# The sum of three numbers is 52. The first number is 8 less than the second. the third number is 2 times the second. What are the numbers?

Jun 9, 2018

The numbers are: $7 , 15 \mathmr{and} 30$

#### Explanation:

First write an expression for each of the three numbers,
We know the relationship between them so we can use one variable. Choose $x$ as the smallest one.

Let the first number be $x$
The second number is $x + 8$
The third number is $2 \left(x + 8\right)$

Their sum is $52$

$x + x + 8 + 2 \left(x + 8\right) = 52$

$x + x + 8 + 2 x + 16 = 52$

$4 x + 24 = 52$

$4 x = 52 - 24$

$4 x = 28$

$x = 7$

The numbers are: $7 , 15 \mathmr{and} 30$

Check: $7 + 15 + 30 = 52$

Jun 9, 2018

$7$, $15$ and $30$

#### Explanation:

$\left(x - 8\right) + x + 2 x = 52$
$4 x - 8 = 52$
$4 x = 52 + 8$
$4 x = 60$
$x = \frac{60}{4}$
$x = 15$

1st number = $15 - 8 = 7$
2nd number = $15$
3rd number = $15 \cdot 2 = 30$

Checking!
$30 + 15 + 7 = 52$

Jun 9, 2018

The numbers are $7 , 15 , \mathmr{and} 30$

#### Explanation:

"The sum of three numbers is 52" gives you the following equation:

$x + y + z = 52 \text{ [1]}$

"the first number is 8 less than the second" gives you the following equation:

$x = y - 8$

or

$y = x + 8 \text{ [2]}$

"the third number is 2 times the second" gives you the following equation:

$z = 2 y \text{ [3]}$

Substitute equation [3] into equation [1]:

$x + y + 2 y = 52$

Combine like terms:

$x + 3 y = 52 \text{ [1.1]}$

Substitute equation [2] into equation equation [1.1]:

$x + 3 \left(x + 8\right) = 52$

$4 x + 24 = 52$

$4 x = 28$

$x = 7$

Use equation [2] to find the value of y:

$y = 7 + 8$

$y = 15$

Use equation [3] to find the value of z:

$z = 2 \left(15\right)$

$z = 30$

Check:

$7 + 15 + 30 = 52$

$52 = 52$

This checks