# A is elder to B by two years. A's father F is twice as old as A and B is twice as old as his sister S. If the ages of the father and the sister differs by 40 years, find the age of A?

Sep 3, 2015

$A$ is 26 years old.

#### Explanation:

The idea here is that you can use the information given to you to write four equations with four unknows, $A$, $B$, $C$, and $D$.

So, you know that $A$ is two years older than $B$, which means that you can write

$A = B + 2 \text{ } \textcolor{b l u e}{\left(1\right)}$

You also know that $A$'s father, $F$, is twice as old as $A$

$F = 2 \cdot A \text{ } \textcolor{b l u e}{\left(2\right)}$

The same can be said for $B$ and his sister $S$

$B = 2 \cdot S \text{ } \textcolor{b l u e}{\left(3\right)}$

FInally, you know that the difference between the ge of the father nad the age of the sister is 40 years old, so you can write

$F - S = 40 \text{ } \textcolor{b l u e}{\left(4\right)}$

Now you can replace $F$ from equation $\textcolor{b l u e}{\left(2\right)}$ into equation $\textcolor{b l u e}{\left(4\right)}$ and $B$ from equation $\textcolor{b l u e}{\left(3\right)}$ into equation $\textcolor{b l u e}{\left(1\right)}$ to write

$\left\{\begin{matrix}A = 2 S + 2 \\ 2 A - S = 40\end{matrix}\right.$

Replace the value of $A$ from the first equation into the second one to get

$2 \cdot \left(2 S + 2\right) - S = 40$

$4 S - S + 4 = 40$

$3 S = 36 \implies S = \frac{36}{3} = 12$

This means that the age of $A$ is

$A = 2 \cdot S + 2$

$A = 2 \cdot 12 + 2 = \textcolor{g r e e n}{26}$