# Average of numbers in a specific range

The question is :

What is the arithmetic mean of all multiples of 10 from 10 to 190 inclusive.

Now I know how many nos there are by using $\frac{190-10}{10}+1 = 19$ but how do I get their sum ?

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Try dividing every term in the sum by 10! – Tom Hutchcroft Jul 24 '12 at 4:49
@TomHutchcroft I dont understand how ? I dont know the sum ? – Rajeshwar Jul 24 '12 at 4:51
Well, the answer is 10*(1+2+3+...+19). Summing the numbers from 1 to n is standard, entbut if you haven't seen it before, try paring up the first and last number (1 + 19 = 20) then the second and second-to-last (2+18=20) and so on, each time we get the same answer, 20. There are 19/2 pairs (10 is half a pair) so we get 19/2*20=190 – Tom Hutchcroft Jul 24 '12 at 4:57
Thanks that helps – Rajeshwar Jul 24 '12 at 4:59

180+10 = 190; 170+20 = 190; 160+30 = 190; ...

how many numbers are there? 18.. we always add two numbers, so the sum is 190 * 9. now we add the last odd number - 190. so it's 190*10 -> 1900

gauss came up with that. pretty clever. thanks to him, you can do this in your head.

now the mean is just the sum divided by the number of parts, no?

we have 19 parts, and a sum of 1900.

1900/19..

holdon, lemme get muh calculatorz.

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Lolz.. Thanks for clearing that up – Rajeshwar Jul 24 '12 at 4:55

Between two numbers say $a$ and $b$ the arithmetic mean is $\frac{a+b}{2}$. For multiples of any number between two end multiples $a$ and $b$ the arithmetic mean is still $\frac{a+b}{2}$.

In your question 10 and 190 both are multiples of 10 . The arithmetic mean of all multiples of 10 between these two numbers is $\frac{10+190}{2}$ . For any other number say 5; 10 and 190 are again the multiples of 5; so the multiples of 5 between 10 and 190 would again average $\frac{10+190}{2}$.

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