## Great 125

Four different positive whole numbers sum to one hundred twenty-five. If you increase one of these numbers by four, decrease the second by four, multiply the third by four, and divide the last by four, you will produce four equivalent numbers. What is the greatest of these four numbers?
Source: mathcontest.olemiss.edu 10/14/29013

SOLUTION
Let $a,b,c,d$ represent the four whole numbers. We have the following conditions:
$a+b+c+d=125\qquad\left (1\right )$
$a+4=b-4=4c=\dfrac{d}{4}\qquad\left (2\right)$
We use the method of substitution to solve Eq. (1) for $c$.
Add $4$ to both sides of Eq. $\left (1\right )$
$\left (a+4\right )+b+c+d=125+4$
Substitute the value of $a+4=4c$ from Eq. $\left (2\right )$
$4c+b+c+d=125+4$
Subtract $4$ from both sides
$4c+\left (b-4\right )+c+d=125+4-4$
Substitute the value of $b-4=4c$
$4c+4c+c+d=125$
Substitute the value of $d=16c$
$4c+4c+c+16c=125$
Simplify
$25c=125$
$c=5$
Calculate $a$
$a+4=4c=20$
Calculate $b$
$b-4=4c=20$
$b=24$
Calculate $d$
$d=16c=180$ the greatest of the four numbers.

Answer: $80$