# 2005 CEMC Gauss (Grade 7) Problems/Problem 22

## Problem

In a bin at the Gauss Grocery, the ratio of the number of apples to the number of oranges is $1 : 4$, and the ratio of the number of oranges to the number of lemons is $5 : 2$. What is the ratio of the number of apples to the number of lemons?

$\text{(A)}\ 1 : 2 \qquad \text{(B)}\ 4 : 5 \qquad \text{(C)}\ 5 : 8 \qquad \text{(D)}\ 20 : 8 \qquad \text{(E)}\ 2 : 1$

## Solution 1

We start by assuming that there are $20$ oranges. (We pick $20$ since the ratio of apples to oranges is $1 : 4$ and the ratio of oranges to lemons is $5 : 2$, so we pick a number of oranges which is divisible by $4$ and by $5$. Note that we did not have to assume that there were $20$ oranges, but making this assumption makes the calculations much easier.) Since there are $20$ oranges and the ratio of the number of apples to the number of oranges is $1 : 4$, then there are $\frac{1}{4}\times 20 = 5$ apples. Since there are $20$ oranges and the ratio of the number of oranges to the number of lemons is $5 : 2$, then there are $\frac{2}{5}\times 20 = 8$ lemons. Therefore, the ratio of the number of apples to the number of lemons is $5 : 8$. Thus, the answer is $C$.

## Solution 2

Let the number of apples be $x$. Since the ratio of the number of apples to the number of oranges is $1 : 4$, then the number of oranges is $4x$. Since the ratio of the number of oranges to the number of lemons is $5 : 2$, then the number of lemons is $\frac{2}{5}\times 4x = \frac{8}{5}x$. Since the number of apples is $x$ and the number of lemons is $\frac{8}{5}x$, then the ratio of the number of apples to the number of lemons is $1 : \frac{8}{5} = 5 : 8$. Therefore, the answer is $C$.