Question

The sum of two consecutive odd integers is sixteen more than `4` times the smaller integer. What are the two integers?

Answer 1

-7 and -5

Explanation:

Translate the sentence into maths.

sum of two consecutive odd integers means `a + (a+2)` because consecutive is the next one, and odd numbers are 2 apart.

This sum... is sixteen more than 4 times the smaller integer
So `a+(a+2)=16+ (a*4)`

Now solve for `a`

`a + (a+2) = 4a +16`
`2a+2=4a+16`
`-14=2a`
`a=-7`

`:. a=-7, -5`

Answer 2

The smaller is `-9` and the larger `-11`

Explanation:

Denote the smaller integer by `n`

If `n > 0` then the larger integer is `n+2` and:

`n+(n+2) = 4n+16`

That is:

`2n+2 = 4n+16`

Subtracting `2n+16` from both sides:

`-14 = 2n`

Hence

`color(red)(cancel(color(black)(n = -7)))`

If `n < 0` then the larger integer is `n-2` and:

`n+(n-2) = 4n+16`

That is:

`2n-2 = 4n+16`

Subtracting `2n+16` from both sides:

`-18 = 2n`

Hence:

`n = -9`