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 $48$

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

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

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

Finally we are told

$1/2 n$ is the same as $2/3 n - 8$ or:

$1/2 n = 2/3 n - 8$

We can now solve for $n$ :

$1/2n - 2/3n = 2/3n - 8 - 2/3n$

$1/2n - 2/3n = -8$

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

$(3/3) 1/2n - (2/2) 2/3n = -8$

$3/6n - 4/6n = -8$

$-1/6 n = -8$

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

$-6 * (-1/6)n = -8 * -6$

$1n = 48$

$n = 48$