86is, and is . If be taken from 8, there remains . B. This article contains only a practical application of the preceding. 3. This example and some of the following contain mixed numbers, but they are quite as easy as the others. The whole numbers may be added separately, and the fractions reduced to a common denominator, and then added as in other cases, and afterwards joined to the whole numbers. 6 and 2 are 8; 1 half and are ğ, making in the whole 8ğ bushels. C. It is difficult to find examples which will aptly illustrate this operation. It can be done more conveniently by the instructor. 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. 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. 1 and 2. ofis of the whole. 3 and 4. of is 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, and is 5 times as much as therefore ofis 78. 83 is 42, 79. 84 is, of 42 is 7. of 86. We may say is, consequently, of is 8, or of 84 is 2, and 24 over, then 2 is hence of 83 is 234. , and of 22 is 4, B. 4. It would take 1 man 4 times 9, or 37 7 men would do it in of that time, that is, in 5 days, and 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. 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. 68 is 27. If of a bushel would last a week, 64 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 given 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. 31. 4 is 21, and 9 is 4. Now 48 times, and 2 is contained only consequently only 2 or 24. 21 is contained in 48 part as many times, B. 1. is; consequently 5 pounds can be bought for of a dollar. 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 11. If a pound had cost of a dollar, 14 pounds could be bought for of a dollar; but since it costs, only as many can be bought; that is, 13 pounds. ut 12. 5. is, and is 1; is contained in 1 15 times is contained only as many times; that is, 3 times Miscellaneous Examples. of a penny is of 4 farthings. Ans. 23 farthings. 7. of 4 quarters is 2 quarters and of a quarter; f Da quarter is of 4 nails, which is quarters, 1 nails. 1g nails. Ans. 2 14. of 24 hours is 14 hours and 3 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. 31. 1 penny is pence in a shilling. 48. 1 farthing is of a shilling, because there are 12 of one shilling. 49. 2 farthings is, or of a shilling. 5 farthings is of a shilling. 51. 1 penny is 1 of 1 pound. 7 pence is of £1 59. 3s. 5d. is 41 pence, which is 75. 1 nail is of a yard. 5 nails is of £1. of a yard. 100. At the end of 1 hour they would be 7 and apart. In 7 hours, 7 times 7, which is 54 miles. miles 121. This is the principle of fellowship; 3 shillings were paid; one paid, the other. 122. One paid, the other §. 126. 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. 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 2, B2, and C, of 88 dollars. 139. The interest of 50 dollars for 1 year and 6 months is 4 dollars and 50 cents, and for 1 month it is 25 cents. The interest of 7 dollars for 19 months (a dollar is of a cent a month) is 66 cents. The whole amounts to 5 dollars and 41 cents. 140. The interest of 200 dollars for 11 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 dol lars and 36 cents. 143. The interest of 100 dollars for 24 years is 13 doi lars 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 21 years is 13 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 24 years is 7 times 13 cents, 11 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 95 146. They would both together do of the work in 1 day, and it would take them of a day to do the other Ans. 1 day. 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 in a week, but the woman alone only; consequently, the man alone would consume; and a bushel would last him 31 weeks. 152. A and B can build of it in 1 day; A, B, and C, can build of it in 1 day; the difference between and į is; therefore C can build of it in 1 day; and it would take him 134 days to build it alone. 164. Find how much they might eat in a day, in order to make it last 1 month, and then it will be easy to fine how much they may eat in a day, to make it last 11 months 167. The money is 7 parts of the whole, and the purs. one part; consequently the money is and the purse o 16. 170. He gave one part for the apple, 2 parts for the orange, and 4 parts for the melon. These make 7 parts. The apple 3 cents, the orange 6 cents, and the melon 12 cents. 175. If to a number half of itself be added, the sum is of that number; hence subtract 2 from 100, and the remainder is of the number of geese that he had. 180. This must be reduced to 6ths. 1 half is, and is, and the number itself is . If therefore to the whole number its half and its third be added, the sum will be 11; hence, 77 is of the number. 181.is; therefore if to a number and of itself be added, the whole number will be ; but when 18 more is added to, the first number is doubled; that is, the number is of the first number; therefore 18 is of the umber. ARITHMETIC. PART III. One Two 66 Three WRITTEN ARITHMETIC. Notation and Numeration. 1. Instead of writing the names of numbers, it is usua) to express them by particular characters, called figures. is written 1 Twenty-six is written 26 66 2 Twenty-seven 66 3 Twenty-eight 66 66 66 66 27 28 4 Twenty-nine 5 Thirty Thirty-one Forty 8 Fifty 10 Seventy Eighty Ninety |