# The sum of 5 consecutive integers is 110. What are the numbers?

Mar 12, 2018

$20 , 21 , 22 , 23 , 24$

#### Explanation:

What, in the first place, are consecutive integers?

They are numbers that come one after another with no numerical gaps. Like these:

$4 , 5 , 6 , 7 , 8$ or these $17 , 18 , 19 , 20 , 21$

We need to find 5 consecutive integers that add up to 110.

Let's call the first integer in the series $N$ for $\text{number}$. The next integer will be $N + 1$ since it is $\text{1 greater}$ than $N$.

The next integers will be $N + 2$, $N + 3$ and $N + 4$ since they are 2, 3 and 4 greater than $N$ respectively.

$N + \left(N + 1\right) + \left(N + 2\right) + \left(N + 3\right) + \left(N + 4\right) = 110$

Now remove parentheses and add like terms:
$\textcolor{b l u e}{N} + \textcolor{b l u e}{N} + 1 + \textcolor{b l u e}{N} + 2 + \textcolor{b l u e}{N} + 3 + \textcolor{b l u e}{N} + 4 = 110$
$\textcolor{b l u e}{\text{5N}} + 10 = 110$

Now finish simplifying:
$5 N + 10 = 110$
$5 N = 100$
$N = 20$

Since $N = 20$ our 5 consecutive numbers are:
$20 , 21 , 22 , 23 , 24$

Mar 12, 2018

$20 , 21 , 22 , 23 , 24$

#### Explanation:

Let the first number be$\text{ } x$
Other numbers will be $x + 1 , x + 2 , x + 3 , x + 4$

$\implies \left(x\right) + \left(x + 1\right) + \left(x + 2\right) + \left(x + 3\right) + \left(x + 4\right) = 110$

$\implies 5 x + 10 = 110$

$\implies 5 x = 110 - 10$

$\implies 5 x = 100$

$\implies x = \frac{100}{5}$

$\implies x = {\cancel{100}}^{20} / {\cancel{5}}^{1}$

$x = 20$

The numbers are $\text{ } 20 , 21 , 22 , 23 , 24$

Mar 12, 2018

#### Explanation:

Let $n$ be the middle number. Then the others are $n - 2$, $n - 1$, $n + 1$, and $n + 2$

The sum is $5 n$, so we need

$5 n = 110$

$n = 22$

The numbers are $20$, $21$, $22$, $23$, and $24$

Mar 12, 2018

20, 21, 22, 23, 24

#### Explanation:

$\textcolor{b r o w n}{\text{The mean value is such that if you multiply it by the count of all the}}$$\textcolor{b r o w n}{\text{numbers you get the sum of those numbers (added them all up).}}$

$\textcolor{b r o w n}{\text{Consecutive means the next, then the next, then the next and so on.}}$

5 times the mean value (average) gives 110

Let the mean value be represented by $\overline{x}$ ( as in statistics)

Then $5 \overline{x} = 110$

divide both sides by 5

$\overline{x} = \frac{110}{5} = 22$

Middle number is $\textcolor{red}{22}$ with two others each side of it giving a total count of 5.

$\textcolor{g r e e n}{\underbrace{22 - 2} , \textcolor{w h i t e}{\text{d")ubrace(22-1),color(white)("d")color(red)(22),color(white)("d")ubrace(22+1),color(white)("d}} \underbrace{22 + 2}}$

$\textcolor{w h i t e}{\text{dd")20,color(white)("dddd")21,color(white)("ddd")22,color(white)("dd")23,color(white)("ddddd}} 24$