C. It is difficult to find examples which will aptly illustrate this operation. It can be done more conveniently by the instructer. Whenever a fraction occurs, which may be reduced to lower terms, if it be suggested to the pupil, he will readily perceive it and do it. This may be done in almost any part of the book, but more especially after studying the 13th section. Perhaps it would be as well to omit this article the first time the pupil goes through the book, and after he has seen the use of the operation, to let him study it. It may be illustrated on Plate III in the following manner.

8. 3. Find all the squares which are divided into 24 parts. There are 4 squares which are divided into 24 parts, viz. the 8th in the 3d row, the 3d in the 8th row, the 6th in the 4th row, and the 4th in the 6th row. Then see if exactly 18 can be found in one or more of the vertical divisions. In the 6th square of the 4th row, there are exactly 18 divisions in three vertical divisions, but those 3 vertical divisions are of the whole square, because it is divided into fourths vertically; therefore are equal to 3.

13. 1. Find the squares which are divided into 56 parts; they are the 8th in the seventh row, and the 7th in the 8th row; see if in either of them, one or more of the vertical divisions contain exactly 42 parts. In the 7th of the Sth row, 6 vertical divisions contain exactly 42; these divisions are g of the square, for it is divided vertically into 8 parts. But may be still reduced to 2, as may be seen by looking on the 3d square of the 4th row; therefore # is equal to 3.

SECTION XIV.

A. THIS section contains the division of fractions by whole numbers, and the multiplication of one fraction by another. Though these operations sometimes appear to be division, and sometimes multiplication, yet there is actually no difference in the operations. The practical examples will generally show how the operations are to be performed, but it will be well to use the plate for young pupils.

1 and 2. In the second row, the 2d square is divided vertically into halves, and each of the halves is divided into halves by the horizontal line; } of { is therefore of the whole.

3 and 4. In the third row, the 2d square shows that of is.

16 and 17. In the 5th row, the 3d square shows that of is of the whole.

33. Since

of a share signify 3 parts of a share,

it is evident that

is .

39. that

of the three parts is 1 part, that

signify 9 pieces or parts, and it is evident of 9 parts is 3 parts, that is .

43. We cannot take of 5 pieces, therefore wo must takeof, which is, and is 5 times as much as, therefore of is. This may be readily seen on the plate. In the sixth row, third square, find by the vertical division, then these being divided each into three parts by the hori zontal division, and of each being taken, you will have.

183

13

52. In the 4th row, the 3d square shows that of is, and must be twice as much, or .

56. In the fifth row, the 3d square shows that of is, but must be twice as much as, thereforeof, are.

say

731.

of 83 is 2, and 2 over, then of 22 is 2, hence of 8 is 24. is 233, and is 3 times as much, or

B. 4. It would take 1 man 4 times 93, or 374 days, and 7 men would do it in 4 of that time, that is in 519 days.

SECTION XV.

A. THIS section contains the divisions of whole numbers by fractions, and fractions by fractions.

1. Since there are in 2, it is evident that he could give them to 6 boys if he gave them apiece, but if he gave them apiece, he could give them to only one half as many, or 3 boys.

ი

5. If of a barrel would last them one month, it is evident that 4 barrels would last 20 months, but since it takes of a barrel, it will last them but one half as long, or 10 months.

7. 63 is 27. If of a bushel would last a week, 62 bushels would last 27 weeks; but since it takes 3, it will last only of the time, or 9 weeks.

13. If he had given of a bushel apiece, he might have given it to 17 persons, but since he gave 3 halves apiece, he could give it to only of thất number, that is to 5 persons, and he would have 1 bushel left, which would be 3 of enough for another. 23. 9 is, and 14 is 4. If it had been only of a dollar a barrel, he might have bought 60 barrels for 9 dollars, but since it was a bar

rel, he could buy only of that number, that is, 6 barrels.

25 and 26. Ans. 94.

31. 4 is, and 93 is 4.

in 48 times, and

Now is contained

is contained only

part as

many times, consequently only 2 or 24. B. 1. is; consequently 5 pounds bought for of a dollar.

can be

3. is, and is. If he had given only apiece, he could have given it to 9 persons, but since he gave he could give it to only 1 half as many, or 4 persons.

5. is, and is. If a pound had cost r of a dollar, 14 pounds could be bought for of a dollar, but since it costs, only as many can be bought; that is, 43 pounds.

9. is, and 15 is 85. If a bushel had cost ' of a dollar, 65 bushels might have been bought, but since it cost 18, only part as much could be bought; that is, 4 bushels.

12. is, and is, is contained in § 15 times, but is contained only as many times;

that is, 33 times,

Miscellaneous Example.

5.

of a penny is

of 4 farthings. Ans. 27

farthings.

6.

of 12 pence. Ans. 10 pence.

7. of 4 quarters is 2 quarters and of a quarter; of a quarter is of 4 nails, which is 13 nails. Ans. 2 quarters, 13 nails.

13. § of 24 hours is 15 hours. 14. of 24 hours is 14 hours of 60 minutes is 24 minutes. minutes.

and 3 of an hour; Ans. 14 hours, 24

28. There being 4 farthings in a penny, 1 farpart of a penny.

thing is

30. 3 farthings is of a penny.

31. I penny is

12 pence in a shilling.

of a shilling, because there are

49. 2 farthings is, or of a shilling. 5 farthings is of a shilling.

51. 1 penny is of 1 pound. 7 pence is t

100. At the end of 1 hour they would be 7 and miles apart. In 7 hours, 7 times 73, which is 543 miles.

121. This is the principle of fellowship; 3 shillings were paid; one paid, the other. 122. One paid, the other §.

123. 20 dollars were paid in the whole, one paid 2, another, and the third.

121, 3 and 4 and 5, are 12. The first put in ; the second; the third

129. 4 dollars for 2 months, is the same as 8 dollars, for 1 month; 3 dollars for 3 months, is the same as 9 dollars for 1 month; and 2 dollars for 4 months, is the same as 8 dollars for 1 month. The question is the same as if A had put in 8 dollars, B 9 dollars, and C 8 dollars. A must have. B 75, and C, of 100 dollars.

131. A's money was in 4 times as long as C's It is the same as if A had put in 8 dollars for the