1550.36 – Splitting up Eighty-Eight

Can you divide the number 88 into two parts so that when the two parts are divided by four and the two quotients are added, then the sum is one-fourth of the original eighty-eight?

Solution

Let one of the two parts be xx. Then the other part is 88−x88 - x. We want x4+88−x4\dfrac{x}{4} + \dfrac{88 - x}{4} to equal 884=22\dfrac{88}{4} = 22. A little algebra never hurt anyone:

x4+88−x4=x4+884−x4=x4+22−x4=22\begin{align*}\dfrac{x}{4} + \dfrac{88 - x}{4} &= \dfrac{x}{4}+\dfrac{88}{4}-\dfrac{x}{4} \\[1em]&=\dfrac{x}{4}+22-\dfrac{x}{4}\\[1em]&=22\end{align*}

So we get an identity: 22=2222 = 22. So xx can be any number and the answer is YES!