Changing Averages To Sums

On standardized tests such as the ACT, SAT, and GRE, a problem involving averages often becomes much easier when we first convert the averages to sums. We can easily change an average to a sum using the following simple formula.

Sum = Average · Number

Many problems with averages involve one or more conversions to sums, followed by a subtraction.

Try to answer the following question using this strategy. Do not check the solution until you have attempted this question yourself.

Example 1

The average (arithmetic mean) of three numbers is 100. If two of the numbers are 80 and 130, what is the third number?

(A) 70
(B) 80
(C) 90
(D) 100
(E) 110

In this case we are averaging 3 numbers. So the Number is 3. The Average is given to be 100. Thus, the Sum of the 3 numbers is 10 · 3 = 300. Since two of the numbers are 80 and 130, the third number is 300 – 80 – 130 = 90, choice (C).

Got it? Well here is another, slightly more advanced, problem where this technique is useful:

Example 2

The average of x, y, z  and w is 15 and the average of z and w is 11. What is the average of x and y?

The Sum of x, y, z and w is 15 · 4 = 60. The Sum of z and w is 11 · 2 = 22. Thus, the Sum of x and y is 60 – 22 = 38. Finally the Average of x and y is 38/2 = 19.

Notice how we actually used the formula “Sum = Average · Number” twice here. For those of you that require further clarification, here are the calculations in more detail.

So (x +y)/2 = 38/2 = 19.

Thoughts on this? Still have questions? Post them on my Facebook wall:

If you liked this article, please share it with your Facebook friends:

Thank you all for your support!

Comments

comments