183. REDUCE TO EQUIVALENT FRACTIONS HAVING A COMMON DENOMINATOR. x 1. and y ab a2d Since the common denominator must be exactly divisible by each of the given denomin. ators, it must contain all the prime factors* found in either of the given denominators. The new denominator 'nust therefore be a xa x b× d = a2 bd; a2 bd ÷ ab = ad; a2bd ÷ u'd = b. xx ad ab x ad = adx y x b by a2d xb a2bd Give any values you please to a, b, d, x, and y, and verify. The common denominator is 5 a. Reduce and verify. *Since the numerical values of the letters are unknown, each must be regarded as prime to all the others. The prime factors, then, in the first denominator are a and b; in the second, a, a, and d. GEOMETRY. 184. QUADRILATERALS.-REVIEW. 1. All the geometrical figures on this page are quadrilaterals; that is, each has four sides. 2. The first four figures are parallelograms; that is, the opposite sides of each figure are parallel. Square. a b 1 C d Oblong. a b 2 d 185. MISCELLANEOUS REVIEW. 1. The difference of two numbers is 37411; the smaller number is 24317. 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+3x= 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 xbx 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 factors has exact integral divisors. 60 8. Change to 60ths. Is ğ more or less than 37? 9. Change to 100ths. 10. Change to 100ths. 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 = 3210 Difference 2511 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. 31+2 = 11. 23-18=17. 2 units from 7, (8 — 1), units = 5 units. 3 tens from 5 tens: tens. (a) Find the sum of the ten differencer |