Friday, March 19, 2010

CP #9


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


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:
xy = 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
(xy)/(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 xy = 12, substitute 12:
12/(xy) = 4/5
60 = 4xy
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.