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# The ratio of the sum of the reciprocals of x and y to the pr

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The ratio of the sum of the reciprocals of x and y to the pr [#permalink]  12 Apr 2019, 03:26
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The ratio of the sum of the reciprocals of x and y to the product of the reciprocals of x and y is 1 : 3. What is sum of the numbers x and y?

(A) $$\frac{1}{3}$$

(B) $$\frac{1}{2}$$

(C) 1

(D) 2

(E) 4
[Reveal] Spoiler: OA

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Re: The ratio of the sum of the reciprocals of x and y to the pr [#permalink]  13 Apr 2019, 02:34
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The reciprocal of x is $$\frac{1}{x}$$. So the sum of the reciprocals of x and y is $$\frac{1}{x}+\frac{1}{y} = \frac{y}{xy}+\frac{x}{xy} = \frac{x + y}{xy}$$.

The product of the reciprocals of x and y is $$\frac{1}{x} \times \frac{1}{y} = \frac{1}{xy}$$

The ratio of the sum to the product is 1 to 3. This will be a bit messy:

$$\frac{\frac{x+y}{xy}}{\frac{1}{xy}} = \frac{1}{3}$$

Luckily this question basically solves itself from here. Note that both the sum and the product have a denominator of xy. So we can multiply the left side by $$\frac{xy}{xy}$$ to get a much simpler equation:

$$\frac{x+y}{1} = \frac{1}{3}$$

The left side reduces to simply x + y. And of course, x + y is the value we're looking for. So the answer is just $$\frac{1}{3}$$.
Re: The ratio of the sum of the reciprocals of x and y to the pr   [#permalink] 13 Apr 2019, 02:34
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