185. MISCELLANEOUS REVIEW. 1. The difference of two numbers is 37411; the smaller number is 243. What is the larger number? * 2. The difference of two numbers is a; the smaller number is b. What is the larger number? 3. James had a certain number of dollars and John had three times as many; together they had 196 dollars. How many had each ? (x + 3 x = 196) † 4. William had a certain number of marbles; Henry had twice as many as William, and George had twice as many as Henry; together they had 161. How many had each? (x + 2x + 4 x = 161)† 5. Divide 140 dollars between two men, giving to one man 30 dollars more than to the other. (x + x + 30 = 140)† 6. By what integral numbers is 30, (2 × 3 × 5), exactly divisible besides itself and 1 ? 7. By what is abc, (a × b× c), exactly divisible besides itself and 1? (1) How many times is a contained in abc? Observe that a number composed of three different prime exact integral divisors. factors has 8. Change to 60ths. 9. Change to 100ths. 10. Change to 100ths. Is more or less than 37? Change to 100ths. Change to 100ths. *The difference of two numbers is 5; the smaller number is 4. What is the larger number? + See page 78. 186. TO SUBTRACT COMMON FRACTIONS. RULE.-Reduce the fractions if necessary to equivalent fractions having a common denominator, find the difference of their numerators, and write it over the common denominator. Fractions. 187. To subtract one mixed number from another when the fraction in the subtrahend is greater than the fraction in the minuend. Operation. 58% = 582 32 = 321 Difference 2517 EXAMPLE. From 58 take 32. Explanation. is greater than, therefore we take 1 unit from the 8 units, change it to 24ths, and add it to the 9 24ths. 2 units from 7, (81), units = 5 units. 3 tens from 5 tens = 2 (a) Find the sum of the ten differences. 1. Observe that by the first operation we obtain ; that in 12 there are 6 times as many parts as there are in and that the parts are of the same size as those in 4. 2. Observe that by the second operation we obtain 7; that in there are the same number of parts as there are in, and that the parts are 6 times as great as those in 7. Note 1.-The 7 of 7 may be regarded as a dividend; the 24, as a divisor, and itself, as a quotient. In, we have a dividend 6 times as great as that in, the divisor remaining unchanged. In 1, we have a divisor 1 sixth as great as that in 4, the dividend remaining unchanged. Multiplying the dividend or dividing the divisor by any number, multiplies the quotient by the same number. Note 2.-The 7 of may be regarded as the antecedent of a couplet; the 24, as itself as the ratio. Multiplying the antecedent or dividing the consequent of any couplet multiplies the ratio by the same number. the consequent, and RULE. To multiply a fraction by an integer, multiply its numerator or divide its denominator by the integer. (a) Find the sum of the twelve products. (b) Find the sum of the twelve products. |