10. What will a barrel of flour cost, if 19 barrels can be purchased for $ 1073 ? Ans. $ 5.65 36 11. If 15 pounds of raisins can be obtained for $ 34, what will 1 pound cost ? Ans. $ 0.2111 12. If 12 quarts of wine cost $ 3.751, what will a quart cost? 13. If $ 19 will buy 37514 acres of land, how much can be bought for $1? Ans. 1935 acres. REDUCTION OF COMPLEX FRACTIONS. are 241. A COMPLEX fraction is one having a fraction in its 3 numerator or denominator, or in both. Thus, and 21 导 § complex fractions. 242. To reduce complex to simple fractions. } Ans. 16 is the dividend, and the denominator Š x D the divisor (Art. 216), we divide the numerator, f, by the denominator, $, as in division of fractions (Art. 239). 7 2. Reduce to a simple fraction. 41 oder con OPERATION. 臺 oder Icon 15, Ans. Ans. 41 OPERATION R1 = 41, Ans. merator, 7, and the We reduce the nu7 1 X 1 등 denominator, lg, to improper fractions, and then proceed as in Ex. 1. Hence, to reduce complex to simple fractions, Consider the denominator as a divisor, and the numerator as a dividend, and proceed as in division of fractions (Art. 239). NOTE. — Another and often a ready method of reducing a complex fraction is to multiply both its terms by the least common multiple of their denomi nators. Or, multiply by the least common multiple of the denominators, 1 64 X 4 26 X 4 Ans. Ans. 12 Ans. 15 5} 12. Reduce to a mixed number. 을 t 13. Reduce to a simple fraction. 64 3 14. Reduce to a mixed number. 21 31 15. Reduce to a simple fraction. Ans. 18 9 114 16. Reduce to a simple fraction. Ans. 34. 17. If 7 were to be the denominator of the fraction whose 77 numerator is what would be its value ? 11% 18. If is the numerator of the fraction whose denominator is what is its value ? Ans. 614. 123 errorles 243. Complex fractions, after being reduced to simple ones, may be added, subtracted, multiplied, and divided, according to the respective rules for simple fractions. EXAMPLES 34 . 498 1. Add of of 28 3 Ans. 670 44 105 45 473 2. Add 4, 25, and together. Ans. 311387894. 9411 3143 343 3. What is the difference between and - ? 97 145 i Ans. 9147025. 4. What is the continued product of the following numbers: 27 874 % 817 and ? 37% 98% 24 128 5. Divide 4 of 74 by of 11 fr. Ans .. 6. Divide 4 of 91 by % of 87. Ans. 1844 MISCELLANEOUS EXAMPLES IN MULTIPLICATION AND DIVISION OF FRACTIONS. 1. At 24 bushels to an acre, how many bushels of wheat will be required to sow 74 acres ? Ans. 174 2. Bought 81 bushels of apples for $ 4.684; what did they cost per bushel ? Ans. $0.574. 3. Bought a bale of cloth for $ 963; I dispose of it for of the cost, and by so doing I lose $2 on a yard; required the number of yards in the bale. Ans. 18 yd. 4. If a dividend be 184 times f and a quotient 63 times f, what was the divisor ? 5. By what number must be multiplied, that the product shall just equal 1? Ans. $. 6. Bought a horse and chaise for $ 250, and paid for the harness of what I paid for the horse. The chaise cost 11 the value of the horse. What was the price of each? Ans. Horse, $ 1303$; chaise, $ 1193}; harness, $8375g. 7. S. Walker has engaged to work at yearly wages of $ 200 and a suit of clothes. At the end of 9 months, falling sick, and being unable to labor longer, he receives the suit of clothes and $ 144, as the amount justly due. What was the cost of the clothes ? Ans. $ 24. 8. What will be the result if of of 34 be multiplied by : of itself, and the product divided by ?? 9. Bought 13} acres of land at $ 254 per acre, and paid for it in wheat at $ 23 per bushel. How many bushels did it require ? Ans. 13777& bushels. 10. How long will it take a man to travel 553 miles, provided he travels 34 miles per hour, and 97 hours 11. If $ 1% per cord is paid E. Holmes for sawing into three pieces wood that is 4 feet long, how much more should he receive per cord for sawing into pieces of the same length wood that is 8 feet long ? Ans. $ 0.221. 12. A steamboat leaves New Orleans, January 1st, bound up the river to a place distant 23174 miles. Her forward motion is at the rate of 94 miles per hour for 164 hours each day, and she lies at anchor in the night for fear of running upon a snag. But having lost her anchor on the fifth day, she each succeeding night drifts backward, at the rate of 2 miles per hour. On what day of January will she reach her point of destination ? Ans. 15th day. per day? A PROPOSED NUMERATOR, OR DENOMINATOR. 244. To reduce one fraction to another of equal value, having a proposed numerator, or denominator. Ex. 1. Reduce 4 to an equivalent fraction having 4 for a 4 numerator. Ans. 5¥ The proposed numerator, 4, is such a part of the given numerator as 4 divided 5 * $ 4 by 5, or 6. Now, as the numerator pro Ans. posed is only as large as the given 4 53 numerator, in order that the value of the 5 X 7 two fractions be the same, the denominator of the proposed fraction should be only $ as large as the denominator of the given fraction. Taking $ of the given denominator, 7, we obtain OPERATION as the OPERATION 5}, which, written under the proposed numerator, gives 53 fraction required. 2. Reduce g to a fraction of equal value having 12 for a denominator. Since the proposed denominator, 12, 12 is 1,2 of the given denominator, 9, we X 8 finding of the given numerator, 8, for 9 numerator of the proposed fraction ; Ans. 12 12 The of 8 -- 103, which, written over X9 103 the proposed denominator, gives 12 as the fraction required. RULE. Take of both terms of the given fraction such a fractional part as the proposed numerator, or denominator, is of the given numera'or, or denominator, and the result will be the required fraction. 103 EXAMPLES. 3. Change it to a fraction whose numerator shall be 34. Ans. 24 Ans. 15. 4. Change 39 to a fraction whose numerator shall be 9. 9 Ans. 21 5. Reduce 4 to a fraction whose numerator shall be 5. 6. Reduce il to a fraction having 12 for its denominator. 7. Change to fifteenths. 8. Reduce { to halves. 331 9. Reduce 18 to thirty-fifths. Ans. 35 10. J. Holton owns 14 of a wood-lot, and his brother OT of the same lot; what fraction whose denominator shall be 12 will express the part each owns ? Ans. • A COMMON NUMERATOR. 245.° A COMMON numerator of two or more fractions is a common multiple of their numerators. 246.° To reduce fractions to a common numerator. Ex. 1. Change 1, 5, 4, and is to other fractions of the same value, having a common numerator. Ans. $ 45, 19, 16. |