# Five consecutive integers add up to 85. What are the numbers?

Jun 1, 2016

The five consecutive integers are $15 , 16 , 17 , 18 , 19$

#### Explanation:

To identify five consecutive integers we begin by giving them each a variable expression

$1 s t = x$
$2 n d = x + 1$
$3 r d = x + 2$
$4 t h = x + 3$
$5 t h = x + 4$

Now we set these equal to a sum of 85

$x + x + 1 + x + 2 + x + 3 + x + 4 = 85$

$5 x + 10 = 85$

$5 x \cancel{+ 10} \cancel{- 10} = 85 - 10$

$5 x = 75$

$\frac{\cancel{5} x}{\cancel{5}} = \frac{75}{5}$

$x = 15$

$1 s t = x = 15$
$2 n d = x + 1 = 16$
$3 r d = x + 2 = 17$
$4 t h = x + 3 = 18$
$5 t h = x + 4 = 19$

Jun 1, 2016

$15 , 16 , 17 , 18 , 19$

#### Explanation:

If the middle integer is $n$, then we are given:

$85 = \left(n - 2\right) + \left(n - 1\right) + n + \left(n + 1\right) + \left(n + 2\right) = 5 n$

Divide both ends by $5$ and transpose to find:

$n = 17$

So the five integers are:

$15 , 16 , 17 , 18 , 19$