Subtracting the less numerator 96 from the greater 105, and placing the remainder 9 over the common denominator 180, we find=180 2. To subtract 15 from 37. 1 20 Reducing these mixed numbers to improper fractions, Reducing these improper fractions to a common denominawe have 235, and 189=567; tor, then 37-15-567—235=332=2215, and 189=567. Otherwise. Taking the fractions and, contained in the two mixed numbers, and reducing them to a common denominator, we shall find 153=1518, and 374 371. Then subtracting 19 from 1, and 15 from 37 we have 371-1519=22% as before. 3. To subtract 26 from 1393. Reducing the fractions contained in these two mixed numbers, to a common denominator, we shall find 1393=1398 and 263 261 1021 As cannot be subtracted from, we add a unit, that is, 12 1 to, making, and say from leaves 1. We then add 1 to the 6, and say 7 from 9 leaves 2, &c. (§ 34.) 4. To subtract 18431 from 2745. 2745(1) 901 Here we annex, mentally, 15, equal to a unit, to the upper number, and say 13 from 1 leaves. Then adding 1 to the 3, we say 4 from 5 leaves 1, &c. (§ 34). 5742. Ans. 225. 625§. Ans. 1048. 720. Ans. 433. 3735. Ans. 560. Ans. 891%. 6. Find the difference between 349 and 7. Find the difference between 730 and 8. Find the difference between 287 and 9. Find the difference between 934 and 10. Find the difference between 870 and 780. 11. If flour were bought at 4 dollars per barrel, and sold at 5 dollars per barrel, what would be the gain per barrel? Ans. 15 dollars. 12. From a barrel of wine which contained 31 gallons, 13 gallons were drawn. What quantity remained in the barrel? Ans. 173 gallons. 13. A person who had to make a journey of 500 miles, has traveled 275 miles on his way. How far has he yet to go? Ans. 2243 miles. 14. A farmer having 1000 acres of land, sells to one of his neighbors 4791% acres. How many has he remaining? Ans. 520 acres. 15. A manufacturer who had on hand 700§ yards of cloth has sold 534 yards of it. What quantity remains on hand? Ans. 166 yards. 16. A merchant bought a quantity of bacon for 15 dollars, and a quantity of pork for 23 dollars. He sold the whole for 48 dollars; what did he gain by the sale? Ans. 9 dollars. 17. Bought at one time 147 bushels of coal, and at another time 320 bushels. Having consumed 200 bushels, I desire to know what quantity is still on hand? Ans. 2671 bushels. 18. A bought of B 75 yards of cloth; of which he sold to C 18 yards, and to D 20 yards. How many yards has he left? Ans. 35 yards. 19. A gentleman having 3000 dollars to divide among his three sons, gives 753 dollars to the first, 1284 dollars to the second, and the remainder to the third. What sum does the third receive? Ans. 962 dollars. cotton each containyards. How many Ans. 50 yards. 20. A merchant bought two pieces of ing 34 yards, of which he has sold 18 yards has he left? 21. If I should collect from A 200 dollars, from B and C each 175 dollars, and then pay to D 567 dollars, and to E the remainder of the sum collected, how many dollars would E receive? Ans. 494 dollars. 22. Bought of A 40 cords of wood for 814 dollars, of which I sold to B 20 cords for 45 dollars. If I sell the rest of the wood to C for 432 dollars, what sum will I gain? Ans. 74 dollars. 23. If a quantity of cloth be purchased for 3214 dollars, a quantity of silk for 137 dollars, and a quantity of linen for 934 dollars, what will be the gain or loss if the whole be sold for 600 dollars? Ans. Gain 47 dollars. 24. A person having 100 dollars on hand, laid out 17 dollars for provisions, and paid taxes amounting to 213 dollars; what sum had he remaining? Ans. 60 dollars. 25. From the sum of 1500 dollars which I deposited in bank, having drawn, at different times, 200 dollars, 1371⁄2 dollars, 3134 dollars, and 79 dollars; what sum have I yet in bank? Ans. 769 dollars. 26. Bought a quantity of iron for 95 dollars, and of coal for 81 dollars. The iron was sold for 115 dollars, and the coal for 100 dollars; what profit was made on both commodities? Ans. 38 dollars. 27. Bought 350 acres of land for 43274 dollars. Having sold 137 acres for 1387 dollars, I desire to know how many acres remain, and for what sum the remainder should be sold to make a profit of 500 dollars on the whole? Ans. 212 acres: 3440 dollars. 28. A merchant bought one piece of cloth containing 53 yards, another containing 393 yards, and another containing 40 yards. Having sold 13 yards from the first piece, 24 from the second, and 19 from the third, the merchant wishes to know the whole number of yards he has remaining. Ans. 76 yards. 29. A speculator bought 1000 acres of land for 15873 dollars, and 500 acres for 7377 dollars. Having sold 945 acres for 2000 dollars, he wishes to know what quantity of land he has remaining, and for what sum he could afford to sell the remainder, so as to lose nothing on the whole. Ans. 554 acres; 3251 dollars. MULTIPLICATION OF FRACTIONS. $ 116. Multiplying by a fraction consists in finding such a part of the multiplicand as is expressed by the multiplier. For example, 12 multiplied by & is 5 times of 12, which is of 12, equal to 10. How many is of 6, or 6 multiplied by1⁄2? 6 multiplied by? of 12, or 12×? of 35, or 35X? of 40, or 40X? How many is 12 multiplied by 3; that is, 3 times 12, together with of 12? 8 multiplied by 44? 10×62? 12×55? How many is 20 multiplied by 44: that is, 4 times 20, together with of 20? 24 multiplied by 3? 8×93? 12×105 ? Compound Fractions. § 117. A fraction multiplied by another fraction, or divided by an integer, may be expressed as a compound fraction, that is, a fraction of a fraction." Thus X is of §, (§ 116 ;) and §÷÷4 is of §, (§ 51). The expressions of, and of are called compound fractions, in contradistinction to simple fractions, which consist of a single numerator and denominator. Hence, § 118. Multiplying two or more fractions together is equivalent to reducing a compound to a simple fraction. § 119. To multiply a fraction by a fraction. 1. Multiply the numerators together for a numerator, and the denominators together for a denominator. 2. An integer and a fraction are multiplied together, by multiplying the numerator, or dividing the denominator, of the fraction, by the integer. 3. A mixed number may be used in multiplication under the form of an improper fraction; or the integer and fraction in it may be taken separately in multiplying,-observing to add together the separate products. EXAMPLES. 1. To multiply by ; that is, to find 4 of 4. 5X3 2. To multiply by 6; that is to find 6 times. Multiplying the numerator by the integer, 5 we have 4×6=24-÷6=4=14; as before. 3. To multiply 5 by 21; that is, to find twice 5, together with of 5. Reducing the two mixed numbers to improper fractions, we have 5, and 2=5. Then 5×2×==14. Otherwise. Taking the integer and fraction in each mixed number, separately, we shall find 5×2=113. and of 5=25. Then 113+25=141. (§ 23). In multiplying, we say twice is, equal to 1. Setting down, and carrying 1,—twice 5 is 10, and 1 is 11. Next, of 5 is 2, with 1 over; this 1, equal to }, added to the makes; then of § is §§. Recurring to the first example, we remark, first, that of is. For if any quantity were divided into 7 sevenths, and each one of these sevenths were divided into 4 equal parts, these last parts would be 28ths of the quantity, since 7 times 4-28. That is, 1 fourth of 1 seventh is. 1 fourth of 5 sevenths is, therefore, and 3 fourths of 5 sevenths is 3 times as much as 1 fourth of it, that is, 3 times, which is 5. This demonstration discovers two principles, as involved in the Rule for multiplying a fraction by a fraction. First. Multiplying the denominator of a fraction, finds such a part of the fraction as is expressed by the reciprocal of the multiplier; and, then, secondly, multiplying the numerator finds as many of such parts as are expressed by the multiplier. Thus, multiplying the denominator of then, multiplying the numerator of by 4, finds of ; of 4.00 |