'a xa x a. Algebraic Fractions. 6 180. The expressions õ' A' ď are algebraic fractions. a The above expressions are read, a divided by b; x divided by 4; 6 divided by cd. REDUCTION OF ALGEBRAIC FRACTIONS. 181. Reduce to lowest terms: ахь oxomt). abc + 60 d Let a = 2, b = 3, 6 = 5, and d 7, and verify. Observe that to reduce a fraction to its lowest terms we have only to strike out the factors that are common to its numerator and denominator. a2b 4. What factors are common to both numerator and ac denominator ? Reduce and verify. Algebraic Fractions. 182. Reduce to HIGHER Terms. 2 a 1. Change to a fraction whose denominator is abc, bc 2a x a 2 a? Let a= 2, b = 3, and c= : 5, and verify the abc reduction. bc xa 3 x 2. Change to a fraction whose denominator is 2 ay. 2 ay 3.xy Give any values you please to a, x, and y 2 ay xy 2 ay and verify the reduction. 3 x X y 183. REDUCE TO EQUIVALENT FRACTIONS HAVING A COMMON DENOMINATOR. 1. Since the common denominator must be and у ab ad 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 must therefore be a X a X 6 X d= a?bd ; abd • ab.= ad ; a2bd • a’d = b. x x ad adx by a'd xū apbd у хь ab x ad Give any values you please to a, b, d, x, and y, and verify. 4 2. 3 and bc The common denominator must contain the factors a, b, b, c, c. Reduce and verify. a b2 The common denominator is 5 a. Reduce and verify. * Since the numerical values of the letters are uuknown, 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. Square. 1 a b 185. MISCELLANEOUS REVIEW. 1. The difference of two numbers is 37411; the smaller number is 243}7. 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 + 2 x + 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 x 3 x 5), exactly divisible besides itself and 1 ? 7. By what is abc, (a x bxc), 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. 8. Change 5 to 60ths. Is 3 more or less than 7 ? 9. Change & to 100ths. Change š to 100ths. 10. 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. FRACTIONS. Art. 179, continued from page 86. Find the sum of-1. and 2 6. 11, , and 1% 2. 13 and 18 7. 11, }, and is 3. U1 and 28 8. 11, 4, and is 4. i and 1 9. , , and 5. i and to 10. 1, 1, and i (a) Find the sum of the ten sums. 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. Find the difference of 1. and is 3. and 4. 91 and t 5. $- |