WRITTEN EXERCISES. 1. Divide 10 by the fraction of SOLUTION.-10 divided by one equals 10, hence OPERATION 10 divided by t equals 6 times 10, and 10 divided 10:45 by 5 sixths equals } of 6 times 10, which is f 10x =12 Ans. times 10, which equals 12. Hence the Rule I.-Multiply the dividend by the denominator of the fraction, and divide the product by the numerator. Rule II.-Invert the divisor and proceed as in multiplrcation of fractions. Divide 2. 5 by Ans. 84. 8. 15 by 84. Ans. 14. 3. 9 by 4. Ans. 154. 9. 256 by 337. Ans. 80. 4. 12 by 4. Ans. 14. 10. 552 by 23. Ans. 232. 5. 27 by 21 Ans. 12. 11. 558 by 89. Ans. 63. 6. 30 by . Ans. 48. 12. 638 by 19. Ans. 33. 7. 144 by 24. Ans. 54. 13. 729 by 224. Ans. 324. CASE III. 182. To divide a fraction by a fraction. MENTAL EXERCISES. 1. How many times is contained in ? SOLUTION.-One is contained in $, & times, and if 1 ls contained in to Limes, & is contained in , 4 times 8 times, which is 20, or fe times; and to contained in $ 1 of , or ye times. Therefore, etc. How many times is 2. contained in /? 6. & contained in 18? 3. f contained in ? 7. $ contained in 2? 4. $ contained in ? 8. I contained in 84? 5. / contained in tf? 9. $ contained in 98? 10 In dividing Å by & we multiply by what, and divide the pro duct by what? 11. How then may we divide a fraction by a fraction without the analysis ? WRITTEN EXERCISES. OPERATIOR. 1. Divide by SOLUTION.— divided by one equals 1, hence divided by equals 6 times ], and divided by Ď sixths equals 1 of 6 times ], which is f times 1, which equals 1, or it. Therefore we have the following Rule I.-Multiply the dividend by the denominator of the divisor, and divide by the numerator. Rule II.-Invert the divisor and proceed as in multipli cation of fractions. NOTE.—Reduce mixed numbers to simple fractions. When the divisor is 8 compound fraction, invert each term and multiply, cancelling when pos sible, Divide 2. by 4. Ans. 14. 11. 7% by 12%. Ans. 4464 3. Yo by : Ans. 11. 12. 1518 by 84. Ane. 148 4. 34 by Ans. 14. 13. 177 by 10%. Ans. 1881 5. H by 1H. Ans. % 14. 21% by 12 Ans. 14. 6. by 2035 Ans. 2. 15. 254 by 154r. Ans. 1921 7.4 by 44 Ans. 14. 16. 331 by 123. Ans. 2881. 8. by to Ans. 14. 17. by z of 19 Ans. 115 9. He hy 4. Ans. 14. 18. 4 of 7% by H. Ans. 648. 10. Hai by 175 Ans. 4. 19. 16+ by 4x11. Ans. 14. . MENTAL EXERCISES. 1. If a yard of cloth cost of a dollar, how many yards can you buy for 12 dollars ? 2. Mary gave $124 for silk at the rate of $2} a yard ; how many yards did she buy? 3. How long will it require for a man to earn $33, at the rate of $24 a day? 4. If 24 barrels of apples cost $15, how many barrels can be bought for 12 dollars ? 5. How many building lots of of an acre each, can we make out of 12 acres ? 6. A lady distributed $29 equally among some poor people, giving them $54 apiece ; how many did she aid ? 7. A school girl bought 6 yards of ribbon worth 5} cents a yard ; how many apples worth 1} cents each would cost the same ? 8. Mrs. Bear exchanged 20 pounds of butter, at 15 cents a pound. for calico, worth 12} cents a yard ; how many yards did she receive? 9. A school-boy shared 14 apples equally with his companions, giving to each 34 apples ; required the number of his companions. 10. A grocer sold 8 bushels of potatoes, worth ${ a busbel, and received for them eggs at the rate of $4 a dozen; how many eggs did be get? per barrel ? per ton ? WRITTEN EXERCISES. 1. If a pound of brown sugar cost 770 cents, how many pounds can you buy for 254 cents ? Ans. 31. SUGGESTION.— The analysis gives 1X14, which by cancelling and multiplying equals 3f. 2. If a quart of vinegar cost 18} cents, how many quarts can be bought for 521 cents ? Ans. 24 3. If 7 yards of book muslin cost $587, bow much will 5 yards cost ? Ans. $4.10. 4. If 15% yards of ribbon cost 45 cents, what will 6 yards cost? Ans. 17 f cts. 5. If 21% pounds of tea cost $1838, what will 1 pound cost ? Ans. $0.85. 6. If 227 barrels of sugar cost $30870, what is the price Ans. $137 7. If 284 tons of Lehigh coal cost $1835, what is the price Ans. $65. 8. Divide of 1 of % by of Hof 14 of 5. Ans. An 9. Find the value of 4X41X1 + 111. Ans. 10. 10. Find the value of (HX)+(*1X34) Ans. 44. 11. Find the value of (447)X7, and the product divided by (61-51). Ans. 335. 12. Find the value of (11+34) X97 and the product divided by (63-411). Ans. 1611. 13. Find the value of (7731-44%)X(61-5) divided by (8}+74311). Ans. 24. 14. If an errand boy earn $7 in a veek, how long will it require him to earn $201? Ans. 24 weeks. 16. A lady distributed $231 among the poor, giving $283 to each person; how many were there? Ans. 8 persons. 16. Mr. B divided $4161 equally among five persons; how much did each receive? Ans. $831 17. By his father's will, Henry shared $9600 equally with his five brothers; how much did he receive? Ans. $1600. 18. The product of two numbers, diminished by 1124, is 1274, and one number is 15; required the other. Ans. 1676 19. A boy shared 102 apples with his companions, giving to each 6 apples; required the number of companions. Ans. 15. 20. A man divided 304 pounds of flour among the poor, giving to each 24 pounds; how many persons were there? Ans. 11 persons. 21. A seamstress bought a sewing-machine for $85.50, and paid $40 down; how much must she save each month, to pay for it in 7 months ? Ans. $6 50. 22. A man's wages are $35 a day and bis daily expenses are $1}; how many days must he labor to enable him to buy & suit of clothes worth $46} ? Ans. 24 days. 23. Mr. Landis gained $228 on the sale of 67 acres of land; how much more would be have gained on each acre, if he had gained $382 on the whole ? Ans. $22.40. REDUCTION OF COMPLEX FRACTIONS. 183. The Reduction of Complex Fractions is the process of changing them to simple fractions. NOTE.- A complex fraction is not really a fraction, according to the de finition of a fraction. It is rather & complex fractional expression of one fraction divided by another. 至 1. Reduce to a simple fraction. OPERATION. -1%, Ans SOLUTION.—This expression means that I OPERATION. is to be divided by }, and inverting the divisor and multiplying, we have $X$, which ==={xt=% equals to SOLUTION 2D.-Multiplying both terms of the complex fraction by 12, the least common 4 x12 multiple of the denominators of the terms, and reducing the resulting fraction to its low- $$x12 est terms, we have . Rule I.-- Multiply the numerator of the complex fraction by its denominator inverted. Rule II.-Multiply both terms of the complex fraction by the least common multiple of the denominators. Ans. H. Ans. H ti WRITTEN EXERCISES. Reduce to simple fractions 21 2. Ans. It 8. 3 65 8. Ans. 4. 9. 91 1014 Ans. 24 10. H 121 8x2 $+4 6. Ans. U 11. HXH $+ 18x 31-24 6. Ans. 114. 12. Hx 41-21 Hx86 12137 7. Ans. 1985 13. 1x 18 41167 RELATION OF NUMBERS. 184. The Relation of Numbers is their relative value as compared with one another. NOTE.—This subject is equivalent to Ratio, but is presented here as affording an excellent illustration of the analysis of numbers. The treat ment of the subject under Ratio is demonstrative rather than analytic. CASK 1. 185. To find the relation of an integer to an integer. 1. 27 is how many times 6, or what is the relation of 27 to 6? SOLUTION.-One is of 6, and if 1 is of 6, 27 is 27 times one-sixth of 6, which is y, or , or 41 times 6. Therefore 27 is 44 times 6. Hence we have the following Rule.- Divide the number which you compare by the number with which it is compared. NOTE –The rule is the same for each case, and need not be repeated. MENTAL EXERCISES. 1. 9 is how many times 6? 25 is how many times 107 64 is how many times 36? 2. 16 is what part of 38 ? 18 is what part of 40? 32 is what part of 72? 90 is what part of 108 ? 8. A watch cost $40, and a chain cost $12; what part of the cost of the watch equals the cost of the chain ? |