Two numbers are such that their difference, sum, and their product are in the ratio of respectively. What is the product of the two numbers?

Source: mathcontest.olemiss.edu 3/18/2013

**SOLUTION
**Let be the two numbers. We are given the following proportion

By the properties of proportions

Simplify

Cross-multiply the first and last ratios

First trial

satisfies Eq.

Verify the ratio

is not the desired ratio

is not a solution

Second trial

satisfies Eq.

Verify the ratio

Divide the ratio by 2

is the desired ratio

is a solution

**Answer**:

a + b = 7(a – b) => 6a = 8b => a = 4b/3 => a – b = b/3.

ab = 24(a – b) = 24b/3 = 8b => a = 8. (Ignoring b = 0, which implies a = 0.)

Thus b = 3a/4 = 6.

Therefore ab = 8×6 = 48.