What is the average of all multiples of 5 from 5 to 50 inclusive?

Answer 1

Well, you can work it out by brute force by saying it's equal to:

(5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50) / 10

But that might be more counting than you'd care to do. Another way to do it is to realize that you can factor five out of all of those terms, and move it outside the brackets as a common coefficient:

5(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10) / 10

Which is a bit easier to solve, but what if we were given a bigger problem? What if we were asked for the average of all multiples of 5 between 5 and 500? That's far more addition than I'd like to do. We can solve it more easily though. Let's make it a bit more abstract; let's say:

"What is the average of the first N multiples of 5, from 5 to 5*N?"

Our equation then would look more like:

5(1 + 2 + 3 + ... + n - 2 + n - 1 + n) / n

That series in the middle can be converted to a simple expression though. The sum of all numbers from 1 to n is equal to n(n + 1) / 2. Our equation then becomes:

5[n(n + 1) / 2] / n

which we can simplify:

= 5n(n + 1) / 2n

= 5(n + 1) / 2

So the average of all multiples of 5 from 5 to 50 would be:

5(10 + 1) / 2

= 55 / 2

= 27.5

The beauty of this is that you an expand it to any number. For instance, what is the average of the first 20 multiples of seven? Why, that's easy! It'd be:

7(20 + 1) / 2

= 147 / 2

= 73.5