#
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 \(1: \frac{8}{5} = 5:8\).