# Rose is two thirds as old as Johnny, in 5yrs the sum of their ages will be 50yrs, how do you find their present ages?

Sep 4, 2015

Rose is 16
Johnny is 24

#### Explanation:

Define
$\textcolor{w h i t e}{\text{XXX}} r$: Rose's age
$\textcolor{w h i t e}{\text{XXX}} j$: Johnny's age

We are told
$\textcolor{w h i t e}{\text{XXX}} r = \frac{2}{3} j$

$\textcolor{w h i t e}{\text{XXX")rarrcolor(white)("XXX")"[1]"color(white)("XXX}} 3 r - 2 j = 0$

and
$\textcolor{w h i t e}{\text{XXX}} \left(r + 5\right) + \left(j + 5\right) = 50$

$\textcolor{w h i t e}{\text{XXX")rarrcolor(white)("XXX")"[2]"color(white)("XXX}} r + j = 40$

Multiplying [2] by 2
$\textcolor{w h i t e}{\text{XXXXXXXX")[3]color(white)("XXX}} 2 r + 2 j = 80$

$\textcolor{w h i t e}{\text{XXXXXXXX")[4]color(white)("XXX}} 5 r = 80$
Dividing [4] by $5$
$\textcolor{w h i t e}{\text{XXXXXXXX")[5]color(white)("XXX}} r = 16$
Substituting $16$ for $r$ in [1] (and simplifying
$\textcolor{w h i t e}{\text{XXXXXXXX")[6]color(white)("XXX}} j = 24$