## Question

Two positive numbers differ by 12 and their reciprocals differ by 4/5. What is their product?

(A) 2/15

(B) 48/5

(C) 15

(D) 42

(E) 60

## Solution

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*) = 4/5Note 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 = 4

*xy*

15 =

*xy*

**The correct answer is (C) 15.**

**-------------------------------------------**

It was a very simple question. I've highlighted where I made mistakes. Both were such stupid errors. Even if I hadn't figured the 2nd one, I would've solved it by an extremely long method.