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

2 Answers
Feb 27, 2018

-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.

Feb 27, 2018

"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 !!