p and q are positive integers and p/q + q/p = 3, then (p^2/q^2) + (q^2/p^2) =

A. 3 B. 7 C. 9 D. 11

Question

p and q are positive integers and p/q + q/p = 3, then (p^2/q^2) + (q^2/p^2) =

A. 3  B. 7  C. 9  D. 11

Accepted Answer

(p^2/q^2) + (q^2/p^2) = 7 (Option B). Let t = p/q. The given condition becomes t + 1/t = 3. The target expression (p^2/q^2) + (q^2/p^2) equals t^2 + 1/t^2. A standard algebraic identity connects t^2 + 1/t^2 to the square of (t + 1/t): squaring gives t^2 + 2 + 1/t^2 = 9, so t^2 + 1/t^2 = 7. This is a clean example of using the 'square then subtract' trick on symmetric rational expressions.