Yesterday, Manhattan GMAT posted a 700 level GMAT question on our blog. Today, they have followed up with the answer:
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 4/5” can be written as:
1/y – 1/x = 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 1/y is the larger reciprocal and 1/x the smaller, and so arranged them in that order to write 1/y – 1/x = 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:
x/(xy) – y/(xy) = 4/5
(x – y)/(xy) = 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:
12/(xy) = 4/5
60 = 4xy
15 = xy
The correct answer is (C).