What part of a Ans. Ti Ans. 11. John Jones owns of a share in a railroad valued at $117; this he bequeathes to his five children. 14. James Page's estate is valued at $10,000, and he has given of it to the Seamen's Society; of the remainder he gave to his good minister; and the remainder he divided equally among his 4 sons and 3 daughters. What sum will each of his children receive? Ans. $6804. ART. 160. To divide a whole number by a fraction. Ex. 1. How many times will 13 contain . Ans. 30. For convenience, we invert the terms of the divisor, and then multiply the whole number by 30 the original denominator, and divide the product by the numer ator. The reason of this operation is evident, since 13 will contain as many times as there are sevenths in 13, equal 91 sevenths. Now if 13 contain 1 seventh 91 times, it will contain as many times as 91 -will contain 3, equal to 30. RULE. Multiply the whole number by the denominator of the fraction, and divide the product by the numerator. QUESTIONS. Art. 160. What is the rule for dividing a whole number by a fraction ? Give the reason for the rule. ART. 161. To divide a mixed number by a whole number. Ex. 1. Divide 178 by 6. 6)17 OPERATION. 43 43 X6 48 2+13=213. Ans. 24. Having divided the whole number as in simple division, we have a remainder of 5, which we reduce to an improper fraction, and divide it by the divisor as in Art. 159. Annexing this fraction to the quotient 2, we obtain 243 for the answer. RULE.-Divide as in whole numbers, as far as the division can be carried, and, if the remainder is a mixed number, reduce it to an improper fraction, and then divide it by the divisor; but if the remainder is a simple fraction only, merely divide it by the divisor. (Art. 159.) 12. Divide $1077 among 4 boys and 3 girls, and give the girls twice as much as the boys. Ans. Boy's share $104; Girl's share $212. 13. If $14 will purchase of a ton of copperas, what quantity will $1 purchase? Ans. lcwt. Oqr. 24lb. QUESTIONS. Art. 161. What is the rule for dividing a mixed number by a whole number? ART. 162. To divide a whole number by a mixed number. Ans. 519. Ex. 1. Divide 25 by 4g. We first reduce the divisor and dividend to fifths, and then divide as in whole numbers. The reason why the answer is in whole numbers, and not in fifths, is because the divisor and dividend were both multiplied by the same number, 5, and therefore their relation to each other is the same as before, and the quotient will not be altered. RULE. Reduce the divisor and dividend to the same parts as are denoted by the denominator of the fraction in the divisor, and then divide as in whole numbers. EXAMPLES FOR PRACTICE. 2. Divide 36 by 97. 3. Divide 97 by 1311. 4. Divide 113 by 214. 5. Divide 342 by 14 Ans. 23r 6. There is a board 19 feet in length, which I wish to saw into pieces 2 feet long; what will be the number of pieces, and how many feet will remain ? Ans. 7 pieces, 2 feet. ART. 163. To divide a fraction by a fraction. The reason of this process will be seen, when we consider that the divisor is an expression denoting that 4 is to be divided by 9. Now, regarding 4 as a whole number, we divide the fraction 7 7 by it, by multiplying the denominator; thus, 8x432 But the divisor 4 is 9 times too great, since it was to be divided by 9, as seen in the original fraction; therefore the quotient, 32, is 9 times too small, and must be multiplied by 9; thus, 32 13. By this operation, we have multiplied the denominator of the dividend by the numerator of the divisor, and the numerator of the dividend by the denominator of the divisor. Hence the = 7 X 9 63 = QUESTIONS. Art. 162. What is the rule for dividing a whole by a mixed number? How does it appear that this process does not alter the quotient? Art. 163. How do you divide a fraction by a fraction? Give the reason why this process divides the fraction of the dividend. RULE. Invert the divisor, and cancel all the factors common to the numerators and denominators, and then proceed as in multiplication of fractions. NOTE. 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 the numerator of the divisor. 10. Divide of by of. 11. Divide off of 76 by of of. 12. Divide of of by of § of f2. 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.-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? 151300 COMPLEX FRACTIONS. ART. 165. To reduce complex to simple fractions. Ex. 1. Reduce 13984 to a simple fraction. OPERATION. 27 Ans. 27. 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. 41 (Art. 163.) Ans. 17. We reduce the numerator, = 16 = 17 8, and the denominator, 41, to improper fractions, and then proceed as in Ex. 1. OPERATION. = H = 14 Ans. 14. We here reduce the denominator, of, to a simple fraction, and then proceed as before. From the preceding illustrations we deduce the following RULE. - Reduce whole and mixed numbers to improper fractions, and compound fractions to simple ones, and then divide the numerator of the complex fraction by the denominator, according to the rule for the division of fractions. QUESTIONS. Art. 165. What is the rule for reducing complex to simple fractions? How does this process differ from division of fractions ? |