESSON II.
1. A man gave to some poor persons $3, which was
2-fifths of his money : how much had he left ?
2. A pole stands 5 -sevenths in the air, and 5 feet in
the ground : how long is the pole ?
3. A man spent 2-fifths of his money, and had $10
left : how much did he spend ?
4. At 3 yards for 2 cents, how many yards of tape
can be bought for 7 cents ?
5. At 5 lemons for 3 cents, how many lemons can be
bought for 12 cents ?
6. If 4-fifths of a yard of calico cost 8 cents, how
many yards can be purchased for 25 cents ?
7c If 1 and 1 -fourth tuns of hay cost $8, what is the
price of 1 tun ?
8. If 6-sevenths of a yard of cloth cost $4, how many
yards can be purchased for 12 ?
9. A jockey, by selling a horse for $45, gained
1 -eighth of the cost: what was the cost?
10. If 1 and 1 -third yards of cloth cost $5, how much
can be purchased for $12 ?
11. By selling 5 yards of cloth for $12, I gained
1 -third of the cost: what did I pay per yard ?
12. By selling 7 yards of cloth for $21, I made
2-fifths of the cost : I paid for it with wheat, at $| per
bushel : how many bushels did I give ?
13. If 3-fifths of an apple cost 2-thirds of a cent, what
will 3 apples cost ?
14. What will be the cost of 11 yards of cloth, if 5
and 1-half yards cost $4| ?
15. By selling cloth at $8 a yard, 1-fifth of the cost
was lost : what part would have been gained, if the clotb
had been sold for $11 ?
TABLES. 115
SECTION XXIIL TABLES.
UNITED STATES MONEY.
10 mills, marked m., . . make 1 cent, . . marked ct.
10 cents " 1 dime, .
"
d.
10 dimes or 100 cents. " 1 dollar, .
"
$.
10 dollars
" 1 eagle, .
" E.
1. Repeat the table of United States Money.
2. How many cents in a half dime ? in a quarter of
a dollar ? in a half dollar ?
3. In 1 cent how many mills ? in 2 ? in 3 ?
4. In 1 dime how many cents ? in 2 ? in 4 ?
5. In 1 dollar how many cents ? in 2 ? in 4 ?
6. In 20 mills how many cents? in 30? in 50?
7. In 20 cents how many dimes ? in 25 ? in 30 ?
8. At 20 cents a yard, what will 3 and 3-fourths
yards of calico cost ? how many dimes ?
9. At 25 cents a yard, what will 3 and 1 -third yards
r}f muslin cost ? how many dimes ?
ENGLISH MONEY.
4 farthings (far.) . make 1 penny, . . marked d.
12 pence
" 1 shilling, .
"
s.
20 shillings . . . .
" 1 pound, . . .
1. Repeat the table of English Money.
2. In 1 penny how many farthings ?in2?3? 5?
3. In 1 shilling how many pence ? in 2? 3? 4?
4. How many shillings in 1 pound ? in 2? 3? 4?
5. How many pence in 8 farthings ? in 12? 16?
6. How many shillings in 24 pence ? in 36 ? 56 ?
7. How many pounds in 40 shillings ? in 60 ? 72 ?
U6 RAY'S INTELLECTUAL ARITHMETIC.
TROY WEIGHT.
24 grains (gr.) make 1 pennyweight, . marked pwt.
20 pennyweights
" 1 ounce, .... "
oz.
12 ounces ... " 1 pound,. ... "
Ib.
1. Repeat the table of Troy Weight,
2. In 1 pennyweight how many grains ? in 2? 3?
3. In 1 ounce how many pennyweights ? in 2 ? 3 ?
4. In 1 pound how many ounces ? in 3 ? 5 ?
5. In 24 oz. how many Ib.? in 36 ? 48? 28? 56?
APOTHECARIES WEIGHT.
20 grains (gr.) . . make 1 scruple, . . marked 3.
3 scruples. ... " 1 dram, ... "
3-
8 drams .... " 1 ounce, ... "
5-
12 ounces .... " 1 pound, ... " Ib.
1. Repeat the table of Apothecaries Weight.
2. In 1 scruple how many grains ? in 2 ? 3 ?
3. In 1 dram how many scruples ? in 3 ? 7 ?
4. In 1 ounce how many drams ? in 2 ? 3 ?
5. In 1 pound how many ounces? in 3 ? 7?
6. In 15 oz. how many Ib. ? in 24 ? in 35 ? 48 ? 50 ?
AVOIRDUPOIS WEIGHT.
16 drams (dr.). . make 1 ounce, . . marked oz.
16 ounces, . . . .
" 1 pound, . .
" Ib.
25 pounds, . . . ,
" 1 quarter, .
"
qr.
4 quarters or 100 Ib. " 1 hundred weight, cwt.
BO hundred weight,
" 1 tun, ... " T.
1. Repeat the table of Avoirdupois Weight.
2. In 2 pounds how many ounces? in 3? in 4?
TABLES.
3. In 2 quarters how many pounds ? in 3 ? in 4 ?
4. In 2 tuns how many cwt. ? in 3 ? in 4 ? in 5 ?
5. At 2 cents an ounce, what will 2 Ib. cost?
6. At 2 dollars a quarter, what will 2 cwt. cost?
DRY MEASURE.
2 pints (pt.) . . . make 1 quart, . . . marked qt.
8 quarts
" 1 peck, ... "
pk.
4 pecks
" 1 bushel, . .
" bu.
1. Repeat the table of Dry Measure.
2. In 1 quart how many pints ? in 2? in 3? 5?
3. In 1 peck how many quarts ? in 2 ? jn 4 ? 7 ?
4. In 1 bushel how many pecks ?in2? in 3? 5?
5. In 2 pints how many quarts ?in5? in 8? 9?
6. In 8 quarts how many pecks ? in 16 ? 24 ? 32 ?
7. In 4 pecks how many bushels ? in 12 ? in 20 ?
8. At 5 cents a pt., what will 2 qt. of beans cost ?
9. At 6 cents a qt., what will 1 peck of corn cost?
10. At 3 cts. a qt., what cost 3 and 3-fourths pk. of salt?
LIQUID OR WINE MEASURE.
marked pt.
"
qt.
"
gal.
bl.
hhd.
" T.
1. Repeat the table of Wine Measure.
2. In 1 pint how many gills ? in 2 ? in 3 ? in 5 ?
3. In 2 quarts how many pints ?in3? 5? 6? 8?
2d Bk. 8
4 gills (gi.) . . . i
2 pints
118 RAY'S INTELLECTUAL ARITHMETIC.
4. In 1 gal. how many quarts? in 2 ? 3 ? 7 ? 10?
5. In 1 tun how many hogsheads ? in 3? in 5? 8?
6. In 4 gi. how many pints ? in 7 ? in 8 ? in 9 ?
7. In 3 pt. how many quarts ? in 6? in 8? 11?
8. In 4 qt. how many gallons ? in 7 ? in 11 ? 12 ?
9. At 12 cents for one pint, what will 1 and 3-fourtha
gallons of Port wine cost ?
LONG MEASURE.
12 inches (in.) . . make 1 foot, . . . marked ft.
3 feet " 1 yard, . .
"
yd.
5 yards or 16J ft.
" 1 rod (or pole),
" rd.
40 rods or 220 yd.
" 1 furlong, .
" fur.
8 furlongs ... " 1 mile,. . .
" mi.
1. Eepeat the table of Long Measure.
2. In 2 feet how many inches ? in 3 ? in 5 ? in 8 ?
3. In 2 yards how many feet ? in 3 ? in 4 ? in 5 ?
4. In 2 rods how many yards ? in 3 ? in 4 ? in 5 ?
5. In 1 furlong how many rods ? how many yards ?
6. In 1 mile how many furlongs ? in 3? 4? 5?
7. How many inches in 8 and 5-sixths feet?
8. In 2 rods how many feet ? in 3 rods ? 4 rods ?
5 rods ? 6 rods ?
SQUARE MEASURE.
144 square inches . make 1 square foot, marked sq. ft.
9 square feet . .
" 1 square yard,.
"
sq. yd.
30J square yards .
" 1 square rod, .
"
sq. rd.
40 square rods . .
" 1 rood, . . . .
" R.
4 roods .... " 1 acre, . . . .
" A.
640 acres " 1 square mile, .
"
sq. mi
TABLES. 119
1. Repeat the table of Square Measure.
2. How many square feet in 3 square yards ? in 5 ?
3. How many roods in 4 acres ? in 6 ? in 8 ? in 10 ?
CLOTH MEASURE.
4 nails (na.) . . make 1 quarter, . . marked qr.
120 BAY'S INTELLECTUAL ARITHMETIC.
The following table shows the names of the different
months of the year, and the number of days embraced
in each.
January, 1st mon., 31 da.
February, 2d " 28
March, 3d 31
April, 4th 30
May, 5th 31
June, 6th 30
July, 7th mon., 31 da.
August, 8th " 31
September, 9th " 30 "
October, 10th " 31
November, llth " 30 "
December, 12th " 31 "
The number of days in each month of the year may
be retained in the mind by committing the following lines
to memory :
Thirty days has September,
April, June, and November;
Other months have thirty-on-e,
Except the second month alone;
To this we twenty-eight assign,
Till leap-year gives it twenty-nine.
1. Repeat the table. "What make minutes ? days ?
2. In 14 days how many weeks ? in 21 ? in 28 ?
3. How many minutes in 1 hour and a quarter ?
4. There are 12 months in 1 year : what part of a
year is 1 month ? what part are 2 months ? 3 months ?
i months? 5? 6? 8? 9? 10?
5. If there are 30 days in 1 month, what part of a
month are 3 days? what part are 5 days? 6 days?,
10 days? 12 days? 15 days? 18 days? 20 days?
84 days? 27 days?
GENERAL REVIEW.
APPLICATIONS OF MENTAL ARITHMETIC,
FOE ADVANCED CLASSES.
SECTION XXIV.
THM preceding sections contain all the elementary forms of
Anatysiw. Those who are properly acquainted with them, will
find bnt little difficulty in these applications.
LESSON I. ADDITION.
SUGGESTION. Where the numbers are large, it is better to add
by 10' s or 100's, as the case may be.
To add 25 ai*d 35 : 20 and 80 are 50 ; 5 and 5 are 10 ; 50 and
10 are 60. Or, 5 and 30 are 55; 55 and 5 are 60.
OP THE SIGNS.
The sign +, called plus, means more. The numbers
between which it is placed are to be added.
Thus, 2+4, (2 plus 4), shows that 2 and 4 are to be
added together.
The sign of equality, =, shows that the number of
units on the right and left of it are equal to each other.
Thus, 2 + 4 = 6, means that 2 added to 4 equal 6 j and
is read, 2 and 4 are 6, or 2 plus 4 equal 6.
EXAMPLES. 4 + 2 = how many ?
3 + 5 = how many ?
2 + 3 + 4 = how many ?
1 + 3 + 5 + 7 = how many ?
$4 + $3 + $1 + $2 = how many $'s ?
(121)
122 KAY'S INTELLECTUAL ARITHMETIC.
1. What is the sum of 4+ 5+ 6+7?
2. What is the sum of 9 + 10 + 11 + 8 ?
3. What is the sum of 20 + 12 + 9 + 11 ?
4. What is the sum of 24 + 20 + 12 + 30 ?
5. What is the sum of 35 + 40 + 15 + 20 ?
6. What is the sum of 50 + 60 + 70 + 80 ?
7. What is the sum of 54 + 20 + 13 + 12 ?
8. What is the sum of $21+$16+$13+$20?
9. Bought at one time 33 gal. of oil, at another 20,
at another 40, at another 50, and at another 62 : how
many gal. did I buy ?
10. A lady paid 23 for a dress, $18 for a shawl, and
69 for a bonnet : how much did she spend ?
11. I owe A 850, B $75, C $40, and D $20: how
much money do I owe ?
12. I collected of one man $110, of another $90, of
another $75, and of another $50 : how much in all ?
13. Thirty-two plus 16 + 20 + 21 + 18 = what ?
14. Fifty-nine plus 21 + 32 + 15 + 11 = what?
LESSON II. SUBTRACTION.
The sign , called minus, means less. Placed between
two numbers, it shows that the one on the right, is to be
taken (subtracted), from the one on the left.
Thus, 4 2=2; read> 4 minus (less) 2 equal 2.
EXAMPLES. 5 2 = how many ?
3 + 9 8 = how many ?
1 + 5 + 9 10= how many ?
2 + 3 + 4+8 11= how many ?
$5 + $9 + $7 + $4 $12 = how many $'s ?
REVIEW. MULTIPLICATION. 1 23
1. What number equals 75 less 40 ? 160 120 ?
100 45? 110 90? 120 95?
2. A boy having 75 ct, purchased 55 ct. worth of
goods : how much change did he receive ?
3. Having $92, I purchased a watch for $73: how
much had I left?
4. Bought a horse for $110, and sold him for $145:
how much did I make?
5. George bought candles for 25 ct., soap for 10 ct.,
sugar for 35 ct., and starch for 3 ct. : he gave $1, and
received 30 ct. change : was this correct ?
6. A boy had $5, from which he took at one time $1
and 50 ct.
; at another, 40 ct.
; at another, $1 and 10 ct.:
how much had he left ?
7. Ten plus 22 + 19 less 8 + 3 + 9 + 6=what ?
8. 42 + 19 + 13 + 15 12 17 20 4=?
9. $37 + $33 + $45 + $25 $35 $20 $40= ?
10. $125+ $140 + $20 $100 $50 $8 $30= ?
11. $160+$80 +130 $210 $30 $10 $40==?
LESSON III. MULTIPLICATION.
The sign X, denotes multiplication, read, multiplied
ly. When placed between two numbers, it shows that
ihey are to be multiplied together.
Thus, 3x5, means 3 multiplied by 5.
EXAMPLES. 3X5 = how many ?
2X5X7 = how many ?
2X4X6X3 = how many ?
$8 + $12 $7 X 10 = how many $'s ?
124 RAY'S INTELLECTUAL ARITHMETIC.
1. What is the product of 2+6X2? 25X2? 16X3?
20X5? 22X6? 40X5? 38X3? 60X4? 45x5?
24X6? 53X9? 65X8?
2. What is the product of 14x6? 4X7X5?
5X6x7? 9X10X5? 6x8X5?
3. What will be the cost of 5 yd. of cloth, at $2
and 50 ct. a yd. ?
4. A man traveling at the rate of 5 mi. an hr., meets
a stage going at the rate of 9 mi. an hr. : how far from
the man will the stage be in 10 hr. ?
5. What cost 9 boxes, at 7 dimes each ?
6. What cost 8 Ib. and 4 oz. of sugar at 12 ct. a Ib. ?
7. What cost 75 ft. of lumber, at 3 ct. a ft. ?
8. If 1 A. of land produce 85 bu. of corn, how many
bu. will 11 A. produce ?
9. Bought 15 Ib. of coffee at 10 ct. a Ib., and 13 Ib.
at 9 ct. a Ib, : what did the whole cost ?
10. Henry has 19 ct ; George 3 times as many, lacking
10 : how many have beth?
11. How many yd. in 3 bales of cloth, each containing
6 pieces of 35 yd. each ?
LESSON IV. DIVISION.
The sign -7-, is read, divided ~by. When placed between
two numbers, it shows that the first is to be divided
by the second.
Thus, 6-7-2, means that 6 is to be divided by 2.
EXAMPLES. 6-
4X5
4 + 10X 2
$3 + $12 $5X4
2 = how many ?
2 = how many ?
7 = how many ?
8 = how many $'
REVIEW. DIVISION. 1 25
1. How often is 3 contained in 48 ? in 51 ? in 60 ?
In 75 ? in 81 ? in 90 ? in 144 ?
2. Divide 125 by 5; multiply the result by 10; the**
divide by 2 : what is the last quotient ?
3. Multiply 14 by 20, and divide the product by 7*
4. What does 12 X 13 -=- 2 -f- 6 = ?
5. What does 15 X 12 -*- 3 -f- 12 3 == ?
6. What does 27 + 9 -f- 12 + 20 17 -f-2 X 5 =?
7. What is 1-half of 28 + 1-third of 72 ?
8. What are 9-fifteenths of 120 + of 60 ?
9o If a boat sail 48 mi. in 12 hr., how far will she
sail in 6 hr. ?
10. At 15 ct. a lb., what quantity of beef can be purchased
for $6 ?
11. Three men bought a horse for $90 : after keeping
him 6 wk.j at $3 a wk., they sold him for $99 : what did
each man lose ?
12. Seven multiplied by 9, divided by 3, 2 added, 13
subtracted, and divided by 5, will = what ?
13. Add 10 to 12 X 3-7-6 + 5-7-8+10 X 5-7-6-^
2 = what ?
14. Seventeen + 6 8 -r- 3 X 8 6 + 14 -r- 4 X 8~
8+12 --10 + 6 + 7 12X 6= what?
15. What number added to itself will give a sum equal
to 14?
Explanation. If a number be added to itself, the sum will
be 2 times the number : 14, then, is 2 times what number ?
16. What number added to itself 3 times, will make 32 ?
17. Divide 16 into 2 parts, so that the second pari
will be 3 times the first.
Explanation. The sum of the parts (16), will be 4 time* the
first part
126 RAY'S INTELLECTUAL ARITHMETIC.
18. Divide 48 into 2 such parts that the second shall
be 7 times the first. 48 is 8 times what number ?
19. Divide 24 into 3 parts, so that the second shall
be 2 times the first, and the third 3 times.
20. A boy being asked the number of ct. he had, replied
:
" Five times the number I have, is just 40 less
than 10 times the number :" how many had he ?
21. Find a number which, being multiplied by 2 -7- 8
X 3 9-f-3X 3 + 11 X 3 1-5-10, equals 5.
Explanation. Begin with the last number mentioned, and reverse
every operation indicated by the signs : thus, 5 X 10 = 50 ;
60+1=51; 51 -J- 3 = 17, &c.
22. What does 12X5 + 3-r-7+ll-f-5 1 + 10
+ 14 .^3 + 19 + 8 9 + 17 + 8=?
23. What does 13 + 27 + 14 + 10 -*- 8 + 21 + 13 -*-
7 + 14 + 20 + 23 + 3 -f- 11 = ?
24. What does 19 + 2 13x6-f-4 + 7 12X5
--7 + 15 11 X 8 --12 + 15 14=?
LESSON V. PRINCIPLES.
1. When 10 was taken from a number, only 2-thirds
of the number remained : what was the number?
2. The sum of two numbers is 12 ; if 6 be added to
the sum, the result will be twice the greater number : what
are the numbers?
3. The sum of two numbers is 16 more than their
difference : if their difference be 4, and 8 one of the numbers,
what is the other number ?
4. The sum of two numbers diminished by the less
gives 15 : if 10 is the less number, what is their difference ?
REVIEW. PRINCIPLES. 127
5. If 6 be taken from the difference of two numbers,
the remainder will be 2 : if 4 is one of the numbers,
what is the other ?
6. If 10 be added to the difference of two numbers,
the sum will be 6 more than the greater number, which
is 19 : what is the less number ?
7. If 10 be taken from the sum of two numbers, of
which 5 is one, there will be 8 left: what is the other
number ?
8. By what part of 6 must 4 be X to = f of 20 ?'
9. What number -=- 8 will give 13 for a quotient ?
10. What number X 12, will give 156 for a product ?
11. If 15 be multiplied by some number and 20 added
to the product, the sum will be 200 : what is the multiplier
?
12. A certain number X 2, gives a result as much less
than 20 as the number is greater than 7 ; but when it is
subtracted from 11, it leaves the same remainder as 5
from 7 : what is the number ? why ?
13. Six is contained in a certain number 12 times, with
a remainder of 5 : what is the number ?
14. If 12 be added to a certain number, 7 will be contained
9 times in the sum, with a remainder of 1 : what
Is the number?
15. If 13 be taken from a certain number, 8 will be
contained 10 times in the difference, with a remainder of
3: what is the number ?
16. When the divisor of 132 was increased by 6, the
quotient was found to be 11 : what was the divisor?
17. When the divisor of 72 was multiplied by 2, the
quotient was 9 : what was the divisor ?
18. When the divisor of 84 was divided by 3, the quotient
was 4 : what was the divisor ?
128 RAY'S INTELLECTUAL ARITHMETIC.
19. If 1 be added to the number of times a certain
number is contained in 60, the result will be 11 : what is
that number ?
20. Twice the greater of two numbers, 2 = their
sum, which is 20 : what are the numbers ?
21. A boy received of his father 3 ct.
; of his mother
twice as many less 1 ; if he had received from his father
5 ct. more, his father would have given him 4 times as
many as his sister : how many ct. did he receive ?
SECTION XXV. QUESTIONS.
LESSON I.
1. If I of a yd. of cloth cost $2, what cost ^ of a yd. ?
2. If | of a yd. of cloth cost $5, what cost | of a yd. ?
ANALYSIS. \-tliird of a yd. will cost \-Tialf as much as
2-thirds ; and 3-thirds, or a whole yd., will cost 3 times as
much as I-third ; and,
One-fourth of a yd. will cost l-fourth as much as 1 yd.,
and ^-fourths, 3 times as much as l-fourth.
3. If | of a bl. of flour cost 3, what cost f of a bl. ?
4. If 4 of a yd. of muslin cost 24 ct., what will y\ of
a yd. cost ?
5. If | of a tun of hay cost $15, what will one-half a
tun cost?
6. If | of an orchard contain 30 fruit trees, how many
trees are there in T
7
^ of it ?
7. If 1| yd. of cloth cost $14, what cost 2^ yd. ?
8. If 14, bl. of flour cost $5J, what cost 2J bl. ?
9. If 3J Ib. of cheese cost 20 ct., what cost 2| Ib. ?
REVIEW. QUESTIONS. 1 29
10. A traveled 30 mi. in 3f hr. : at that rate, how far
can he travel in 7J hr. ?
11. If a man earn $1J in 10 hr., how much can he
earn in 11 hr. ?
12. A can earn $9? in 6 da., of 8 hr. each : how much
tan he earn in 7 da., of 9 hr. each ?
13. If 5| bu. of wheat cost $94, what cost 3| bu. ?
14. If 8J is f of a number, what is f of it?
15. If 3^ is 2 times some number, what is 2J times
that number ?
16. If | of a bl. of flour cost $4i, what cost f of a bl.?
17. If f of a yd. of lace cost $f, what cost | of a yd. ?
18. If the wages of 3 men for 5 da, is $30, what will
be the wages of 4 men for 7 da. ?
SUGGESTION. First find the wages of one man for one day.
19. If 6 persons spend $36 in 8 da., how much, at that
rate, would 5 men spend in 12 da. ?
20. If 3 men can build 12 rd. of wall in 8 da., how
many rd. can 5 men build in 3 da. ?
21. If 6 horses eat 36 bu. of oats in 10 da., how many
bu. will 5 horses eat in 9 da. ?
22. If 5 oxen eat 2 A. of grass in 6 da., in how many
da. will 12 oxen eat 8 A. ?
23. If a family of 8 persons spend $400 in 5 mon.,
how much would maintain them 8 mon., if 3 more persons
were added ?
ANALYSIS. $400 for 5 mon. is $80 for 1 mon. : if 8 persons
spend $80 in 1 mon., I person spends $10 in I mon.
Hence, 1 1 persons spend $1 10 in 1 mow., and $880 in 8 mon.
24. If 10 oxen can be kept on 5 A. for 3 mon., how
many sheep can be kept on 15 A. for 5 mon., if 7 sheep
eat as much as 1 ox ?
130 RAY'S INTELLECTUAL ARITHMETIC.
LESSON II.
1. If 5 men can do a piece of work in 18 da., how
many men will do it in 9 da. ?
2. If 8 men can do a piece of work in 15 da., how
many men can do it in 12 da. ?
3. If 9 pipes fill a cistern in 2^-hr., in what time
will 5 such pipes fill it ?
4. If 5 men do a piece of work in 6 da., how many
can do a piece twice as large, in 1 -fifth the time?
5. If 8 men can do a piece of work in 5 da., in what
time can 5 men do it ?
6. If 6 men can do a piece of work in 5 da., in what
time can they do it, if they receive the assistance of 3
additional men when the work is half completed ?
7. If 7 men can do a piece of work in 4 da., in what
time can it be done, if 3 of the men leave when the work
is half completed ?
'8. If 20 Ib. of flour afford 8 five ct. loaves, how many
one ct. loaves will it furnish ? how many four ct. loaves ?
how many ten ct. loaves ?
9. If 10 Ib. of flour afford 6 five ct. loaves, how
many 3 ct. loaves will it furnish ?
ANALYSIS. It will afford 5 times as many 1 ct. loaves as 5
ct. loaves : l-third as many 3 ct. loaves as 1 ct. loaves.
10. If a sack of flour make 20 three ct. loaves, how
many 4 ct. loaves will it make ? 5 ct. loaves ?
11. If the 5 ct. loaf weigh 8 oz. when flour is $3 a
bl., what should it weigh when flour is 1 a bl. ? what if
flour is 2 a bl. ? $4 a bl. ?
12. A 4 ct. loaf weighs 10 oz. when flour is $6 a bl. ;
what will it weigh when flour is $5 a bl. ?
REVIEW. QUESTIONS. 1 3 1
ANALYSIS. Ifflour were $1 a bl, it ought to weigh 6 times
as much as when flour is $6 a bl. ; that is, 60 oz. : and.
When flour is $5 a bl., it ought to weigh J as much as when
it is $1 a bl., that is, 12 oz.
13. If the 3 ct. loaf weigh 7 oz. when flour is $3J a bl.,
what ought it to weigh when flour is $2J a bl. ?
14. If 6 men can mow a field in 5J da., how muck
time would be saved by employing 4 more men ?
LESSON III.
1. A and B hired a pasture for $45 : A pastured 4
cows, and B 5 cows : what should each pay ?
ANALYSIS. They together pastured 9 cows, of which ^ were
As, and f J3's; hence, A should pay | of $45, which are
$20; and B f of $45, which are $25.
2. William had 3 ct., Thomas 4 ct., and John 5 ct.
;
they bought 36 peaches : what was the share of each ?
3. Two men paid $3 for 7^ dozen oysters : the first
paid $2, and the second $1 : how many should each have ?
4. A and B bought a horse for $40; A paid $25,
and B the rest : they sold him for $56 : what should each
receive ?
5. A boat worth $860, of which ^ belonged to A, J to
*
B, and the rest to C, was entirely lost : what loss will
each sustain, it having been insured for $500 ?
6. Two boys bought a silver watch for $7 : the first
paid $2^ ; the second, $4A ; they sold it for $21 : what
was each one's share ?
7. A man failing, paid 80 ct. on each dollar of his
indebtedness : what did I receive, if he owed me $60 ?
ANALYSIS. 80 ct. are f of $1 ; he therefore paid me | of
">,or$48.
132 KAY'S INTELLECTUAL ARITHMETIC
8. A grocer failing, pays 60 ct. on the dollar : what
mil B receive to whom he owes $25 ?
9. A trader failing, pays only 15 ct. on the dollar:
what will C receive to whom he owes $80 ?
10. A and B rent a pasture for $25 : A puts in 27
oxen, and B 180 sheep : what should each pay, supposing
an ox to eat as much as 10 sheep ?
11. A and B rent a pasture for $60 : A puts in 14
horses, and B 15 cows : what should each pay, if 2 horses
eat as much as 3 cows ?
12. A and B rent a pasture for $75 : A puts in 8
horses ; B 15 oxen and 120 sheep : what should each pay,
if a horse eat as much as 20 sheep, and 2 horses as much
as 3 oxen ?
13. A and B rent a pasture for $35 ; A puts in 4 horses
2 wk. ; B, 3 horses 4 wk. : what ought each to pay ?
ANALYSIS. 4 horses for 2 wk. = 1 horse for 8 wk. ; and
3 horses for 4 wk. 1 horse for 12 wk. : 8 wk. and 12 wk.
are 20 wk ; hence,
A must pay ^ or f of the rent,=$l4; and B , or f of
the rent, = $21.
14. C and D join their stocks in trade ; puts in $50
for 4 mon., and D $60 for 5 mon. : they gain $45 : what
is the share of each?
15. Two masons, A and B, built a wall for $81 ; A
sent 3 men for 4 da., and B 5 men for 3 da. : what ought
each to receive ?
16. A and B traded in company ; A put in $2 as often
as B put in $3 ; A's money was employed 5 mon., and
B's 4 mon. ; they gained $55 : what was each man's share ?
17. E and F rented a field for $27 ; E put in 4 horses
for 5 mon., and F 10 cows for 6 mon. : what ought each
to pay, if 2 horses eat as much as 3 cows ?
EEVIEW. QUESTIONS. 133
LESSON IV.
1. Divide 20 apples between A and B, so that A may
as often as B gets 3.
ANALYSIS. Of each 5 apples, A must get 2 and B 3J
In 20 apples there are 4 times 5 apples ; hence,
A must get 4 times 2 apples, or 8 apples; and B 4 times 3
zpples, or 12 apples.
2. Divide 28 ct. between John and James, so that
John may get 3 as often as James gets 4.
3. Divide 45 ct. between A, B, and C, so that A may
get 4 ct. as often as B gets 3, and gets 2.
4. In an orchard of 96 trees, there are 5 apple-trees
for 3 peach-trees : how many of each kind ?
5. On a farm there are 60 animals horses, cows, and
sheep ; for each horse there are 3 cows, and for each cow
there are 2 sheep : how many animals of each kind ?
6. A school of 35 pupils has 2 boys for 3 girls : how
many of each in the school ?
7. What number is that which being added to 3 times
itself will make 48 ?
8. Divide 42 plums between A, B, and C, so that B
may get twice, and C three times as many as A.
9. Mary has 25 yd. of ribbon : she wishes to divide
it into two parts, so that one shall be 4 times the length
of the other : what will be the length of each part 1
10. Divide 35 cherries between Emma, Agnes, and
Sarah, so that Agnes shall have twice as many as Emma,
and Sarah twice as many as Agnes.
11. Divide 28 into two parts, so that one shall be 3
fcimes a certain number, and the other 4 times.
2d Bk. 9
134 RAY'S INTELLECTUAL ARITHMETIC.
LESSON V.
1. What part of 8 is 2? what part is 4? is 1 ?
2. How many times does 10 contain 2 ? 2 is what
part of 10?
3. Twelve is how many times 2 ? 2 is what part of
12 ? what is the ratio of 2 to 12 ?
EXPLANATION. When two numbers are compared, to
see how many times greater one is than the other, what
do you find? Ans. The ratio of the numbers.
How do you find the ratio or relation of two numbers?
Ans. Divide the second number by the first.
4. How many times does 18 contain 9 ? what is the
ratio of 9 to 18?
5. What is the ratio of 12 to 36 ? 9 to 45 ? 11 to 66?
13 to 52 ? 2 to 1 ? 4 to 3 ?
6. What is the ratio of 2i to 5 ? 6] to 12-^ ? \ to \ ?
of \ to ? | to i ? i to f?
is b o O xv o
7. If the ratio of two numbers is 5, and 6 is the less
number, what is the greater ?
8. The ratio of 7 to 21 is equal to the ratio of some
number to 36 : what is the number ?
9. Five less than the ratio of 2 to 20 is | of the ratio
of 2 to what ?
10. The ratio of 2 to 18, plus 3, is 7 less than the
ratio of 2 to what?
11. The ratio of 9 to 27, increased by 5, is equal to
the ratio of 2 i to what?
12. Divide 25 ct. between John and George, so that
their shares shall be in the ratio of 2 to 3.
Explanation. When two numbers are in the ratio of 2 to 3,
one will contain 2 as often as the other contains 3.
REVIEW. QUESTIONS. 135
13. Divide the number 48 into two parts that shall be
in the ratio of 5 to 7.
14. Divide the number 60 into three parts that shall
be to each other as 3, 4, and 5.
15. Divide the number 70 into 4 parts that shall be
to each other as 1, 2, 3, and 4.
16. Divide the number 22 into two parts that shall be
to each other as 2| to 3.
ANALYSIS. 2^ and 3 are 5J units. The first part will
therefore be as many times 2^, and the second as many times
3, as 5J are contained times in 22.
17. Divide 16 apples between Henry and Oliver, so
that their shares shall be in the ratio of 1J to 2J.
18. Divide the number 39 into three parts that shall
be to each other as ^, J-, and J.
19. Divide 14 ct. between A and B, so that B may
have 1J times as many as A.
ANALYSIS. As often as A gets 1 ct., B gets 1J; that is, of
each 2J ct. A gets 1, and B \\ct. ; Jience,
A gets as many times 1 ct. and B as many times 1| ct.,
as 2J ct. are contained times in 14 ct. Ans. A 6 ct., B 8 ct.
20. John and James together have 33 marbles; James
has If times as many as John : how many has each ?
21. William's age is 1| times Frank's age; the sum of
their ages is 32 yr. : what is the age of each ?
22. A basket contains 30 apples : the number of those
which are sound, is 2J times the number of those not
sound : how many are there of each ?
23. Two men built 27 ft. of wall : how much did eacli
build, if one built J as much as the other ?
24. A, B, and C, have $42 ; B has half as many as
A, and C half as many as B : how many has each ?
136 RAY'S INTELLECTUAL ARITHMETIC.
25. On a, farm there are 104 animals hogs, sheep, and
cows ; there are f as many sheep as hogs, and | as many
cows as sheep : how many are there of each ?
LESSON VI.
1. Divide 15 ct. between A and B, so that B may
have 3 more than A.
ANALYSIS. Reserving 3 ct. for the number that B receives
more than A, there, are 12 ct. left', dividing these equally, A
will get 6 <tf. ; and B, 6 ct. plus 3 ct. reserved, or 9 ct.
2. Thomas has 5 apples more than James, and they
both together have 19: how many has each?
3. The sum of two numbers is 31, and the greater
exceeds the less by 7 : what are the numbers ?
4. Thomas and James each had the same number of
ct., when Thomas found 8 ct. more ; they then had together
32 ct. : how many had each ?
5. Thomas and William each bought the same number
of peaches ; after Thomas ate 4, and William 6, they both
together had 20 left : how many had each remaining ?
6. Mary bought twice as many cherries as Sarah ; and
after Mary ate 7, and Sarah 5, they had only 24 left:
how many had each left ?
7. If 5 be added to the treble of a certain number,
the sum will be 50 : what is the number ?
8. If | of a certain number be increased by 10, the
gum will be 31 : what is the number ?
9. If | of a number be diminished by 7, the remainder
will be 21 : what is the number ?
10. James is 4 yr. older than Henry, and Henry is
E yr. younger than Oliver j the sum of their ages is 37 yr. :
what is the age of each ?
REVIEW. QUESTIONS. 137
11. Mary has 8 ct. more than Jane, and Sarah 3 less
than Mary ; they all have 43 ct. : how many has each ?
12. The sum of the ages of Mary and Frank is 42 yr.;
Mary is twice as old as Frank, less 3 yr. : what is the
age of each ?
13. I bought a watch, a chain, and a ring, for $62 ;
the chain cost $5 less than the ring, and the watch $12
more than the chain : what did I pay for each ?
14. Thirty ct. are 6 ct. less than of ^ of my money :
how much have I ?
15. John has twice as much money as James, + $3 ;
Frank has as much as John and James, + $7 ; together
they have $55 : how much has each ?
LESSON VII.
1. Divide the number 15 into two parts, so that the
less part may be | of the greater.
ANALYSIS. The greater part being 3-thirds, and the less,
2-thirds, their sum will be 5-thirds : 15, then, is 5-thirds of
what number ?
2. Thomas and John have $60 to pay ; John has | as
much to pay as Thomas : what must each pay ?
3. The time past from noon is equal to half the time
to midnight : what o'clock is it ?
4. The time elapsed since noon is | of the time to
midnight : what is the hour ?
5. I had 56 mi. to travel in 2 da. ; the second da., I
went | as far as the first : how far did I travel each da. ?
6. Divide 100 into two such parts, that
If of the first
less 8 will = the second.
7. Divide the number 45 into three such parts, that
the second shall be J, and the third J of the first part.
The sum of all the parts will be nine-fourths of the first part
138 RAY'S INTELLECTUAL AKITHMETIC.
8. A, B, and C, together have 40 ct.
; B has f as
many as A, and C f as many as B : how many has each ?
9. A tree 70 ft. long was broken into 3 pieces; the
middle part was | of the top part ; the lower part was |
of the middle part : what was the length of each ?
10. I bought a hat, coat, and vest, for 34 ; the hat
cost | of the price of the coat, and the vest f the price
of the hat : what was the cost of each ?
11. In a field containing 55 sheep and cows, A of the
cows = of the sheep : how many are there of each ?
12. The sum of two numbers is 100 ; and A of the less
equals | of the greater : what are the numbers ?
13. One-fourth of Mary's age = J of Sarah's, and the
sum of their ages is 14 yr. : what the age of each ?
14. Divide 38 ct. between A and B, so that f of A's
share may be equal to f of B's.
ANALYSIS. If f of As share = f of B's, then i of As
share = ^ of | ; that is. T
3
Q of It's ; and the whole of A's =
3 times T
3
, that is, -^ of B's ; and,
Hence, {g of B's + -^ of B's = i of B's= 38 ct. : 38 ct,
then, are || of what number ?
15. Divide the number 51 into two such parts, that | of
the first will equal | of the second.
16. In an orchard of 65 apple and peach-trees, f of the
apple-trees = 4 Of the peach-trees : how many are there
of each ?
17. From C to D is 66 mi. ; A left C at the same time
B left D ; when they met, f of the distance A had traveled
=g of the distance B had traveled: how much
farther did B travel than A ?
18. The time past noon, + 3 hr., is equal to of the
time to midnight : what is the hour ?
REVIEW. QUESTIONS. 139
19. What is the hour in the afternoon, when the time
past noon is equal to J of the time past midnight ?
Explanation. Since the time past noon is one-fifth of the whole
time from midnight, the time from midnight to noon, which is 12
hr., must equal the remaining four-fifths of the time.
20. What is the hour in the afternoon, when the time
past noon is | of the time past midnight ?
21. What is the hour of the day, when ^ of the time
past noon is ^ of the time past midnight ?
LESSON VIII.
1. What number is that to which, if its half be added,
the sum will be 15 ?
ANALYSIS. The number + its ^ = f of the number. Now}
if | 15, J5, and f, or the whole number, = 10.
2. What number is that to which if its f be added,
the sum will be 20?
3. If to Mary's age its f be added, the sum will be
21 yr. : what is her age ?
4. What number is that which being doubled, and increased
by its |, the sum will be 52 ?
5. What number is that which being doubled, and
diminished by its ^, the remainder will be 40 ?
6. What number is that which being trebled, and
diminished by its f , the remainder will be 48 ?
7. If to David's age you add its ^ and its |, the sum
will be 26 : what is his age ?
8. If to Sarah's age you add its |, its J-, + 10 yr.,
the sum will be twice her age : how old is she ?
9. Thomas spent | of his money, and had 30 ct. left ;
how much had he at first ?
140 RAY'S INTELLECTUAL ARITHMETIC.
10. If to a certain number you add its J, its | + 27,
the number will be trebled : what is the number ?
11. A father is 40 yr. older than his son ; the son's
age is T
3
T of the father's age : what is the age of each ?
12. If to Susan's age you add its 4 + 18 yr., the sum
will be 3 times her age : how old is she ?
13. A piece of flannel, losing | of its length by shrinkage,
measured 28 yd. : what was its length ?
14. Tha distance from A to B is ^ the distance from
C to D, and f of the distance from A to B, + 20 mi., =
the distance from C to D : what is the distance from A
to B, and from C to D ?
15. My age + its ^, its i, and its f = 94 : what is
my age ?
LESSON IX.
1. If A can do a piece of work in 2 da., what part
of it can he do in 1 da. ?
2. A can drink a keg of cider in 4 da. : what part
of it can he drink in 1 da. ?
3. B can do a piece of work in 4 a da. : how many
times the work can he do in 1 da. ?
4. C can mow a certain lot in | of a da. : how many
such lots can he mow in a da. ?
5. A can mow a certain field in 2^ da. : what part of
it can he mow in 1 da. ?
ANALYSIS. If he can mow the field in 2^ da., he could mow
twice the field in 5 da., and of twice the field, that is, f of
ihe field in 1 da.
6. B can dig a trench in 3^- da. : what part of it can
he dig in 1 da, ?
7. C can walk from Cincinnati to Dayton in 3^ da. :
what part of the distance can he walk in 2 da. ?
REVIEW. QUESTIONS. 141
8. A can do -J of a piece of work in 1 da., and B J of
It : what part of the work can both do in a da. ?
9. A can do 4, B -j, and C I of a piece of work in 1
da. : what part of it can they all do in a da. ?
10. If A can do a piece of work in 2 da., and B in 3 da. :
in what time can they both together do it ?
ANALYSIS. If A does it in 2 da., he can do J of it in 1 da. ;
and if B does it in 3 da., he can do i of it in 1 da. ; hence,
Both can do | + \ f in 1 da. : it will require them both
as many da. as | are contained times in 1, (the whole work).
Ana.
11. A can dig a trench in 6 da., and B in 12 da. : in
what time can they both together do it ?
12. C alone can do a piece of work in 5 da., and B
in 7 da. : in what time can both do it ?
13. A can do a piece of work in 2 da., B in 3 da., and
C in 6 da. : in what time can all three do it ?
14. A cistern has 3 pipes; the first will empty it in
3 hr., the second in 5 hr., and the third in 6 hr. : in what
time will all three empty it ?
15. A and B mow a field in 4 da.; B can mow it alone
in 12 da. : in what time can A mow it?
Explanation. Both mow one-fourth in 1 da., and B one-twelfth
in 1 da.; therefore, A can mow one-fourth less one-twelfth, which
is one-sixth in 1 da. ; hence, he can mow it in 6 da.
16. A man and his wife can drink a keg of beer in 12
da. ; when the man is away, it lasts the woman 30 da. :
in what time can the man drink it alone ?
17. Three men, A, B, and C, can together reap a field
of wheat in 4 da. ; A can reap it alone in 8 da., and B
in 12 da. : in what time can C reap it ?
18. A can do a piece of work in J a da., and B in J of
a da. : how long will it take both to do it ?
142 RAY'S INTELLECTUAL ARITHMETIC.
ANALYSIS. If A does the work in J of a da., he can do 2
times the work in I da. ; and
If B does the work in\ of a da., he can do 3 times the work
in I da. ; hence,
They both together can do 5 times the work in 1 da., or the
whole work in of a da.
19. A cistern has two pipes ; by the 1st it may be
emptied in * of an hr., and by the 2nd in 1 of an hr. :
in what time will it be emptied by both together ?
20. A can alone dig a cellar in 2\ da., and B in 3J da. :
in what time can they both together dig it ?
Explanation. A digs 2-fifths in 1 da., and B 3-tenths in 1 da. ;
hence, they both dig 7-tenths in 1 da.
21. C can reap a field of wheat in 5 da., and D can
reap it in 3J da, : in what time can both reap it ?
22. A can do a piece of work in 3 da,, and B a piece
3 times as large in 7 da. : in what time can they together
do a piece 5 times as large as the piece done in 3 da. ?
23. A cistern of 100 gal., is emptied in 20 min. by 3
pipes ; the 1st discharges \ a gal. in a min. ; the 2nd, \\
gal. : how much does the 3rd discharge ?
LESSON X.
1. Two-thirds of 14 is f of what number?
ANALYSIS. 1 \ is f ; f of f is twice i of f ; J of f is f, and
twice | are |; if i is % of some number, of f, or is %of
the number ; | is % of seven times |, which are M.
2. | of 5| is | of what number ?
3. 4 f 4f is j^ of what number ?
4. f of 5| is T
7
n of what number ?
5. | of 2| is \ of how many times 2 ?
REVIEW. QUESTIONS. 143
6. Three-fifths of li is f of how many times 4?
7. Three-fourths of 3J- is j of how many times 3?
8. John has 10 marbles, and f- of what John has is
the T
8
7 of what James has : how many has James ?
9. Jane received f of 60 plums ; she gave away | of
her | : how many were left ?
10. James has a given distance to travel ; after going
35 mi., there remains f of the distance : when he has
gone I of the remainder, how far must he then go ?
11. A horse cost $40 ; f of the price of the horse = f of
the price of the cart : what did the cart cost ?
12. B's coat cost $27, and his hat $8; of the cost
of the coat + f that of the hat, = f of the cost of his
watch : what did the watch cost ?
13. Mary lost f of her plums ; she gave f of the remainder
to Sarah, and had 6 plums left : how many had
she at first ?
14. John has 12 ct. ; f of his money = J of of
William's money : how much has William ?
15. From A to B is 36 mi. ; f of this is f of the distance
from C to D : what is the distance from C to D ?
16. On counting their money, it was found that A
had 12 ct. more than B ; and that ^ of B's = f of A's :
how much had each ?
17. In an orchard, J are apple-trees, J are pear-trees,
TV2 are plum-trees, and the remainder, which is 32, cherrytrees
: how many trees are thm*e of each kind ?
18. In an orchard of apple and pear-trees, the latter
are f of the whole ; the apple-trees are 25 more than the
pear-trees : how many are there of each ?
19. In an orchard of apple, plum, and cherry-trees, 69
in all, the plum-trees = J of the apple-trees, and the
cherry-trees = J of the apple-trees + i of the plum :
how many trees are there of each kind ?
144 RAY'S INTELLECTUAL ARITHMETIC.
20. The age of Jane is | of the age of Sarah, and | of
both their ages is | of the age of Mary, which is 12 yr. ;
what are the ages of Jane and Sarah ?
21. How many times T
3
T of 55, is twice that number
of which | of 30 is | ?
22. John's money is f of Charles's ; and | of John's + 33 = Charles's : how much has each ?
LESSON XI.
1. A hare takes 4 leaps while a hound takes 3 ; 2 of
the hound's leaps = 3 of the hare's : how many leaps
must the hound take to gain the length of a hare's leap
on the hare ?
ANALYSIS. Since 2 of the hound's leaps 3 of the hare's,
1 of the hound's leaps = l^of the hares, and 3 of the hound's
leaps 4^ of the hares; hence.
In taking 3 leaps the hound gains % the length of a hare's
leap on the hare; therefore,
The hound must take 6 leaps to gain 1 leap on the hare.
2. Henry takes 6 steps while John takes 5 ; but 4 of
John's steps = 5 of Henry's : how many steps must John
take, to gain one of Henry's steps on him?
3. Henry is 30 steps before John, but John takes 7
steps while Henry takes 5 : supposing the length of their
steps to be equal, how many steps must John take to
overtake Henry ?
4. A hare is 10 leaps before a hound, and takes 4 leaps
while the hound takes 3 ; but 2 of the hound's leaps = 3 of
the hare's : how many leaps must the hound take to
catch the hare ?
ANALYSIS. Since the hound takes 3 leaps while the hare
takes 4, the hound will take 1 leap while the hare takes 1|
leaps, and 2 leaps, while the hare takes 2| ; but,
REVIEW. QUESTIONS. 1 45
By the second condition, 2 of the hounds leaps 3 of the
hare's ; therefore,
In making 2 leaps the hound gains | of a hards leap on the
hare ; that is, 3 leaps 2f leaps ; hence,
To gain 10 of the hares leaps, the hound must make as many
times 2 leaps as is contained times in 1 0, that is, 60 leaps.
5. A hare is 100 leaps before a hound, and takes 5
leaps while the hound takes 3 ; but 3 leaps of the hound
= 10 of the hare : how many leaps must the hound take
to catch the hare ?
6. N is 35 steps ahead of M, and takes 7 steps while
M takes 5 ; but 4 of M's steps = 7 of N's : how many
steps must M take to overtake N ? How many more will
N have made, when he is overtaken ?
7/* A hare is 8 leaps before a hound, and takes 3 leaps
while the hound takes 2 ; but 2 of the hound's leaps are
equal to 3 of the hare's : will the hound catch the hare,
and if not, why ?
8. A fish's head is 6 in. long ; its tail is as long as
the head and half of its body ; and the body is as long
as both head and tail : what is the length of the fish ?
ANALYSIS. Since the tail is as long as the head and half
the body, the tail is 6 in. + ^ the body ; but,
The body = the head and tail ; therefore, the body = 6 in. +
6 in. + ^ the body ; that is,
The body = 12 in. + i the body ; therefore, J the body is 12
in., and the body is 24 in. long.
The tail 6 in. + J of 24 in.= 18 in. Therefore, the whole
length is 6 in. + 24 in. + 18 in. = 48 in.
9. A trout's head is 4 in. long ; its tail is as long as
its head and | of its body ; the body is as long as its
head and tail : what is its length ?
10. A has 10 ct. ; C has as many as A, + 1 as many
as B ; B has as many as both A and C : how many ct.
have B and C each ?
146 RAY'S INTELLECTUAL ARITHMETIC.
11. A man bought a sheep, cow, and horse ; the sheep
cost $8; the cow, as much as the sheep, and 1 as much
as the horse ; and the horse cost twice as much as both
the sheep and cow : what did each cost ?
12. The head of a fish weighs 8 Ib. ; the tail weighs 3 Ib.
more than the head and half the body ; and the body
weighs as much as both the head and tail : what is the
weight of the fish ?
13. B is pursuing A, who is some distance in advance ; B goes 4 steps, while A goes 5, but 3 of A's steps = 2
of B's ; B goes 30 ft. before overtaking A : how many ft.
is A in advance of B ? Ans. 5 ft.
LESSON XII.
1. A gentleman meeting some beggars, found that if
he gave each of them 3 ct., he would have 12 ct. left, but
if he gave each of them 5 ct., he would not have money
enough by 8 ct. : how many beggars were there ?
ANALYSIS. Each beggar will receive 2 ct. more when there
are 5 ct. given to each, than when there are 3 ct. ; but,
Since the money to be distributed is 12 ct. more than 3 ct.
for each beggar, and 5 ct. for each beggar is S ct. more than
the money, it will, therefore, require 20 ct. more to give 5 ct. to
each, than to give 3 ct. ; hence,
As each beggar gets 2 ct. more, it will take as many beggars
to get 20 ct. more, as 2 ct. are contained times in 20 c, which
are 10. Ans. 10 beggars.
2. A father wishes to distribute some peaches among
his children ; if he gives each of them 2 peaches, he will
have 9 left ; but if he gives each 4 peaches, he will have
3 left : how many children has he ?
Explanation. The difference between having 3 peaches left ay
9 left, is 6.
REVIEW. QUESTIONS. 1 47
3. Mary wishes to divide some cherries among her
playmates ; she finds that if she gives each of them 5, she
will have 21 left; but if she gives each 8, she will have
none left : what is the number of her playmates ?
4. A lady wished to buy a certain number of yd of
silk for a dress ; if she paid $1 a yd., she would have
$5 left ; but if she paid $H per yd., it would take all
her money : how many yd. did she want ?
5. To buy a certain number of oranges at 8 ct. each,
requires 6 ct. more than all the money James has ; but
if he buys the same number of lemons, at 3 ct. each, he
will have 29 ct. left : how much money has he ?
6. There are two pieces of muslin, each containing the
same number of yd. ; to buy the first at 12^ ct. a yd.,
requires 40 ct. more than to buy the second at 10 ct. a yd. :
how many yd. in each ?
7. Five times a certain number is 16 more than 3
times the same number : what is the number ?
ANALYSIS. 5 times any number less 3 times the same number,
is 2 times the number; therefore, 2 times the number is 16.
'
8. Thomas's age is three times that of James, and the
difference of their ages is 10 yr. : what is the age of each ?
9. The age of A is 5 times the age of B ; and the
age of B is twice the age of C ; A is 45 yr. older than
C : what is the age of each ?
10. A has ^- as much money as B ; B has | as much
as C ; C has $15 more than A : how much money has each?
11. A farmer's sheep are in 3 fields; the second contains
4 times as many as the first; the third 3 times aa
many as the second, and 70 more than both the first and
second : how many sheep are there in each field ?
12. The age of A is the age of B ; twice the age of
A is | the age of C; C is 20 yr. older than B : what is
the age of each ?
148 KAY'S INTELLECTUAL ARITHMETIC.
13. A father, who had as many sons as daughters,
divided 818 among them, giving to each daughter 82{
and to each son 1 : how many children had he ?
14. A man agreed to pay a laborer 82 for every da.
he worked ; and the laborer, for every da. he was idle,
was to forfeit 81 ; at the expiration of 20 da., the laborer
received 825 : how many da. was he idle ?
ANALYSIS. Had he worked every da. he would have received,
at the expiration of the 20 da,, $.40; but as he received only
$25, he lost $15 by being idle.
Each da. he was idle he received $3 less than if he had
worked, $2 wages, and the $1 forfeited; hence,
He was idle as many da. as $3 are contained times in $15,
that is, 5 da.
15. James was hired for 30 da. ; for every da. he
worked, he was to receive 30 ct., and for every day he
Was idle, he was to pay 20 ct. ; at the end of the time,
he received 85 : how many da. did he work ?
16. When A and B entered school, the age of A was 3
times that of B ; but in 5 yr., A's age was only twice
B's : what were their ages at first ?
ANALYSIS. Since the age of A is three times that of B, and
when 5 yr. are added to each, the age of A is only twice that
of B, therefore,
Three times the age of B, (which is the age of A,) increased
by 5 yr. = twice the age of B and 5 yr ; that is, =
twice the age of B, and twice 5 yr. ; hence,
The age of B must be 5yr.; and that of A, 15 yr.
17. Four yr. ago, the age of A was 3 times the age
of B ; but 4 yr. hence, it will be only twice his age : what
were their ages 4 yr. ago ?
18. B's age is twice A's ; in 10 yr., A's will be f of
B's : what are their ages ?
REVIEW. QUESTIONS. 149
19. A person has 2 watches, and a chain worth $10 ;
the first watch and chain are worth half as much as the
second; the chain and second watch, are worth 3 times
as much as the first : what is the value of each ?
ANALYSIS, Since the 1st watch and chain are together
worth 1-halfofthe 2nd, the 2nd must be worth twice the 1st +
twice the chain. But,
The 2nd watch and chain = 3 times the first; or the 2nd
watch= 3 times the 1st the chain; hence,
Three times the 1st the chain 2 times the first + 2 times
the chain; or the 1st= 3 times the chain, or $30; and
the 2nd = $80.
20. A person has 2 horses, and a saddle worth $12 :
the 1st horse and saddle are worth ^ of the 2nd horse ;
but the 2nd horse and saddle are worth four times the
1st horse : what is the value of each horse ?
21. If a herring and a half cost 2 pence and a half,
how many can you buy for 9 pence ?
LESSON XIIL
1. If 12 peaches are worth 84 apples, and 8 apples
24 plums, how many plums shall I give for 5 peaches ?
2. If 1 ox is worth 8 sheep, and 3 oxen are worth
2 horses, what is the value of 1 horse, if a sheep is
worth $5 ?
3. A walks 10 mi. in 1-| da., and B 8 mi. in If da. :
how far will B travel while A is traveling 20 mi. ?
4. A bought a number of apples at 2 for 3 ct., and
as many more at 2 for 5 ct.
; he sold them at the rate
of 3 for 7 ct. : how much per dozen did he gain ?
5. C bought a number of eggs at 2 ct. each, and twice
as many at 3 ct. each ; he sold them at the rate of 3 for
10 ct. : how much per dozen did he gain ?
2d Bk. 10
150 AAY'S INTELLECTUAL ARITHMETIC.
If he had sold them at the rate of 4 for 10 ct., how
much per dozen would he have gained or lost ?
6. Bought a number of pears at 2 for 1 ct., and as
many more at 4 for 1 ct. ; by selling 5 for 3 ct., I gained
18 ct. : how many pears did I buy ?
7. A poulterer bought a number of ducks, at the rate
of 6 for 81, and twice as many chickens, at the rate of
8 for 81 ; by selling 2 chickens and 1 duck for $, he
gained $2^ : how many of each did he buy ?
8. If 3 men can perform a piece of work in 4 da.,
working 10 hr. a da., in how many da. can 8 men perform
the^same job, working 6 hr. a da. ?
9. Divide 32 peaches between Mary, James, and Lucy,
giving Mary 2, and Lucy 3 more than James.
10. If 10 gal. of water per hr. run into a vessel containing
15 gal., and 17 gal. run out in 2 hr., how long will
the vessel be in filling ?
11. A can do a piece of work in 4^ da., and A and B
together in 2| da. : in what time can B do it alone ?
12. A, B, and C, together, can do a piece of work in
5 da. ; A and B, in 8 da. ; and B and C, in 9 da. : in what
time can each of them do it alone ?
13. If 5 men or 7 women can do a piece of work in
35 da., in what time can 5 men and 7 women do it?
14. If 2 men and 4 women can do a piece of work in
28 da., in what time can 1 man and 1 woman do it, if a
woman does | of a man's work ?
15. A man and his wife consume a sack of meal in
15 da.; after living together 6 da., the woman alone conr
sumed the remainder in 30 da. : how long would a ?ack
last either of them alone ?
16. A man had 80 eggs, which he intended to sell as
follows: 36 at 3 for 4 ct., 24 at 4 for 3 ct., and the rest
at 10 for 17 ct. : but having mixed them, how must he
sell them per dozen to get the intended price ?
REVIEW. QUESTIONS. 151
17. If | of James's money be increased by $6. the sum
will equal what Thomas has; both together have $34:
how much has each ?
18. If $5 be taken from the | of A's money, the remainder
will equal B's ; both together have $51 : how
much has each ?
19. The age of A is twice the age of B ; and f of B's
age + 44 years, = 2^ times the age of A : what is the
age of each ?
20. A has not $40 ; but if he had half as many more,
and $2^ besides, he would have $40 : how many has he ?
21. Two and a half times a number + 2 = 100: What
is the number ?
22. A farmer sold | of his sheep, but soon afterward
purchased 4 as many as he had left, when he had 65
sheep : how many sheep had he at first?
23. John had $50 in silver and gold j | of the silver,
increased by $10, is equal to 1| times the gold : what
amount has he of each ?
24. One-half of A's money, diminished by $3, is equal
to J- of B's, increased by $5, and both together they have
$56 : how much money has each ?
25. A started from C the same time that B started
from D ; when they met, | of the distance A traveled = | of the distance B traveled ; from C to D is 86 mi. :
what was the distance each traveled ?
26. Two-thirds of A's money = f of B's, and | of
their difference is $15 : how much money has each ?
27. My watch and chain cost f as much as my watch ;
3 times the price of my chain + twice the price of my
watch = $100 : what did each cost?
28. Three towns, A, B, and C, are situated on the same
road ; the distance from A to B is 24 mi. ; and | of the
distance from A to B = f of the distance from B to C
how far is it from A to C?
152 KAY'S INTELLECTUAL ARITHMETIC.
29. A, B, and C, rent a pasture for 92 ; A puts in 4
horses for 2 mon., B 9 cows for 3 mon., and C 20 sheep
for 5 mon. : what should each pay, if 2 horses eat as much
as 3 cows, and 3 cows as much as 10 sheep ?
30. John bought 5 melons for 5 ct., and James 3
melons for 3 ct. ; they then joined Thomas, and each one
ate an equal part of the melons ; when Thomas left, he
gave them 8 ct. : how should this be divided ?
31. A person having 3 sons, A, B, and C, devised f of
his estate to A, ^ to B, and the remainder to C ; the difference
of the legacies of A and C was 160 : what amount
did each receive ?
32. The age of A is f of the age of B ; and the sum
of their ages -f- half the age of B = twice the age of
A 2 yr. : what is the age of each ?
33. A and B together can do a job in 16 da. ; they
work 4 da., when A leaves, and B finishes the work in
36 da. more : in how many da. can each do it ?
34. Three persons, A, B, and C, are to share a certain
sum of money, of which A's part is $12, which is % of the
sum of the shares of B and C; and | of C's share is equal
to T
3 of the sum of the shares of A and B : what are the
shares of each ?
35. If 9 men mow a field in 12 da., how many men can
mow of it in of the time ?
36. Two men formed a partnership for 1 yr. ; the 1st
put in 100, and the 2nd, 200 : how much must the
first put in at the end of 6 mon., to entitle him to 1-half
of the profits ?
37. A and B had 24 ct. ; A said to B, Give me 2 of
your ct., and I shall have twice as many as you ;" B replied,
" Give me two of yours, and I shall have as many
as you :" how many had each?
38. A gentleman being asked his age, replied :
" The
excess of f of 50 above *my age, is equal to the difference
between my ag~ and 10 yr. :" what was his age ?
PERCENTAGE. 153
39. If I sell my eggs at 6 ct. a dozen, I will lose 12 ct.
;
but if I sell them at 10 ct., I will gain 18 ct. : what did
they cost per dozen ?
40. If I sell my sugar at a certain price per lb., I will
lose $1, but if I increase the price 3 ct. per lb., I will
gain 50 ct. : how many lb. have I ?
41. If the labor of 1 man is equal to that of 2 women,
and the labor of 1 woman is equal to that of 3 boys, how
many men would it take to do in 1 da., what 12 boys are
a wk. in doing ?
42. If sugar worth 3i ct. a lb., be mixed in equal
quantities with sugar worth 6i ct., what will ^ a lb. of
the mixture be worth, and how many lb. must be given
for $1 ?
43. If | of the gain = 7\ of the selling price, for how
much will 3| yd. of cloth be sold, that cost $4 a yd.?
44. When sugar is worth 7 ct. a lb., a package was sold
for 24 ct., gaining 3 ct. : for how much should a package
weighing twice as much be sold, to gain 5 ct., when sugar
costs 8 ct. a lb. ?
SECTION XXVI. PERCENTAGE.
LESSON I. GAIN AND LOSS.
Explanation. The terras Percentage and Per cent, mean a certain
number of parts out of each hundred parts ; that is, a certain
number of hundredths of the sum considered.
One per cent, of any number is 1-hundredth of that number,
that is, 1 of each 100 parts.
Thus, 1 per cent, of $100 is $1 : 1 per cent, of $200 i* $2 , of
$50, 50 cents: of $1, 1 cent.
1. What is 1 per cent, of $1 ? $2 ? $5 ?
2. What is 2 per cent, of $3 ? $4 ? $6 ?
154 RAY'S INTELLECTUAL ARITHMETIC.
3. What is 3 per cent, of $10 ? $20 ? $60 ?
4. What is 4 per cent, of $25 ? $45 ? $75 ?
5. What is 5 per cent, of $100 ? $300 ? $700 ?
6. What is 6 per cent, of $150 ? $250 ? $350 ?
7. What is 2^ per cent, of $100 ? $200 ? $500 ?
8. I bought a piece of cloth for $15, and in selling
it gained 5 per cent, of the cost : what did I gain ?
ANALYSIS. 5 per cent, of 1 is five \-JiundredtTis; 5 per cent,
of $1 is 5 ct. ; of $15 it will be 15 times 5 ctf., wfo'cA are
75 c Ansstfb cZ.
9. A grocer bought a bl. of sugar for $10, and in
selling it gained 10 per cent. : how much did he gain ?
10. A farmer having a flock of 40 sheep, lost 5 per
cent, of them : how many had he left ?
11. A flock of 50 sheep increases 10 per cent, in one
year : how many are then in the flock ?
12. A lady having $20, spent 10 per cent, for muslin,
and 10 per cent, of the remainder for calico : how much
did she pay for both ?
One per cent, of anything is T^ part of it; two per
cent, is T|^ = J^ ; four per cent. T^ = r,
1-.
5 per cent. = ^
8| per cent. = j
1
^
10 per cent. = -L
12^ per cent. = J
16| per cent. = 1
20 per cent, =
25 per cent. =
33| per cent. =
50 per cent. =
75 per cent. =
13. I paid 30 ct. per yd. for muslin : at what price
must I sell it, to make 10 per cent. ?
ANALYSIS. Ten per cent, being IQ-hundredths, or \-tenth,
I must add to the first cost \-tenih of itself: but,
PERCENTAGE. 155
One-tenth of3Qct. is 3 ct, and 30 ct. adaed to 3e. arc
33 ct Ans. 33 ct. per yd.
14. To make 12| per cent, profit, what must muslin be
sold at that cost 8 ct. per yd.? 25 ct. ?
15. To make 8| per cent, profit, what must sugar be
sold for that cost 6 ct. per Ib. ? 12 ct. ?
16. To make 25 per cent, profit, what must calico be
sold for that cost 12 ct. per yd.? 16 ct? 20 ct.? 35 ct?
LESSON II.
1. A merchant bought cloth at $5 per yd., and sold
it at $7 per yd. : what did he gain on a yd. ? how much
v per cent. ?
ANALYSIS. Since Tie bought at $5 and sold at $7 per yd.j
he gained $2 on every $5, that is,
He gained | of the first cost ; ^ of this is T^j, and are
twice -$j which are T
4^, or 40 per cent.
2. James bought a melon for 4 ct., and sold it for 5 ct. :
what per cent, did he gain ?
3. An orange was bought for 5 ct., and sold for 4 ct. :
what was the per cent, of loss ?
4. Thomas bought a watch for $4, and sold it for $6 :
what per cent, did he gain ?
5. Henry bought a horse for $15, and sold it for $24 :
what per cent, did he gain ?
6. A keg of wine holding 5 gal., lost 6 qt by leakage
: what was the loss per cent. ?
7. By selling citrons at 6 ct. each, John cleared 1 of
the first cost : what per cent, would he have cleared by
selling them at 8 ct. each ?
8. A merchant bought cloth at the rate of 6 yd. for
83, and sold it at the rate of 5 yd. for $4 : what per cent,
did he gain ?
156 RAY'S INTELLECTUAL ARITHMETIC.
9. Henry sold melons at 8 ct. each, and lost i of the
first cost : what per cent, would he have lost by selling
them at 3 for 25 ct. : what per cent would he have gained
by selling them at 2 for 25 ct. ?
10. James bought a lot of lemons, at the rate of 2 for
3 ct. ; but finding them damaged, he sold them at the rate
of 3 for 2 ct. : what per cent, did he lose ?
11. Sold a watch for $12, and gained 20 per cent.: what
was the first cost ?
ANALYSIS. 20 per cent, is -%$ or , hence the gain was equal
to \ of the cost; therefore,
The watch:*oldfor | + I = f of the cost 12 then is f of
what?
12. I sold a piece of cloth for S26, and gained 30 per
cent. : what did the cloth cost me ?
- 13. If there is a gain of 40 per cent, when muslin is
sold at 14 ct. a yd., what is the cost price ?
14. By selling a horse for 45, there was a gain of 12^
per cent. : what did the horse cost ?
15. Sold a horse for 145, and lost 10 per cent. : what
was the cost ?
16. Thomas sold a watch for 21, and gained 75 per
cent. : what did he pay for it ?
17. James sold 10 oranges for 40 ct., and gained 33^
per cent. : how much did each orange cost ?
18. "When an article is sold at | of its cost, what is the
gain per cent. ?
19. When an article is sold at | of its cost, what is the
loss per cent.? at f ? at T% ? at $% ?
LESSON III.
1. When the gain is 20 per cent., what part of the
cost is equal to the gain ? when it is 75 per cent. ?
PERCENTAGE. 157
2. When the gain is 100 per cent., what part of the
cost is equal to the gain ? when it is 150 per cent. ?
3. When the loss is 25 per cent., what part of the
cost is equal to the loss ? when it is 35 per cent. ?
4. What is the loss per cent, when the whole is lost?
What is the gain per cent, when the gain is three times
the cost?
5. When -i of the gain is equal to \ of the cost, what
is the gain per cent. ?
Explanation. If 1-third of the gain is equal to 1-fifth of the
cost, the whole gain is equal to 3-fifths of the cost, and 3-fifths
are 60-hundredths or 60 per cent.
6. A sold a watch, so that of the gain was equal
to 2
6- of the cost : what did he gain per cent. ?
7. When | of the gain is equal to i| of the cost, what
is the gain per cent. ?
8. Sold a watch for $10, by which I gained 25 per
cent. : what per cent, would I have gained by selling it
for $12?
9. By selling muslin at 7 ct. per yd., there is a loss
of 124 per cent. : what will be the loss per cent, by selling
it at 6 ct. per yd. ?
10. By selling my horse for $35, there was a loss
of 16| per cent. : what would have been the gain per cent,
by selling him for $63?
11. I bought a watch for $18, which was 20 per cent,
more than its value : I sold it at 10 per cent, less than its
value : what sum did I lose ?
12. A sold B a watch for $60, and gained 20 per cent. :
afterward B sold it and lost 20 per cent, on what it cost
him : how much did B lose more than A gained ?
13. A watchmaker sold 2 watches for $30 each : on
one he gained 25 per cent., and on the other he lost 25
per cent. : how much did he gain or lose by the sale ?
158 RAY'S INTELLECTUAL ARITHMETIC.
14. By selling 4 apples for 3 ct., a dealer gains 50 pei
cent. : what per cent, will he gain by selling them at the
rate of 5 for 4 ct. ?
15. Sold 5 lemons for 4 ct., and lost 20 per cent. : what
per cent, will I lose by selling 6 for 5 ct. ?
16. Two-thirds of 10 per cent, of 60, is i of what per
cent, of 40 ?
17. One-half of f of 50 per cent, of 120, is 10 less
than 20 per cent, of what ?
iv. INTEREST.
EXPLANATION. Interest is money paid for the use of money.
The Principal is the sum of money which is loaned.
The Amount is the principal and interest added together.
The Rate Per Cent, is so many cents paid on each dollar.
1. If the interest of $1 at 6 per cent, for 1 yr., is 6 ct.,
what will be the interest of $10? of $12? of $15?
of $20 ?
2. What is the interest of $2 for 3 yr., at 5 per cent.?
ANALYSIS. The interest of $1 for 1 yr. at 5 per cent., is 5
ct. ; and for $2 the interest is twice as much as for $1, which
is 2 times 5 ct., equal 1 ct. ; and,
For 3 yr. the interest is 3 times as much as for 1 yr., which
is 3 times 10 ct., equal 30 ct. Ans. 30 ct.
3. Find the interest of $5 for 2 yr., at 6 per cent.
4. Find the interest of $8 for 5 yr., at 5 per cent.
5. Find the interest of $20 for 3 yr., at 8 per cent.
6. Find the interest of $25 for 6 yr., at 4 per cent.
7. Find the interest of $40 for 4 yr., at 5 per cent.
8. Find the interest of $50 for 3 yr., at 6 per cent.
PERCENTAGE. 159
9. Find the interest of $60 for 2 yr., at 7 per cent.
10. Find the interest of $75 for 3 jr., at 4 per cent.
LESSON Y.
1. What the interest of $50 for 5 mon., at 6 per cent.?
ANALYSIS. For 5 mon. the interest will be 5 times as much
a#for\ mon.; andfor 1 mon., \-twelfth as much asfor a yr.
2. Find the interest of $60 for 4 mon., at 5 per cent.
3. Find the interest of $80 for 7 mon., at.6 ;per cent.
4. Find the interest of $40 for 9 mon., at 8 per cent.
5. Find the interest of $75 for 8 mon., at 9 per cent.
What is the interest
6. Of $120 for 6 mon. 15 da., at 5 per cent. ?
7. Of $150 for 10 mon. 10 da., at 4 per cent. ?
8. Of $45 for 11 mon. 23 da., at 8 per cent. ?
9. Of $200 for 4 mon. 24 da., at 6 per cent. ?
10. Of $480 for 9 mon. 18 da., at 5 per cent.?
11. Of $360 for 5 mon. 19 da., at 5 per cent.?
12. Of $144 for 8 mon. 25 da., at 4 per cent.?
13. Of $40 for 1 yr. 4 mon., at 6 per cent. ?
14. Of $60 for 2 yr. 3 mon., at 5 per cent. ?
15. Of $75 for 1 yr. 3 mon. 6 da., at 4 per cent. ?
What is the amount
16. Of $25 for 3 yr., at 4 per cent. ?
17. Of $40 for 2 yr., at 5 per cent. ?
18. Of $55 for 3 yr., at 8 per cent. ?
19. Of $30 for 1 yr. 4 mon., at 7 per cent. ?
160 RAY'S INTELLECTUAL ARITHMETIC.
20. Of $50 for 2 yr. 3 mon. 6 da., at 6 per cent. ?
21. Of 90 for 1 yr. 3 inon. 6 da., at 8 per cent. ?
LESSON VI.
1. The interest of a certain principal for 2 yr., at 6
per cent., is $3 : what is the principal ?
ANALYSIS. The interest of $1 for 2 yr., at 6 per cent., is
12 ct. ; and the principal must be as many times $1 as 12 ct.
are contained times in $3. Ans. $25.
2. The interest of a certain principal for 3 yr., at 4
per cent., is $6 : what is the principal ?
3. What principal at interest for 4 yr., at 5 per cefct.,
will produce $12 interest ?
4. What principal at interest for 5 yr., at 8 per cent.,
will produce $30 interest?
5. What principal at interest for 4 yr., at 7J per cent.,
will produce $42 interest ?
6. What principal at interest for 2 yr. 6 mon., at 6
per cent., will produce $36 interest?
7. What principal at interest for 3 yr. 4 mon., at 6
per cent., will produce $70 interest ?
8. A father wishes to place such a sum at interest
at 5 per cent., as will produce for his son an annual income
of $200 : what sum must he invest ?
LESSON VII.
1. What principal on interest for 2 yr., at 5 per cent.,
will amount to $55 ?
ANALYSIS. The amount of $1 for 2 yr. at 5 per cent., is
$1.10, and it will require as many times $1 to amount to $55
as $1.10 is contained times in $55.
PERCENTAGE. 161
What principal on interest,
2. At 6 per cent., for 3 yr., will amount to $236 ?
3. At 5 per cent., for 4 yr., will amount to $600 ?
4. At 10 per cent., for 5 yr., will amount to $375 ?
5. At 6 per cent., for 5 yr., will amount to $390 ?
6. The amount due on a note which has been on interest
3 yr. 4 mon., at 6 per cent., is $30 : what is the
face of the note ?
7. Two-fifths of A's money on interest for 2 yr. 6
mon., at 8 per cent., is $60: what is his whole money ?
LESSON VIII.
1. In what time, at 6 per cent., will $50 give $10
interest ?
ANALYSIS. The interest of $50 for 1 yr., at 6 per cent, is
$3 ; and it will require $50 as many yr., to give $10 interest,
as $3 is contained times in $10, which is 3J.
In what time,
2. At 5 per cent., will $40 give $8 interest ?
3. At 8 per cent., will $75 give $15 interest?
4. At 10 per cent., will $60 give $16 interest?
5. At 5 per cent., will $14$ give $24 interest ?
6. At 6 per cent, will $25 give $10 interest?
7. In what time, at 4 f>er cent., will any given principal
double itself?
ANALYSIS. At 1 per cent, any given sum, as $100, will
double itself in 100 yr. ; and,
At 4 per cent, it will double itself in J of the time that it
will at 1 per cent : \ of lOOyr. is 25 yr. Ans. 25 yr.
162 KAY'S INTELLECTUAL ARITHMETIC.
8. In what time will any given principal double itself,
at 2 per cent. ? at 3 per cent. ? at 5 per cent. ? at 6 per
cent. ? at 7 per cent. ? at 8 ? at 10 ? at 12 ?
9. In what time will any given principal treble itself,
at 5 per cent. ?
10. In what time will any given principal treble itselfj
at 8 per cent. ? at 10 per cent. ?
LESSON IX.
1. At what per cent, will 200, in 2 yr., give $24 in.
terest ?
ANALYSIS. At I per cent, for 2 yr., $200 will give $4 in*
terest; and,
It will take as many times 1 per cent, for $200 to give $24
interest, as $4, the interest of $200 at 1 per cent., is contained
times in $24.
At what per cent.,
2. Will $50 in 5 yr., give $20 interest ?
3. Will $75 in 3 yr., give $11J interest ?
4. Will $300 in 3 yr., give $63 interest?
5. Will $300 in 2 yr. 3 mon., give $54 interest ?
6. Will $240 in 3 yr. 4 mon., give $56 interest?
7. Will $200 in 4 yr., amount to $240 ?
8. Will $150 in 3 yr. 8 mon., amount to $183 ?
j,
9. Will any given principal double itself in 20 yr. ?
ANALYSIS. Any principal, as $100, will double itself
in 1 yr., at 100 per cent., and in 20 yr., at ^ of 100 per
cent., or 5 per cent.
At what per cent.,
10. Will any given principal double itself in 12 yr. ?
PERCENTAGE. 163
11. Will any given principal double itself in 10 yr. ?
12. Will any given principal double itself in 8 yr. ?
in 5 yr. ? in 4 yr. ? in 2 yr. ?
LESSON X.
1. What principal, at 5 per cent, for 4 yr., will amount
to $72?
ANALYSIS. $1, at 5 per cent., for 4 yr., will amount to
$1.20; and,
The required principal will be as many times $1, as the
amount of $1 at the given rate for the given time, which is
$1.20,.& contained times in $72.
2. What principal, at 6 per cent., for 5 yr., will
amount to $520 ?
3. What principal, at 4 per cent., for 5 yr., will
amount to $30 ?
4. What principal, at 10 per cent., for 5 yr., will
amount to $750 ?
5. What principal, at 5 per cent., for 3 yr., will
amount to $345 ?
6. What principal, at 6 per cent., for 4 yr., will
amount to $496 ?
7. What is the present worth of $24, due 4 yr. hence>
reckoning interest at 5 per cent. ?
Explanation. The present worth is that principal of which $24
is the amount. The discount is the interest on the present worth.
8. What is the present worth of $65, due 5 yr. hence,
interest at 6 per cent. ? what the discount ?
9. What is the present worth of $55, due 5 yr. hence,
interest at 5 per cent. ? what the discount ?
10. A owes $77, payable 6 yr. 8 mon. hence : what will
he gain by paying it now, money worth 6 per cent. ?
164 RAY'S INTELLECTUAL ARITHMETIC,
LESSON XI.
1. At 6 per cent., for 4 yr. 2 mon., what part of the
principal is equal to the interest ?
2. At 5 per cent., for 5 yr., what part of the amount
is equal to the interest ?
3. When the interest for 2 yr. = 1 of the principal,
what is the rate per cent. ?
4. When the interest for 2 yr. 6 mon. ==
\ of the
principal, what is the rate per cent. ?
5. When the interest, at 10 per cent. = f of the principal,
what is the time ?
6. When 3 times the yearly interest = ^ of the principal,
what is the rate per cent. ?
7. When I of the interest for 2 yr. = ^ of the principal,
what is the rate per cent, ?
8. When f of the interest for 3 yr. = g% of the principal,
what is the rate per cent. ?
9. The interest for 8 mon. is ^ of the principal :
what is the interest of 200 for 1 yr. 4 m$n. ?
10. If the interest for 1 yr. 4 mon., is ^ of the principal,
what the interest of $100 for 1 yr. 8 mon. 18 da. ?
11. In what time will any principal at 5 per cent., give
the same interest as in 4 yr., at 10 per cent. ?
12. The interest of A's and B's money for 3| yr., at 5
per cent., is $40, and A's money is twice that of B's :
what sum has each ?
13. Twice A's money = 3 times B's ; and the interest
at 7 per cent, for 1| yr., of what they both have, is $49 :
how much money has each ?
14. One-half of A's money = f of B's ; and the interest
of | of A's and ^ of B's money, at 4 per cent, for
2 yr. 3 mon., is $18 : how much has each ?
THE END,
.
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