# In 2 years a boy will be 3/4 as old as his sister. 2 years ago he was 2/3 as old as she. How old are they?

Sep 10, 2015

The boy is $10$
and his sister is $14$

#### Explanation:

Define
$\textcolor{w h i t e}{\text{XXX}} b$: boy's current age
$\textcolor{w h i t e}{\text{XXX}} g$: his sister's current age

We are told
[1]$\textcolor{w h i t e}{\text{XXX}} b + 2 = \frac{3}{4} \left(g + 2\right)$
and
[2]$\textcolor{w h i t e}{\text{XXX}} b - 2 = \frac{2}{3} \left(g - 2\right)$

[1] can be simplified into standard form as
[3]$\textcolor{w h i t e}{\text{XXX}} 4 b - 3 g = - 2$

[2] can be simplified into standard form as
[4]$\textcolor{w h i t e}{\text{XXX}} 3 b - 2 g = 2$

Multiply [3] by 2 and [4] by 3
[5]$\textcolor{w h i t e}{\text{XXX}} 8 b - 6 g = - 4$
[6]$\textcolor{w h i t e}{\text{XXX}} 9 b - 6 g = 6$

Subtract [5] from [6]
[7]$\textcolor{w h i t e}{\text{XXX}} b = 10$

Substitute $10$ for $b$ in [3]
[8]$\textcolor{w h i t e}{\text{XXX}} 4 \left(10\right) - 3 g = - 2$

[9]$\textcolor{w h i t e}{\text{XXX}} 3 g = 42$

[10]$\textcolor{w h i t e}{\text{XXX}} g = 14$