# One half of a number increased by 16 is four less than two thirds of the number. What are the numbers?

Oct 24, 2016

The number is $\textcolor{g r e e n}{72}$

#### Explanation:

Let the number be $n$

the number increased by $16$
$\textcolor{w h i t e}{\text{XXX}} n + 16$

one half of the number increased by $16$
$\textcolor{w h i t e}{\text{XXX}} \frac{1}{2} \left(n + 16\right)$

two thirds of the number
$\textcolor{w h i t e}{\text{XXX}} \frac{2}{3} n$

one half of the number increased by $16$
is $4$ less than
two thirds of the number
$\textcolor{w h i t e}{\text{XXX}} \frac{1}{2} \left(n + 16\right) = \frac{2}{3} n - 4$

Multiply both sides by $6$ to get rid of the fractions
$\textcolor{w h i t e}{\text{XXX}} 3 \left(n + 16\right) = 4 n - 24$

Simplify
$\textcolor{w h i t e}{\text{XXX}} 3 n + 48 = 4 n - 24$

Subtract $3 n$ from both sides
$\textcolor{w h i t e}{\text{XXX}} 48 = n - 24$

Add $24$ to both sides
$\textcolor{w h i t e}{\text{XXX}} 72 = n$
or
$\textcolor{w h i t e}{\text{XXX}} n = 72$