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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] New post 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] New post 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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The ratio of the sum of the reciprocals of x and y to the pr

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