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 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 27, or 2, or 4 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 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? 8. 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, 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 numDer. 1. The fraction is what part of 6? SOLUTION.-One is of 6, and if 1 is of 6, is of †, which is A f 6, and is 3 times, which is of 6. MENTAL EXERCISES. 1. What part of 3 is ? Of 9 is $? Of 4 is ? Of 7 is §? 2. What part of 5 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 8 yards; what part of the silk then remained? 6. A bought 6 oranges, and B bought 4 oranges; what did B ay 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? Ans. f. CASE III. 187. To find the relation of a number to a frac tion. 1. 6 is how many times ? SOLUTION.-is of, and since is of, or 1 is 4 times, or of, and 6 is 6 times, or 44, which equals 8. MENTAL EXERCISES. 1. 8 is how many times ? 14 is how many times 7? 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 $, 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? 8. Mary had 12 minus of a quart; how many times Sarah's number equals Ans. 1. Mary's? quarts of nuts and Sarah had as many, CASE IV. 188. To find the relation of a fraction to a frac tion. 3. A had $ and spent $; 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? REMARK.-He sold, and hence there remained, and equals troo 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. . 12. Two boys had each of a bushel of walnuts, when he 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. 618 ninths. NOTE.-For Greatest Common Divisor and Least Common Multiple of Fractions see Manual. ARITHMETICAL ANALYSIS. 189. Analysis is the process of solving problems by comparison of their elements. In comparing, we reason to the unit and from the unit; the unit being the basis of the reasoning process. CASE I. 190. To pass from one integer to another integer. 1. If 5 oranges cost 12 cents, what will 8 oranges com at the same rate? SOLUTION.-If 5 oranges cost 12 or 45 cents, 1 orange cost of, which is § cents; and if 1 orange cost cents, 8 oranges will cost 8 times cents, or 20 cents. OPERATION. 5 cost 2 66 8 " x8=20, Ans. MENTAL EXERCISES. 1. If 4 apples cost 5 cents, what will 5 apples cost at the same rate? 2. If 3 girls earn $21 in a day, how much could 5 girls earn at the same rate? 3. How long will it take a boy to plow 8 furrows, if it takes him 13 minutes to plow 5 furrows? 4. How much will 9 barrels of flour cost, if 2 barrels cost $103 ? 5. If 5 hats can be bought for $114, how much will 8 hats cost? 6. What cost 5 pounds of butter at the rate of $3 for 10 pounds? WRITTEN EXERCISES. 2. If 6 hens cost $3.54, what will 7 hens cost at the same rate? Ans. $4.13. 3. If 8 turkeys cost $194, what will 10 turkey cost at the same rate? Ans. $24. 4. How much must I pay for 22 yards of muslin, at the rate of 5 yards for $18? Ans. $4.18. 5. How much will 21 books cost at the rate of 9 books for Ans. $36. 154? 6. What must I pay for 8 arithmetics, at the rate of 5 arithmetics for $4.75? Ans. $7.60. 7. If 3 tons of hay cost $47, how much will 12 tons cost at the same rate? Ans. $189. 8. Required the cost of 21 harrels of flour, at the rate of $17.50 for 3 barrels Ana $122.50 |