# Problem of the Week Problem C and Solution Chip to Chip

## Problem

Mr. Chips has a bin full of bingo chips. The ratio of the number of red chips to the number of blue chips is $$1:4$$, and the ratio of the number of blue chips to the number of green chips is $$5:2$$.

What is the ratio of the number of red chips to the number of green chips?

## Solution

Solution 1

We start by assuming that there are $$20$$ blue chips. (We pick $$20$$ since the ratio of red chips to blue chips is $$1:4$$ and the ratio of blue chips to green chips is $$5:2$$, so we pick a number of blue chips which is divisible by $$4$$ and by $$5$$. Note that we did not have to assume that there were $$20$$ blue chips, but making this assumption makes the calculations much easier.)

Since there are $$20$$ blue chips and the ratio of the number of red chips to the number of blue chips is $$1:4$$, then there are $$\frac{1}{4} \times 20 = 5$$ red chips.

Since there are $$20$$ blue chips and the ratio of the number of blue chips to the number of green chips is $$5:2$$, then there are $$\frac{5}{2} \times 20 = 8$$ green chips.

Therefore, the ratio of the number of red chips to the number of green chips is $$5:8$$.

Solution 2

Let $$r$$ represent the number of red chips.

Since the ratio of the number of red chips to the number of blue chips is $$1:4$$, then the number of blue chips is $$4r$$.

Since the ratio of the number of blue chips to the number of green chips is $$5:2$$, then the number of green chips is $$\frac{2}{5} \times 4r = \frac{8}{5}r$$.

Since the number of red chips is $$r$$ and the number of green chips is $$\frac{8}{5}r$$, then the ratio of the number of red chips to the number of green chips is $$1: \frac{8}{5} = 5:8$$.