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Applications of Solving Equations

Mean Average Problems

Problems

Problems

The mean average (or simply the mean) of a group of values is the sum of the values divided by the total number of values. For example, the mean average of {5, 15, 25, 10, 15} is = = 14 .

It is often useful, given a group of values, to know what value will yield a certain average when added to the group. For example, we might ask, "What number, when added to the set {5, 15, 25, 10, 15} , will yield an average of 16?" or "For what value of x does the set {5, 15, 25, 10, 15, x} have an average of 16?"

To solve such a problem, simply set the new average equal to the desired average: = 16 . Remember that there is one more value now, so the denominator should be 1 greater. Then, solve the algebraic equation:

= 16
= 16
×6 = 16×6
70 + x = 96
70 + x - 70 = 96 - 70
x = 26

Check: = 16 ? Yes!


Example 1: Sally receives the following scores on her math tests: 78, 92, 83, 99 . What score does she need on the next test in order to have an average of 90 on her math tests?

= 90
= 90
×5 = 90×5
352 + x = 450
352 + x - 352 = 450 - 352
x = 98

Check: = 90 ? Yes!

Thus, Sally needs a 98 on her next math test.


Example 2: Sam receives the following scores on his English tests: 63, 84, 96 . What average score does he need on the last two tests in order to maintain an 85 average?

First, since only the sum of the last two scores matters when computing the overall average, one need only know the average of the last two scores, and the problem makes sense. Therefore, we can suppose that the average of the last two scores is x , and calculate as if both of the last two scores are equal to x .

= 85
= 85
×5 = 85×5
243 + 2x = 425
243 + 2x - 243 = 425 - 243
2x = 182
=
x = 91

Check: = 85 ? Yes!

Thus, Sam needs an average of 91 on his next two English tests.

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