Official Solution:Two positive numbers differ by 12 and their reciprocals differ by \(\frac{4}{5}\). What is their product?A. \(\frac{2}{15}\)
B. \(\frac{48}{5}\)
C. \(15\)
D. \(42\)
E. \(60\)
Don’t be afraid to assign variables even when none are given in the problem. "Two positive numbers differ by 12" can be written as:
\(x - y = 12\)
And "their reciprocals differ by \(\frac{4}{5}\)" can be written as:
\(\frac{1}{y} - \frac{1}{x} = \frac{4}{5}\)
(Note: Here, we’ve assigned \(x\) as the bigger of the two numbers and \(y\) as the smaller, so we’ve intuited that \(\frac{1}{y}\) is the larger reciprocal and \(\frac{1}{x}\) the smaller, and so arranged them in that order to write \(\frac{1}{y} - \frac{1}{x} = \frac{4}{5}\)).
Now we have a system of two variables and two equations. Note that it is NOT necessary to solve for \(x\) and \(y\), since we are being asked for the product, \(xy\).
First, let’s simplify the second equation by finding a common denominator for the terms on the left:
\(\frac{x}{xy} - \frac{y}{xy} = \frac{4}{5}\)
\(\frac{x - y}{xy} = \frac{4}{5}\)
Note that the denominator is \(xy\), which is exactly the quantity we want to find.
Since we know from the first equation that \(x - y = 12\), substitute 12:
\(\frac{12}{xy} = \frac{4}{5}\)
\(60 = 4xy\)
\(15 = xy\)
Answer: C.
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