RULE. ·Invert the divisor, and then proceed as in multiplication of fractions. NOTE 1.- Factors common to numerator and denominator should be cancelled. NOTE 2.-When the divisor and dividend have a common denominator, their denominators cancel each other, and the division may be performed by simply dividing the numerator of the dividend by that of the divisor. ART. 164. To divide a mixed number by a mixed number, it is only necessary to reduce them to improper fractions, and proceed as in the foregoing rule. (Art. 163.) QUESTIONS. Ans. 138. Ans. 15 Ans. 2. 55 Ans. 2112 Ans. 833. Ans. 818. Ans. 11. What is the rule for dividing one fraction by another? How may fractions be divided when they have a common denominator? Does this process differ in principle from the other? - Art. 164. How do you divide a mixed number by a mixed number? COMPLEX FRACTIONS. ART. 165. To reduce complex to simple fractions. Ex. 1. Reduce OPERATION. = 2o7 Since the numerator of a fraction is the dividend, and the denominator the divisor (Art. 132), it will be seen by this operation that we simply divide the numerator, §, by the denominator, &, as in division of fractions. (Art. 163.) OPERATION. Ans. 17. We reduce the numerator, † × 3 = ♣ = 17 8, and the denominator, 4, to improper fractions, and then proceed as in Ex. 1. RULE. - Reduce the terms of the complex fraction, if necessary, to the form of a simple fraction. Then divide the numerator of the complex fraction by its denominator. QUESTIONS. Art. 165. What is the rule for reducing complex to simple fractions? How does this process differ from division of fractions? 91 what is its value when reduced to a simple fraction? 127' 11. If 7 is the denominator of the following fraction Ans. 71 12. If is the numerator of the following fraction, what is its value when reduced to a simple fraction? Ans. 2. ART. 166. Complex fractions, after being reduced to simple ones, may be added, subtracted, multiplied, and divided, according to the respective rules for simple fractions. QUESTION. - Art. 166. How do you add, subtract, multiply, and divide, complex fractions? GREATEST COMMON DIVISOR OF FRACTIONS. ART. 167. To find the greatest common divisor of two or more fractions. Ex. 1. What is the greatest common divisor of 3, 3, and 12? Greatest common divisor of the numerators = 2 Least common denominator of the fractions = 45 greatest common divisor required. Having reduced the fractions to equivalent fractions with the least common denominator (Art. 141), we find the greatest common divisor of the numerators 20, 30, and 36, to be 2. (Art. 124.) Now, since the 20, 30, and 36, are forty-fifths, their greatest common divisor is not 2, a whole number, but so many forty-fifths. Therefore we write the 2 over the common denominator 45, and have 45 as the answer. 2 RULE. Reduce the fractions, if necessary, to the least common denominator. Then find the greatest common divisor of the numerators, which, written over the least common denominator, will give the greatest common divisor required. EXAMPLES FOR PRACTICE. 2. What is the greatest common divisor of 2, §, 3. What is the greatest common divisor of 13, 4. What is the greatest common divisor of 15, 4, and 11? Ans. 213 24, 4, and 5? Ans. 5. There is a three-sided lot, of which one side is 1663ft., another side 1561ft., and the third side 2084ft. What must be the length of the longest rails that can be used in fencing it, allowing the end of each rail to lap by the other ift., and all the panels to be of equal length? Ans. 1011ft. LEAST COMMON MULTIPLE OF FRACTIONS. ART. 168. To find the least common multiple of fractions. Ex. 1. What is the least common multiple of, 11, and 51? Ans. 21 = 101. QUESTIONS. Art. 167. What is the rule for finding the greatest common divisor of fractions? Why, in the operation, was the divisor 2 written over the denominator 45? Least common multiple of the numerators Having reduced the fractions to their lowest terms, we find the least common multiple of the numerators, 1, 3, and 21, to be 21. (Art. 128.) Now, since the 1, 3, and 21, are, from the nature of a fraction, dividends of which their respective denominators, 6, 2, and 4, are the divisors (Art. 132), the least common multiple of the fractions is not 21, a whole number, but so many fractional parts of the greatest common divisor of the denominators. This common divisor we find to be 2, which, written as the denominator of the 21, gives 2110 as the least number that can be exactly divided by the given fractions. = RULE. Reduce the fractions, if necessary, to their lowest terms. Then find the least common multiple of the numerators, which, written over the greatest common divisor of the denominators, will give the least common multiple required. EXAMPLES FOR PRACTICE. 2. What is the least common multiple of 18, §, and 15? Ans. 44. 3. What is the least number that can be exactly divided by 5, 2, 5, 6, and? Ans. 95. 4. What is the smallest sum of money for which I could purchase a number of bushels of oats, at $5 a bushel; a number of bushels of corn, at $§ a bushel; a number of bushels of rye, at $1 a bushel; or a number of bushels of wheat, at $21 a bushel; and how many bushels of each could I purchase for that sum? Ans. $221; 72 bushels of oats; 36 bushels of corn; 15 bushels of rye; 10 bushels of wheat. 5. There is an island 10 miles in circuit, around which A can travel in of a day, and B in of a day. Supposing them each to start together from the same point to travel around it in the same direction, how long must they travel before coming together again at the place of departure, and how many miles will each have travelled? Ans. 5 days; A 70 miles; B 60 miles. QUESTIONS. Art. 168. What is the rule for finding the least common multiple of fractions? Why is not the least common multiple of the numerators the least common multiple of the fractions? |