# If one-half of a number is 8 less than two-thirds of the number, what is the number?

Nov 23, 2016

The number is $48$

#### Explanation:

First, let's call the number we are looking for $n$.

We can then write "two-thirds of the number" as $\frac{2}{3} n$

We can also write "one-half of a number" as $\frac{1}{2} n$

Finally we are told

$\frac{1}{2} n$ is the same as $\frac{2}{3} n - 8$ or:

$\frac{1}{2} n = \frac{2}{3} n - 8$

We can now solve for $n$:

$\frac{1}{2} n - \frac{2}{3} n = \frac{2}{3} n - 8 - \frac{2}{3} n$

$\frac{1}{2} n - \frac{2}{3} n = - 8$

We next need to get the fractions over common denominators, in this case $6$ by multiplying each fraction by the necessary for of $1$:

$\left(\frac{3}{3}\right) \frac{1}{2} n - \left(\frac{2}{2}\right) \frac{2}{3} n = - 8$

$\frac{3}{6} n - \frac{4}{6} n = - 8$

$- \frac{1}{6} n = - 8$

We can finally solve for $n$ by multiplying by -6 to isolate $n$

$- 6 \cdot \left(- \frac{1}{6}\right) n = - 8 \cdot - 6$

$1 n = 48$

$n = 48$