# Ann is 12 years old. Her mother is 3 times as old as Ann. How many years ago was her mother 4 times as old as Ann?

May 11, 2018

4 years ago

#### Explanation:

If we call Ann´s age A and mother´s age M:

$A = 12$

$A \cdot 3 = M$

, a few years back ( Y ):

$\left(A - Y\right) \cdot 4 = \left(M - Y\right)$

So, by substitution:

$\left(A - Y\right) \cdot 4 = \left(A \cdot 3 - Y\right)$

$4 A - 4 Y = 3 A - Y$

$4 A - 3 A = 4 Y - Y$

$A = 3 Y$

$Y = \frac{A}{3} = \frac{12}{3} = 4$

May 11, 2018

$4$ years ago.

#### Explanation:

We know both of their present ages.

If Ann is $12$ years old, then her mother is $3 \times 12 = 36$ years old.

A number of years ago,they were both younger.
Let the number of years be $x$.

Ann's age was $12 - x$ and her mother's age was $36 - x$

$36 - x = 4 \left(12 - x\right) \text{ } \leftarrow$ the mom was $4$ times older,

$36 - x = 48 - 4 x$

$4 x - x = 48 - 36$

$3 x = 12$

$x = 4$

It was $4$ years ago.

Check: Ann was $8$ and her mom was $32$

$4 \times 8 = 32$