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

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Answer 1 · 2016-11-23T14:09:33

The number is $48$ $48$

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

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

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

Finally we are told

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

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

We can now solve for $n$ $n$ :

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

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

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

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

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

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

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

$- 6 \cdot (- \frac{1}{6}) n = - 8 \cdot - 6$ $-6 * (-1/6)n = -8 * -6$

$1 n = 48$ $1n = 48$

$n = 48$ $n = 48$