Question

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

Answer 1

`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 `"number"`. The next integer will be `N +1` since it is `"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 + (N+1) + (N+2)+(N+3)+(N+4) = 110`

Now remove parentheses and add like terms:
`color(blue)N + color(blue)N+1 + color(blue)N+2+color(blue)N+3+color(blue)N+4= 110`
`color(blue)"5N" + 10 = 110`

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

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

Answer 2

`20,21,22,23,24`

Explanation:

Let the first number be`" " x`
Other numbers will be `x+1,x+2,x+3,x+4`

`=>(x)+(x+1)+(x+2)+(x+3)+(x+4)=110`

`=>5x+10=110`

`=>5x=110-10`

`=>5x=100`

`=>x=100/5`

`=>x=cancel100^20/cancel5^1`

`x=20`

The numbers are `" "20,21,22,23,24`

Answer 3

Please see below.

Explanation:

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

The sum is `5n`, so we need

`5n = 110`

`n = 22`

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

Answer 4

20, 21, 22, 23, 24

Explanation:

`color(brown)("The mean value is such that if you multiply it by the count of all the")` `color(brown)("numbers you get the sum of those numbers (added them all up).")`

`color(brown)("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 `barx` ( as in statistics)

Then `5barx=110`

divide both sides by 5

`barx=110/5=22`

Middle number is `color(red)(22)` with two others each side of it giving a total count of 5.

`color(green)(ubrace(22-2),color(white)("d")ubrace(22-1),color(white)("d")color(red)(22),color(white)("d")ubrace(22+1),color(white)("d")ubrace(22+2))`

`color(white)("dd")20,color(white)("dddd")21,color(white)("ddd")22,color(white)("dd")23,color(white)("ddddd")24`