5. Divide 5 by 5. Ans. 25. 6. Divide 12 by : Ans. 16. 7. Divide 16 by 1. Ans. 32. 8. Divide 100 by 13. Ans. 11114. 9. I have 50 square yards of cloth; how many yards, iš of a yard wide, will be sufficient to line it? Ans. 83ş yards. 10. A. Poor can walk 371 miles in 60 minutes ; Benjamin can walk 11 as fast as Poor. How long will it take Benjamin to walk the same distance ? _Ans. 734 minutes. OPERATION. 161. To divide a mixed number by a whole number. Ex. 1. Divide 173 by 6. Ans. 213. Having divided the whole 6)173 number as in simple division, we 2, 5 = t; have a remainder of 58, which we reduce to an improper frac2 +48=24%. tion, and divide it by the divisor, as in Art. 159. Annexing this result to the quotient 2, we obtain 24 for the answer. That is, we Divide the integral part of the mixed number; and the remainder, re• duced if necessary to a simple fraction, divide as is in Art. 159. 43 43 48 EXAMPLES FOR PRACTICE. 2. Divide 17by 7. Ans. 216 Ans. 26. 3. Divide 183 by 8. 4. Divide 27 11 by 9. Ans. 3 us. 5. Divide 3115 by 11. Ans. 2016 Ans. 617. 6. Divide 784 by 12. 7. Divide 1891} by 4. Ans. 4738 8. Divide 107 by 3. Ans. 3546 9. Divide $ 145 among 7 men. Ans. $2425 10. Divide $ 106% among 8 boys. Ans. $ 1393. 11. What is the value of 4 of a dollar ? Ans. $0.341. 12. Divide $ 10711 among 4 boys and 3 girls, and give each of the girls twice as much as each of the boys? Ans. Boy's share, $ 104; Girl's share, 2145. 13. If $ 14 will purchase 37 of a ton of copperas, what quan Ans. 1 cwt. tity will $ 1 purchase ? 161. How do you divide a mixed number by a whole number? 162. To divide a whole number by a mixed number. Ex. 1. Divide 25 by 4%. Ans. 519. OPERATION. 43/25 We first reduce the divisor and dividend to fifths, 5 5 and then divide as in whole numbers. The divisor and dividend were both multiplied 23) 125 (514 by the same number, 5; therefore their relation to 115 each other is the same as before, and the quotient 10 is not changed. (Art. 115, Note.) Hence, Reduce the divisor and dividend to the same fractional 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 93. Ans. 375. 3. Divide 97 by 1312. Ans. 6164. 4. Divide 113 by 217. Ans. 51 5. Divide 342 by 1411 Ans. 231 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 ? Ans. 7+4 pieces. 163. To divide a fraction by a fraction. Ex. 1. Divide } by $. Ans. 131 Since 1 is contained in }, } times, $ is contained in 9 times ] times, or for times; and is contained in }, of ee times, which is ft, or ift, times. That is, we have multiplied the denominator of the dividend by the number denoting the numerator of the divisor, and the numerator of the dividend by the number denoting the denominator of the divisor, hence, for convenience, as in the second operation, we can simply invert the terms of the divisor and proceed as in Art. 156. 162. How do you divide a whole by a mixed number? How does it appear that this process does not alter the quotient ? — 163. How do vou divide a fraction by a fraction ? Give the reason why this process divides the fraction of the dividend. RULE. — Invert the divisor, and then proceed as in multiplication of fractions. Note 1.– Factors common to numerator and denominator should be canceled. Note 2. - When the divisor and dividend have a common denominator their denominations cancel each other, and the division may be performed by simply dividing the numerator of the dividend by that of the divisor. EXAMPLES FOR PRACTICE. 2. Divide y by 4. Ans. 138. 3. Divide by : Ans. 31 4. Divide 13 by tt. Ans. 3 5. Divide by s. Ans. 25. 6. Divide by +. Ans. 62. 7. Divide by it. Ans. 43. Ans. 6. 8. Divide 1 by 26 9. Divide 1: by Zoo Ans. 2. 10. Divide of } by off. Ans. 185 11. Divide of a of 76 by of 1 of $. Ans. it Ans. 34 12. Divide if of $ off by of ¢ of 1g. 164. To divide a mixed number by a mixed number. Reduce them to improper fractions, and proceed as in Art. 163. Ex. 1. Divide 7 by 3%. Ans. 26 OPERATION. 7=; 3= 24. X = 4= 236. EXAMPLES FOR PRACTICE. Ans. 136. Ans. 15. Ans. 2152: Ans. 2112 Ans. 83Ans. 8131 Ans. 113 163 The rule for dividing one fraction by another? How may fractione be divided when they have a common denominator ? Does this process differ in principle from the other ? - · 164. How do you divide a mixed number by a mixed number? OPERATION. Ans. 17. Since the numerator of a fraction is the - $ xd 2 옳 dividend, and the denominator the divisor (Art. 132), we simply divide the numerator, š, by the denominator, g, as in division of fractions. (Art. 163.) 8 Ex. 2. Reduce to a simple fraction. 44 We reduce the numerator, 8 우 { x = 1 = 1% 8, and the denominator, 41, to 4} improper fractions, and then proceed as in Ex. 1. 导 Ex. 3. Reduce to a simple fraction. Ans. 14. * of OPERATION. OPERATION. We reduce the denomi유 = $ x S 18 14 nator, t of $, to a simple 1 of fraction, and then proceed as before. 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. Note. — Another method is to multiply both terms of the complex fraction by a common multiple of their denominators. EXAMPLES FOR PRACTICE. 12 4. Reduce to a whole number. 导 Ans. 28. Ans. gg. 5. Reduce to a simple fraction. 14 43 6. Reduce to a simple fraction, 9 Ans. It 165. The rule for reducing complex to simple fractions ? How does this process differ from division of fractions ? Ans. 1: 7. Reduce to a simple fraction. 13 8. Change to a simple fraction. Ans. i $ 123 125 84 9. Change to a mixed number. Ans. 215. 2 를 92 1C. Reduce to a simple fraction. Ans. 25 97 what 11. If 7 is the denominator of the following fraction, is its value when reduced to a simple fraction ? Ans. 721 12. If I is the numerator of the following fraction, what is its value when reduced to a simple fraction ? Ans. 47 166. Complex fractions, after being reduced to simple ones, may be added, subtracted, multiplied, and divided, like them. 166. How do you add, subtract, multiply, and divide complex fractions ? |