18. The product of two numbers, diminished by 112, is 127, and one number is 15; required the other. Ans. 163. 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 30 pounds of flour among the poor. giving to each 2 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 $3 a day and his daily expenses are $13; how many days must he labor to enable him to buy a suit of clothes worth $46} ? Ans. 24 days. 23. Mr. Landis gained $228 on the sale of 63 acres of land; how much more would he 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 definition of a fraction. It is rather a complex fractional expression of one fraction divided by another. 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. 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 treatment of the subject under Ratio is demonstrative rather than analytic. CASE I. 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 1 of 6, and if 1 is of 6, 27 is 27 times one-sixth of 6, which is 27, or 2, or 4 times 6. Therefore 27 is 4 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 10? 64 is how many times 36? 2. 16 is what part of 36? 18 is what part of 40? 32 is what part of 72? 90 is what part of 108? 3. A watch cost $40, and a chain cost $12; what part of the cost of the watch equals the cost of the chain? 4. If 5 yards of cloth cost $21, what will 20 yards cost at the same rate? 5. If 6 tons of coal are bought for $19, what will 30 tons cost at the same rate? 6. A has $20 and B has & as much, plus $3; what part of A's money equals B's money? 10. If 16 apples cost 13 cents, what will 48 apples cost at the same rate? SOLUTION.-If 16 apples cost 13 cents, 48 apples, which are 8, or 3 times 16 apples, cost 3 times 13 cents, or 39 cents. OPERATION. 16 A. cost 13 cts. 48 A. cost 3X13 cts. = 39 cts. 11. If 15 oranges cost 14 cents, what will 60 oranges cost at the same rate? 12. What cost 125 sheep, if 25 sheep cost 137 dollars? Ans. 56 cents. Ans. $685. 13. If 56 cows cost $1000, what will 42 cows cost at the same rate? Ans. $750. 14. If 105 pens cost 77 cents, what will 225 pens cost at the same rate? Ans. $1.65. NOTE.-Classes needing a shorter course may omit the other three cases unless otherwise directed. CASE II. 186. To find the relation of a fraction to a num SOLUTION.-One is of 6, and if 1 is of 6, is of, which is f 6, and is 3 times, which is of 6. MENTAL EXERCISES. 1. What part of 3 is ? 2. What part of 5 is Of 9 is &? Of 4 is ? Of 7 is §? 3. If of a ton of coal cost $24, what will 6 tons cost at the same rate? 4. A hat was bought for $2 and sold at a gain of $; what part of the cost equals the gain? 5. Sarah bought 5 yards of silk and used 3} yards; what part of the silk then remained? 6. A bought 6 oranges, and B bought 4 oranges; what did Bay for his oranges, if A paid 26 cents for his? 10. A having $25, gave $5 to B and of the remainder to C what part of $25 then remained? CASE III. Ans. f. 187. To find the relation of a number to a fraction. 1. 6 is how many times ? SOLUTION. is of, and since is of, or 1 is 4 times }, or of 4, and 6 is 6 times, or 24, which equals 8. MENTAL EXERCISES. 1. 8 is how many times ? 14 is how many times ? 24 is how many times &? 2. 96 is how many times ? 115 is how many times ? 144 is how many times ? 3. If a yard of cloth cost $4, how many yards will $9 buy? 4. A can walk a certain distance in 24 hours; how many times that distance can he walk in 6 hours? 5. A book cost $4 and a slate $; how many times the cost of the slate equals the cost of the book? Ans. 2133. 7. 63504 to 50%? Ans. 54. Ans. 154. Ans. 1260 as many, 8. Mary had 12 minus & of a quart; how many times Sarah's number equals Mary's? Ans. 1. CASE IV. 188. To find the relation of a fraction to a frac 3. A had $ and spent $3; what part of his money did he spend? 4. If 2 yards of silk cost $71, what will 7 yards cost at the same rate? 5. A merchant bought of a cargo of oranges and sold of them ; what part of the cargo did he sell? 6. Edwin and Emma each had $2; Edwin spent of his, and Emma spent of hers; what part of Emma's money then equaled Edwin's? WRITTEN EXERCISES. What is the relation of Ans. . 8. X to &×18? Ans. 188. Ans. 28.9. 4×18 to 18×34? Ans. 1. of of his stock, how many Ans. . 10. If a merchant sold fifths of of his stock remained? REMARK.-He sold, and hence there remained, and equals two ifths of of his stock; hence the answer is . 11. Harry lost of of a certain sum of money; how many sevenths of 3 of the sum remained? Ans. 4. 12. Two boys had each of a bushel of walnuts, when the first sold of his to the second; what part of what the second has equals what the first has? Ans. . 13. James had of a ton of coal and Thomas had § of a ton; if James burns of his coal and Thomas of his how many ninths of a ton will remain? Ans. 61% ninths. NOTE. For Greatest Common Divisor and Least Common Multiple of Practions see Manual. |