# The ratio of Sue's age to Betty's age is 4:1. Twenty years from now, Sue will be twice as old as Betty will be then. How do you find their present age?

Oct 31, 2016

Betty: 10
Sue: 40

#### Explanation:

Let $S$ be Sue's age
Let $B$ be Betty's age

$S : B = 4 : 1$

$\implies 4 B = S$

$S + 20 : B + 20 = 2 : 1$

$\implies S + 20 = 2 \left(B + 20\right)$

$4 B = S$

$S + 20 = 2 \left(B + 20\right)$

$\implies 4 B + 20 = 2 B + 40$

$\implies 2 B = 20$
$\implies B = 10$

$\implies S = 4 B = 40$

Oct 31, 2016

Sue is $40$ and Betty is $10$

#### Explanation:

Let Sue's current age be $S$
and Betty's current age be $B$

The ratio of Sue's age to Betty's age is $4 : 1$
$\rightarrow \frac{S}{B} = \frac{4}{1} \textcolor{w h i t e}{\text{XX")orcolor(white)("XX")S=4Bcolor(white)("XX}}$[1]

Twenty years from now Sue will be twice as old as Betty.
$\rightarrow S + 20 = 2 \times \left(B + 20\right) \textcolor{w h i t e}{\text{XXXX}}$[2]

Substituting $4 B$ for $S$ (from [1]) into [2]
$\textcolor{w h i t e}{\text{XXX}} 4 B + 20 = 2 \left(B + 20\right)$

$\textcolor{w h i t e}{\text{XXX}} 4 B + 20 = 2 B + 40$

$\textcolor{w h i t e}{\text{XXX}} 2 B = 20$

$\textcolor{w h i t e}{\text{XXX}} B = 10$

Substituting $10$ for $B$ back in [1]
$\textcolor{w h i t e}{\text{XXX}} S = 4 \times 10 = 40$