Manhattan GMAT Challenge Problem of the Week - 17 Dec 09

by , Dec 17, 2009

Welcome back to this week's Challenge Problem! As always, the problem and solution below were written by one of our fantastic instructors. Each challenge problem represents a 750+ level question, so do not worry if you cannot solve the problem in a 2 minute time frame. If you are up for the challenge, however, set your timer for 2 minutes and go!

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

Dont 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, weve assigned x as the bigger of the two numbers and y as the smaller, so weve 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, lets 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) 15.

To view the current Challenge Problem, simply visit the Challenge Problem page on Manhattan GMAT's website.