What is the average (arithmetic mean) of all the multiples of ten from 10 to 190 inclusive?

1 Answer
Anjali G
Nov 25, 2016

The average is 100.

Explanation:

The simplest method:
Take the average of the outermost two numbers. This will be the same as the average of all 19 numbers, because the numbers in between are evenly spaced out, in a symmetrical pattern.
#frac{10+20+...+180+190}{19}=frac{10+190}{2}#
#=200/2#
#=100#

The longer method would be to expand the expression, and solve manually, or using Gauss's method:
#frac{10+20+30+40+50+60+70+80+90+100+110+120+130+140+150+160+170+180+190}{19}#

Factor a 10 out of the numerator:
#=frac{10(1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19)}{19}#

To add the whole numbers from 1 to 19, use Gauss's method:
#"Sum from 1 to n" = (n+1)(n/2)#
#"Sum from 1 to 19"=(19+1)(19/2)#
#=(20)(9.5)#
#=190#

So, back to the expression:
#frac{(10)(1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19)}{19}#
#=frac{(10)(190)}{19}#
#=100#

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