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Problem 9

From youngest to oldest, the children in a family are Alfie, Beth, and Cyril. Alfie is two years younger than Beth. In two years' time, Cyril will be twice as old as Beth.

Given that the product of Alfie and Cyril's ages is $80,$ how old are each of the three children?

Find the sum of their ages.

Hint

Let their ages by $a, b$ and $c$

We are told that

a = b - 2

Hint

In two years' time, Cyril will be $c + 2$ , and Beth will be $b + 2$ years old

Solution

Let their ages by $a, b$ and $c$ .

Alfie is two years younger than Beth, so $a = b - 2$ .

In two years' time, Cyril will be twice as old as Beth, so

\begin{aligned} c + 2 & = 2 (b + 2) \\ c & = 2 b + 4 - 2 \\ c & = 2 (b + 1) \end{aligned}

Therefore,

\begin{aligned} a c & = 80 \\ 2 (b + 1) (b - 2) & = 80 \\ (b + 1) (b - 2) & = 40 \\ b^{2} - b - 2 & = 40 \\ b^{2} - b - 42 & = 0 \\ (b + 6) (b - 7) & = 0 \\ b & \in {- 6, 7} \end{aligned}

However, $b > 0$ so $b = 7$ is the only solution.

This also tells us that $a = b - 2 = 5$ and $c = 2 (b + 1) = 16$ .

Problem 9 ​

Problem 9