Question
Two positive numbers differ by 2, and the sum of their
reciprocals is 3/4. What is the larger number?Solution
ATQ, Let the smaller number be x; then the larger is x + 2. Given: 1/x + 1/(x + 2) = 3/4 ( (x + 2) + x ) / [x(x + 2)] = 3/4 (2x + 2) / (x2 + 2x) = 3/4 4(2x + 2) = 3(x2 + 2x) 8x + 8 = 3x2 + 6x 3x2 − 2x − 8 = 0 Solve: Discriminant = (−2)2 + 4·3·8 = 4 + 96 = 100 x = [2 ± 10] / (2·3) ⇒ x = 2, or x = −4/3 (reject negative). So smaller = 2, larger = 4.
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