# 30MSC 2023 #11: Using a Base

Using a base is very helpful in computing averages. The conventional method is to add all the values and  then divide the total by the number of values to get the average, which is called the Arithmetic Mean.

For two numbers, their average is the value midway between them. The average of 100 and 150 is computed by adding 100 and 150 and then dividing  their sum by 2 or (100 + 150)/2 = 250/2 = 125.

But in getting the average of 567,100 and 567,150, adding the two values before dividing them by two would be time consuming. We only note that their difference is 50 and half of 50 is 25. Their average then is 567,100 + 25 = 567,150.

Here, 567,100 is the base and 25 is the average of the differences of the numbers (0 and 50) from the base.

This technique can be used to quickly solve this problem given in the Juniors group of the 2nd International Vedic Mathematics Olympiad (IVMO 2022):

What is the mean (average) of 8283, 8294, 8279, 8276 and 8288?

We can see that all the values are near 8280 so we can use that value as the base. Next we add the differences of the values from the base (3 + 14 – 1 – 4 + 8) = 20 and 20/5 = 4. Thus the average of those values is 8284.