# 2008 iTest Problems/Problem 16

## Problem

In order to encourage the kids to straighten up their closets and the storage shed, Jerry offers his kids some extra spending money for their upcoming vacation. "I don't care what you do, I just want to see everything look clean and organized."

While going through his closet, Joshua finds an old bag of marbles that are either blue or red. The ratio of blue to red marbles in the bag is $17:7$. Alexis also has some marbles of the same colors, but hasn't used them for anything in years. She decides to give Joshua her marbles to put in his marble bag so that all the marbles are in one place. Alexis has twice as many red marbles as blue marbles, and when the twins get all their marbles in one bag, there are exactly as many red marbles and blue marbles, and the total number of marbles is between $200$ and $250$. How many total marbles do the twins have together?

## Solution

Let $17j$ equal the number of blue marbles Joshua has, and let $a$ equal the number of blue marbles Alexis has. Since the twins have the same number of red marbles as blue marbles when combined, we can write an equation. $\[17j+a=7j+2a\]$ $\[10j=a\]$ Making the substitution, the total number of blue marbles is $27j$, so the total number of marbles is $54j$. Since the only multiple of $54$ between $200$ and $250$ is $216$, the twins have a total of $\boxed{216}$ marbles.