Question

The average of five consecutive odd integers is -21. What is the least of these integers?

Answer 1

`-25`

Explanation:

Take `x`. This is the smallest integer. Since these are consecutive odd integers, the second must be `2` greater than the first. The third number must be `2` greater than the second. And so forth.

For example, `1, 3, 5, 7, and 9` are five consecutive odd integers, and they are all two more than the last. So, our five numbers are

`x, x+2, (x+2)+2, ((x+2)+2)+2, and (((x+2)+2)+2)+2`

which means

`x, x+2, x+4, x+6, and x+8`

According to the question, their average is `-21`. So,

`(x+(x+2)+(x+4)+(x+6)+(x+8))/5 = -21`

Therefore, by simplifying,

`(5x+20)/5 = -21`

So

`5x+20 = -105`

Then

`5x = -125`

and

`x=-25`

Shortcut: Since these are odd integers that are consecutive, you can take `-21` as the middle number, `-23` as the second, `-19` to even out the `-23` and maintain the average of `-21`, then `-25` as the first, then `-17` as the last. This is a little hard to explain but makes sense if you really think about it.

Answer 2

`"The answer is:" \qquad \qquad \qquad -25.`

`"[And the 5 integers in question are:"`

`\qquad \qquad \qquad \qquad \qquad \quad -25, -23, -21, -19, -17.]`

Explanation:

`"Let the smallest of these odd integers be:" \qquad \qquad 2 n - 1.`

`"The remaining 4 odd integers are:"`

`\qquad \qquad \qquad \qquad \qquad \quad \ 2 n + 1, \quad 2 n + 3, \quad 2 n + 5, \quad 2 n + 7. \quad`

`"The average of all 5 odd integers is:"`

`{ ( 2 n - 1 ) + ( 2 n + 1 ) + ( 2 n + 3 ) + ( 2 n + 5 ) + ( 2 n + 7 ) } / 5.`

`"The average of all 5 odd integers is given to be -21. Thus:"`

`{ ( 2 n - 1 ) + ( 2 n + 1 ) + ( 2 n + 3 ) + ( 2 n + 5 ) + ( 2 n + 7 ) } / 5`

`\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad = -21.`

`\qquad \qquad \qquad \qquad \qquad { 5 ( 2 n ) - 1 + 1 + 3 + 5 + 7 } / 5 \ = \ -21`

`\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad { 10 n + 15 } / 5 \ = \ -21`

`\qquad \qquad \qquad \qquad \qquad \qquad \qquad \ { color{red}cancel{ 5 } ( 2 n + 3 ) } / color{red}cancel{ 5 } \ = \ -21`

`\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \ 2 n + 3 \ = \ -21`

`\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \ \ 2 n \ = \ -21 - 3`

`\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \ \ 2 n \ = \ -24`

`\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \ \ \ \ n \ = \ -12.`

`"At the start here, we had:"`

`\qquad \qquad \quad \ "the smallest of these odd integers is:" \qquad \quad 2 n - 1.`

`"As we found" \quad n \ = -12, \ "we have:"`

`\qquad \qquad \quad \ "the smallest of these odd integers is:"`

`\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad = \ 2 ( -12 ) - 1`

`\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad = \ -24 - 1`

`\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad = \ -25.`

`"This is our answer:" \qquad \qquad \qquad -25. \qquad \qquad \qquad \qquad \qquad \qquad !!`