## 1-7-24

Two numbers are such that their difference, sum, and their product are in the ratio of $1:7:24$ respectively. What is the product of the two numbers?
Source: mathcontest.olemiss.edu 3/18/2013

SOLUTION
Let $a,b,a>b$ be the two numbers. We are given the following proportion
$\displaystyle\frac{a-b}{1}=\frac{a+b}{7}=\frac{ab}{24}$
By the properties of proportions
$\displaystyle\frac{a-b}{1}=\frac{a+b}{7}=\frac{a-b+a+b}{1+7}$
Simplify
$\displaystyle\frac{a-b}{1}=\frac{a+b}{7}=\frac{2a}{8}$
Cross-multiply the first and last ratios
$8\left (a-b\right )=2a$
$8a-8b=2a$
$6a=8b$
$3a=4b\qquad\left (1\right )$

First trial
$a=4;b=3$ satisfies Eq. $\left (1\right )\!,\;3\left (4\right )=4\left (3\right )=12$
Verify the ratio
$a-b=1:a+b=7:ab=12$ is not the desired ratio
$ab=12$ is not a solution
Second trial
$a=8;b=6$ satisfies Eq. $\left (1\right )\!,\;3\left (8\right )=4\left (6\right )=24$
Verify the ratio
$a-b=2:a+b=14:ab=48$
Divide the ratio by 2
$a-b=1:a+b=7:ab=24$ is the desired ratio
$ab=48$ is a solution

Answer: $48$