1955 AHSME Problems/Problem 42

Problem

If $a, b$ , and $c$ are positive integers, the radicals $\sqrt{a+\frac{b}{c}}$ and $a\sqrt{\frac{b}{c}}$ are equal when and only when:

$\textbf{(A)}\ a=b=c=1\qquad\textbf{(B)}\ a=b\text{ and }c=a=1\qquad\textbf{(C)}\ c=\frac{b(a^2-1)}{a}\\ \textbf{(D)}\ a=b\text{ and }c\text{ is any value}\qquad\textbf{(E)}\ a=b\text{ and }c=a-1$

Solution

Here, we are told that the two quantities are equal.

Squaring both sides, we get: $a+\frac{b}{c}=a^2*\frac{b}{c}$ .

Multiply both sides by $c$ : $ac+b=ba^2$ .

Looking at all answer choices, we can see that $ac=ba^2-b=b(a^2-1)$ .

This means that $c=\frac{b(a^2-1)}{a}$ , and this is option C. Therefore chose $\boxed{C}$ as your answer.

1955 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 41	Followed by Problem 43
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1955 AHSME Problems/Problem 42

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