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. 393, 9411 3143 34 1. Add } of şof 28 to 3 Ans. 6-75. 44 105 45 47 & 2. Add , 25, and together. Ans. 39347891. 493 343 3. What is the difference between and ? 97 145ů Ans. 84.76.51 3180120 4. What is the continued product of the following numbers : 27 87} } and 3798}" 21" 128 5. Divide of 77 by of 11t: Ans. 1 6. Divide fof 91 by of 87. Ans. 4614 81% ? MISCELLANEOUS EXAMPLES IN MULTIPLICATION AND DIVISION OF FRACTIONS. 1. At 2 bushels to an acre, how many bushels of wheat will be required to sow 74 acres ? Ans. 174. 2. Bought 8} bushels of apples for $ 4.684; what did they cost per bushel ? Ans. $0.577. 3. Bought a bale of cloth for $ 963; I dispose of it for ģ of the cost, and by so doing I lose $2 on a yard; required the number of yards in the bale. Ans. 1813 syd. 4. If a dividend be 18} times of and a quotient 64 times , what was the divisor ? 5. By what number must be multiplied, that the product shall just equal 1 ? 6. Bought a horse and chaise for $ 250, and paid for the harness Is 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, $ 11933; harness, $83753 Ans. 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 3} be multiplied by } of itself, and the product divided by 1? 9. Bought 137 acres of land at $ 254 per acre, and paid for it in wheat at $ 2; per bushel. How many bushels did it require ? Ans. 137 178 bushels. 10. How long will it take a man to travel 553 miles, provided he travels 34 miles per hour, and 97 hours per day? 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 23177 miles. Her forward motion is at the rate of 94 miles per hour for 16 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. A PROPOSED NUMERATOR, OR DENOMINATOR. 244. To reduce one fraction to another of equal value, having a proposed numerator, or denominator. Ex. 1. Reduce to an equivalent fraction having 4 for a 4 numerator. Ans. 53 The proposed numerator, 4, is such a 4 x 5 part of the given numerator as 4 divided 4 by 5, or 4. Now, as the numerator pro Ans. posed is only as large as the given 4 54 numerator, in order that the value of the 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. erli el 5 X 7 as the OPERATION. find 5, which, written under the proposed numerator, gives 58 fraction required. 2. Reduce s to a fraction of equal value having 12 for a denominator. Since the proposed denominator, 12, 12 is 1 of the given denominator, 9, we X 8 la of the given numerator, 8, for 9 102 numerator of the proposed fraction ; Ans. 12 12 in of 8 = 103, which, written over X9 the proposed denominator, gives 104 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 numerator, or denominator, and the result will be the required fraction. EXAMPLES. 3. Change it to a fraction whose numerator shall be 34. Ans. 36 4. Change 3ę to a fraction whose numerator shall be 9. 9 Ans. 27 5. Reduce 4 to a fraction whose numerator shall be 5. 6. Reduce 1: to a fraction having 12 for its denominator. 7. Change to fifteenths. 8. Reduce { to halves. 331 9. Reduce to to thirty-fifths. Ans. 35 10. J. Holton owns 19 of a wood-lot, and his brother 2001 of the same lot; what fraction whose denominator shall be 12 will express the part each owns ? Ans. 15. Ans. Ani 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, , and % to other fractions of the same value, having a common numerator. Ans. 39, 45, 46 = common nu 36 36 40 OPERATION. 36, least common multiple of the numerators, 36 of 4 48, new denominator. * 16(merator. Ans. 40, new denominator. 18 We find the least common multiple of all the numerators, which is 36, for the common numerator; and to obtain the several new denoninators we take such a part of the given denominators, respectively, as the common numerator, 36, is of each given numerator. Thus, both terms of each fraction being proportionably increased, its value is not changed. RULE. - Find the least common multiple of the given numerators for a common numerator. Take, for the new denominator of each fraction, respectively, such a part of its given denominator as the common numerator is of its given numerator. Note. — Compound fractions, or whole and mixed numbers, must be reduced to simple fractions, and all to their lowest terms, before finding the common numerator. EXAMPLES. 2. Reduce , 4, 4, and $ to other fractions of equal value having a common numerator. Ans. 34, 35, 36, 34. 3. Change § 27, and 14 to fractions having a common numerator. 4. A can travel round a certain island, which is 50 miles in circumference, in 445 days, B in 63 days, and C in 6 days. If they all set out from the same point, and travel round the island the same way, in how many days will they all meet at the point from which they started, and how many times will each have gone round the island ? Ans. They will meet in 320 days; A will have gone round the island 75 times ; B, 50 times ; and C, 48 times. GREATEST COMMON DIVISOR OF FRACTIONS. 247. The greatest common divisor of two or more fractions is the greatest number that will divide each of them, and give a whole number for the quotient. 248. To find the greatest common divisor of two or more fractions. OPERATION. Greatest com mon divisor Ex. 1. What is the greatest common divisor of 15, 25, and 54 Ans. 5 15, 23, 54 = 15, 29,16 15, P, 46 = 1'}, 403, 43 Greatest common divisor of the numerators 41 Least common denominator of the fractions 45 required. Having reduced the fractions to equivalent fractions having the least common denominator, we find the greatest common divisor of the numerators 12, 100, and 240 to be 4. Now, since the 12, 100, and 240 represent forty-fifths, their greatest common divisor is not 4, a whole number, but 4 forty-fifths; therefore we write the 4 over the least common denominator, 45, and have to as the answer. RULE. - Reduce the fractions, if necessary, to their least common denominator. The greatest common divisor of the numerators, written over the least common denominator, will give the greatest common divisor required. EXAMPLES. 2. What is the greatest common divisor of 6, 4, g, and 3 ? Ans. ts. 3. What is the greatest common divisor of 123, 94, and 87? 4. What is the greatest common divisor of 1, s, š, and & ? Ans. z. 5. What is the greatest common divisor of 34,5 7o, and 235? 6. A farmer has 33 bushels of corn, 671 bushels of rye, 703 bushels of wheat. He wishes to put this grain, without mixing, into the largest bags, each of which shall contain the same quantity. Required the number of bags and the quantity each will contain. Ans. The capacity of each bag, 33 bushels; and the number of bags, 51. 7. I have three fields; the first contains 7311 acres, the second 884 acres, the third 13911 acres. Required the largest-sized house-lots of the same extent into which the three fields can be divided, and also the number of lots. Ans. Size of each lot, 74 acres ; number of lots, 41. LEAST COMMON MULTIPLE OF FRACTIONS. 249. The least common multiple of two or more fractions is the least number that can be divided by each of them, and give a whole number for the quotient. |