# Tanya is 12 years older than Leah. Three years ago, Tanya was five times as old as Leah. How old is Leah?

Apr 30, 2017

See the solution process below:

#### Explanation:

First, let the age of Leah be represented by $l$ and the age of $T a n y a$ be represented by $t$:

The current age differences between Tanya and Leah can be written as:

$t = l + 12$

We can write the age difference three years ago as:

$t - 3 = 5 \left(l - 3\right)$

We can subsitute $l + 12$ from the first equation for $t$ in the second equation and solve for $l$:

$t - 3 = 5 \left(l - 3\right)$ becomes:

$\left(l + 12\right) - 3 = 5 \left(l - 3\right)$

$l + 9 = \left(5 \cdot l\right) - \left(5 \cdot 3\right)$

$l + 9 = 5 l - 15$

$- \textcolor{red}{l} + l + 9 + \textcolor{red}{15} = - \textcolor{red}{l} + 5 l - 15 + \textcolor{red}{15}$

$0 + 24 = - 1 \textcolor{red}{l} + 5 l - 0$

$24 = \left(- 1 + 5\right) l$

$24 = 4 l$

$\frac{24}{\textcolor{red}{4}} = \frac{4 l}{\textcolor{red}{4}}$

$6 = \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{4}}} l}{\cancel{\textcolor{red}{4}}}$

$6 = l$

$l = 6$

Leah is 6 years old.