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

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About mvtrinh

Retired high school math teacher.
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One Response to 1-7-24

  1. Nick says:

    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.

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