What isthe sum of the two-digit multiples of 11?

Question

Mathcounts

Answer 1

In elementary school, Carl Friedrich Gauss was given a problem to keep him busy -- add up the numbers from 1 to 100. He quickly realized that 1+100 = 2+99 = 3+97  = ... = 101 and there are 100 such sums.  This gives twice the desired answer, so, the sum of the numbers from 1 to n is:

     n*(n+1)/2

 

The 2-digit multiples of 11 are 11*1, 11*2,  etc.  This problem is:

        11 + 22 + 33 + 44 + 55 + 66 + 77 + 88 + 99

= 11( 1 +   2 +   3  +  4  +  5  +  6 + 7 + 8  + 9)

= 11(9)(10)/2

= 990/2

= 495

Answer 2

This is a good problem to practice mental math.

You may be able to do most, if not all, of the following steps in your head:

 

11 + 22 + 33 + 44 + 55 + 66 + 77 + 88 + 99 =

11 + 99 + 22 + 88 + 33 + 77 + 44 + 66 + 55 =

110 + 110 + 110 + 110 + 55 =

440 + 55 = 495