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 Jthe 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 TV of the whole. 33. Since £ of a share signify 3 parts of a share, it is evident that \ of the three parts is 1 part, that is 39. $ signify 9 pieces or parts, and it is evident that \ of 9 parts is 3 parts, that is ^. 43. We cannot take \ of 5 pieces, therefore we must take \ of which is yV> and £ is 5 times as much as £, therefore \ of £ is f^. 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 horizontal division, and \ of each bein^ taken, you will have T5T. 52. In the 4th row, the 3d square shows that \ of \ is T*y, and | must be twice as much, or -fa. 56. In the fifth row, the 3d square shows that \ of § is XV but f must be twice as much as £9 therefore | of f, are 78. 8f isy,)oryu h 79. 8f is V, | of | is TV, consequently j of y is f£, or Ijf 86. We may say | of 8f is 2, and 2£ over, then 2$ is V, and £ of y is §f, hence % of 8£ is 2ff. 90. \ of 18| is 2|J, and ^ is 3 times as much, or m B. 4. It would take 1 man 4 times 9f, or 374 days, and 7 men would do it in \ of that time, that is, in 5Jf days. SECTION XV. A. This section contains the divisions of whole numbers by fractions, aaid fractions by fractions. 1. Since there are f in 2, it is evident that he could give them to 6 boys if he gave them £ apiece, but if he g$ve them § apiece, he could give them to only one half as many, or 3 boys. 5. If | of a barrel would last them one month, it is evident that 4 barrels would last 20 months, but since it takes f of a barrel, it will last them but one half as long, or 10 months. 7. 6f is y. If | of a bushel would last a week, 6f bushels would last 27 weeks; but since it takes £, it will last only £ of the time, or 9 weeks. 13. If he had given £ of a bushel apiece, he might Have givej^it to 17 persons, but. since he gave 3 halves apiece, he could give it to only £ of that number, that is to 5 persons, and he would have 1 bushel left, which would be § of enough for another. 23. 9f is \6, and 14 is y. If it had been only I of a dollar a barrel, he might have bought 66 barrels for 9f dollars, but since it was y a barrel, he could buy only yV of that number, that is, 6 barrels. 25 and 26. Ans. Of. 31. 4J is Vt *t is ¥• Now £ is contained in y 48 times, and %l is contained only ^T part as many times, consequently only 2/T or 2$. B. 1. £ is ^; consequently 5 pounds can be ^ bought for | of a dollar. 3. J is T9s, and £ is -fa. If he had given only T*7 apiece, he could have given it to 9 persons, but since he gave y% he could give it to only 1 half as many, or 4£ persons. 5. ^ is /r, and f is £f. If a pound had cost JT of a dollar, 14 pounds could be bought for |f of a dollar, but since it costs ¥3T, only £ as many can be bought; that is, 4| pounds. 9» | is ih and H is H* a bushel had cost TV of a dollar, 65 bushels might have been bought, but since it cost only Ty part as much could be bought; that is, 4T'F bushels. 12. % is T\, and £ is jf, T'T is contained in |f 15 times, but is contained only £ as many times; that is, 3f times. Miscellaneous Examples, 5. £ of ,a penny is f of 4 farthings. jins. 2} farthings. 6. | of 12 pence. Arts. 10 pence. 7. | of 4 quarters is 2 quarters and f of a quarter ; f of a quarter is f of 4 nails, which is If nails. Ans. 2 quarters, lg nails. 13. f of 24 hours is 15 hours. 14. | of 24 hours is 14 hours and § of an hour; £ of 60 minutes is 24 minutes. Ans. 14 hours, 24 minutes. 28. There being 4 farthings in a penny, 1 farthing is \ part of a penny. ^ 30. 3 farthings is f of a penny. 31. 1 penny is y1^ of a shilling, because there are 12 pence in a shilling. 34. 5 pence is T5¥ of a shilling. 41. 1 shilling is ^ of a pound. 43. 3 shillings is ^ of a pound. 48. 1 farthing is of one shilling. 49. 2 farthings is TaF, or ^ of a shillkig. 5 farthings is of a shilling. 51. 1 penny is of 1 pound. 7 pence is of l£. 59. 3s. 5d. is 41 pence, which is -fife of 1£. 75. 1 nail is T!g of a yard, 5 nails is T5F of a yard. 89. 1 oz. is TV of 1 lb. 15 oz. is }£ of 1 lb. 91. 1 lb. is of 1 quarter. 9 lbs. is & of 1 quarter. 100. £t the end of 1 hour they would be 7 and f miles apart. In 7 hours, 7 times 7J, which is 54f miles. 121. This is the principle of fellowship; 3 shillings were paid; one paid the other f. 122. One paid f, the other f. 123. 20 dollars were paid in the whole, one paid another ^ and the third 121. 3 and 4 and 5, are 12. The first put in T\; the second T4*; the third T5^. 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 ¥8J9 B A, and C ¥8T, of 100 doUars. 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 same time, and B 8 dollars for the same time. A must have Jfe, B ^, and C of 88 dollar^ The examples 127, 128, 129, 130, and 491, are double or compound fellowship. 139. The interest of 50 dollars for I year and 6 months is 4 dollars and 50 cents, and for 1 month it is 25 cents. The interest of 7 dollars for 18 months (a dollar is \ of a cent a month) is 63 cents. The whole amounts to 5 dollars and 38 cents. 140. The interest of 200 dollars for \\ years is 16 dollars. The interest of 67 dollars is 67 cents for every 2 months, for 16 months it will be 8 times 67 cents, which are 5 dollars and 36 cents. The whole interest is 21 dollars and 36 cents. 143. The interest of 100 dollars for 2{ years, is 13 dollars and 50 cents. The interest of 100 dollars for 60 days would be 1 dollar, the interest for 20 days will be } of a dollar, or 33£ cents. The interest of 1 dollar for 2| years is 13j cents, for 10 dollars the interest would be 1 dollar and 35 cents, and for 30 dollars, 4 dollars and 5 cents. The interest of 7 dollars for 2} years is 7 times 13£ cents or 94^ cents. The interest of 37 dollars for 60 days would be 37 cents, and for 20 days £ of 37 cents, or 12£ cents. The whole interest is 18 dollars and 95£ cents. 146. They would both together do f of the work in 1 day, and it would take them | of a day to do the other Ans. 1£ day. 147. f would be done in 1 day, and it would take £ of a day to do the other Ans. 1£ days. 149. They both together consume \ of a bushel in a week, but the woman alone consumes only \ of a bushel in a week. That is, they both together consume f ^ in a week, but the woman alone only -jSg-, consequently, the man alone would consume fV» and a bushel would last him 3£ weeks. |