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?
(A) $1: 2$
(B) $4: 5$
(C) $5: 8$
(D) $20: 8$
(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 $5: 8$.

## 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 $4 x$.

Since the ratio of the number of oranges to the number of lemons is $5: 2$, then the number of lemons is $\frac{2}{5} \times 4 x=\frac{8}{5} x$.

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 $1: \frac{8}{5}=5: 8$.

ANSWER: (C)

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