# Irene is one-third the age of Bea. Three years ago, Bea was twice as old as Irene. How old is each?

Jul 17, 2016

There is something wrong with the information in the question. As people age, the relationship between their ages becomes larger, not smaller. Working with the information given produces a negative age. A suggested option is offered below.

#### Explanation:

Choose a variable to represent the age of the younger girl. (Irene)

Let Irene's present age be $x$ years.

Therefore Bea's present age is $3 x$

THREE YEARS AGO, they were both 3 years younger than now.

Irene's age was $\left(x - 3\right)$ and Bea's age was $\left(3 x - 3\right)$

At that time, Bea was twice as old as Irene....

$3 x - 3 = 2 \left(x - 3\right)$

A common mistake is to multiply the wrong age by 3, because that is what the English sounds like. Bea was already older, do not make her even older by multiplying her age by 2.
$3 x - 3 = 2 x - 6$
$x = - 3$ This is not possible

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However, if the question had read " In 3 years time, Bea will be twice as old as Irene, then we proceed as follows:

IN THREE YEARS TIME, they will both be 3 years older.
Irene will be $x + 3$ and Bea will be $3 x + 3$
Then we will have

$\left(3 x + 3\right) = 2 \left(x + 3\right)$
$3 x + 3 = 2 x + 6$

$x = 3$

Check: Now Irene is 3 and Bea is 9.
In 3 years time: Irene will be 6 and Bea will be 12.