RELATION OF NUMBERS. 250. 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. 251. To find the relation of an integer to an integer. 1. What is the relation of 29 to 7? of 7, SOLUTION.-One is of 7, and if 1 is of 7, 29 is 29 times which are 22, or 44 times 7. Therefore, 29 is 4 times 7. 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. 6. At the rate of 15 apples for 12 cents, what will 45 apples cost? SOLUTION.-If 15 apples cost 12 cents, 45 apples, which are 15, or 3 times 15 apples, will cost 3 times 12 cents, or 36 cents. 7. If 12 oranges cost 15 cents, what will 84 oranges cost? Ans. $1.05. 8. If 18 pounds of tea cost $16.20, what will 108 pounds cost? Ans. $97.20. acres, sold 56 9. A man, having a farm containing 80 acres; what part of his farm remained? Ans. broken off by 10. A pole whose height was 80 feet, was the wind 48 feet from the top; what part of the pole was left standing? Ans. . 11. A has 18 cows, B 24 cows, and C 28 cows; if each obtains of the others, what part of A's will equal B's and C's respectively? Ans. B's, 23; C's, CASE II. 252. To find the relation of a fraction to a number. 1. The fraction is what part of 9? SOLUTION.-One is of 9, and if 1 is of 9, is of, which is, 253. To find the relation of a number to a fraction. 1. What is the relation of 7 to ? SOLUTION.-is of, and if is of 3, or 1 is 6 times or § of 3, and 7 is 7 times, or 42, which equals 83. Therefore, etc. 5. 729 to 27? 10. A has 56 11 Ans. 1080. 9. 1020 to of 124? Ans. 2231. bushels of wheat, and B as many, +4 bushels; how many times B's number equals A's? CASE IV. Ans. 1. 254. To find the relation of a fraction to a fraction. 1. What part of 18 is §? SOLUTION. is 15 of 18, and 18, or one, is 16 times, which equals of. If 1 equals 1 of 15, equals of 18, and equals 5 times of, which equals of 18, or 3. 10. A and B had each 15 of a ton of hay; A sold B 1⁄2 of what he had; what part of B's equals A's? Ans.. GREATEST COMMON DIVISOR. 255. The Greatest Common Divisor of two or more fractions is the greatest fraction that will exactly divide each of them. PRINCIPLES. 1. A fraction is a divisor of a given fraction when its numerator is a divisor of the given numerator, and its denominator is a multiple of the given denominator. To divide by a fraction we divide by its numerator and multiply by its denominator; hence, to obtain an integral quotient, the numerator of the divisor must divide the given numerator, and the denominator of the divisor must contain the given denominator. Illustrate with and. 2. A common divisor of several fractions is a fraction whose numerator is a common divisor of their numerators, and whose denominator is a common multiple of their denominators. 3. The greatest common divisor of several fractions is a fraction whose numerator is the greatest common divisor of their numerators, and whose denominator is the least common multiple of their denominators. 6 7 OPERATION. 9 3--14: 3=G. C. D. of Num's 28 L. C. M. of Den's. 1. Find the greatest common divisor of SOLUTION. To be a divisor of each of these fractions the numerator must divide each of the given numerators, and the denominator must contain each of the given denominators; hence, the greatest common divisor must be a fraction whose numerator is the greatest common divisor of the given numerators, and the denominator the least common multiple of the given denominators. The greatest common divisor of the numerators is 3, and the least common multiple of the denominators is 28; hence the greatest common divisor of the given fractions is. .. G. C. D: = =28, Ans. Rule. Reduce the given fractions to simple ones in their lowest terms; then find the G. C. D. of the numerators and divide it by the L. C. M. of the denominators. Find the greatest common divisor of 8. A farmer has 514 bushels of russets, 713 bushels of rambos, 143 bushels of seek-no-farthers, and 357 bushels of pearmains; required the largest bins of equal size which can be filled, each kind being kept by itself; also the number of bins. Ans. 5 bushels; 59 bins. 9. Mr. Johnson has four fields in the outskirts of a growing Western city, containing respectively 63 acres, 7 acres, 10 acres, 8 acres, which he wishes to divide into the larg est possible house-lots of equal size; what will be the size and number of the lots? Ans. of an acre; 332 lots. 10. Mr. Smith has a field whose sides are 335 feet, 397 feet, 3225 feet, and 23511 feet. He wishes to build a fence round it, 5 rails high, the rails overlapping of a foot; what is the longest rail that can be used, and how many rails will be required? Ans. 131 ft.; 520 rails. LEAST COMMON MULTIPLE. 256. The Least Common Multiple of two or more fractions is the least number that will exactly contain each of them. PRINCIPLES. 1. A multiple of a fraction is a fraction whose numerator is a multiple of the given numerator, and whose denominator is a divisor of the given denominator. To divide by a fraction, we divide by its numerator and multiply by its denominator; hence, to give an integral quotient, when we divide a multiple by a fraction, the numerator of the multiple must contain the numerator of the fraction, and the denominator of the multiple must divide the denominator of the fraction. Illustrate with g, a multiple of. 2. A common multiple of several fractions is a fraction whose numerator is a common multiple of their numerators, and whose denominator is a common divisor of their denomi nators. 3. The least common multiple of several fractions is a fraction whose numerator is the least common multiple of their numerators, and whose denominator is the greatest common divisor of their denominators. For, that common multiple is the smallest which has the smallest numerator and the largest denominator. 35 8 32 40° OPERATION. L. C. M. of Num.= 105 1. Find the least common multiple of 7, 35 21 SOLUTION. To be a multiple of each of these fractions, the numerator must contain each of the given numerators, and the denominator divide each of the given denominators; hence the least common multiple must be a fraction whose numerator is the least common multiple of the given numerators, and whose denominator is the greatest common divisor of the given denominators. The least common multiple of the numerators we find to be 105, and the greatest common divisor of the denominators is 8; hence 105, or 13, is the least common multiple of the given fractions. Rule. Reduce the fractions to simple ones in their lowest terms; then find the L. C. M. of the numerators, and divide it by the G. C. D. of the denominators. Find the least common multiple of 3. 33, 4. 11 77° 24 36 3 7 Ans. 84. Ans. 12. 5. 6221, 711, 1111. Ans. 4024. Ans. 9714. 4. 34, 35, 16, 26. Ans. 1004. 7. 63, 77, 97, 10,235. Ans. 180644. 8. The Earth, Mars, and Saturn were in conjunction December, 1875; when will they be again in conjunction at the same point of their orbits, the period of revolution of Mars being 18 years and of Saturn 29 years? Ans. In 1003 yr. 9. A man has a square lot which he wishes to fence, and has rails of four different lengths, namely, 123 feet, 12 feet, 13 feet, and 133 feet, and not enough of either to fence any two sides of the lot; what was the smallest possible side of the lot? Ans. 3712 ft. 10. A, B, and C start at the same place and travel round an island, A making the circuit in of a day, B in of a day, and C in of a day; in how many days will they meet at the starting place, and how many times will each bave gone round the island? Ans. 6 days. 11. A, B, C, and D start at the same place and travel round an island 72 miles in circumference, A traveling 21 miles an hour, B 33 miles an hour, C 33 miles an hour, D 42 miles an hour; how many days before they meet at the starting place, if they travel 10 hours a day, and how far will each have traveled? Ans. 120 days; A, 42 times, etc. |