Question

In a class in which the final course grade depends entirely on the average of four equally weighted 100-point tests, Sarah has scored 86,85, and 84 on the first three. What range of scores on the fourth test will give Sarah a C for the semester (an average between 70 and 79, inclusive)? Assume that all test scores have a non-negative value.
Express your answer in interval notation.

Answer 1

Using the mean concept, it is found that scores in the interval [25,61] will give Sarah a C for the semester.

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• The mean of a data-set is the sum of all it's observations divided by the number of observations.

• There are 4 observations, the scores of 86, 85, 84 and x.
• Thus, the equation for the mean is:

$M = \frac{86 + 85 + 84 + x}{4} = \frac{255 + x}{4}$

For a mean of 70, that is, M = 70.

$70 = \frac{255 + x}{4}$

$255 + x = 280$

$x = 25$

For a mean of 79, that is, M = 79.

$79 = \frac{255 + x}{4}$

$255 + x = 316$

$x = 61$

Scores between 25 and 61 she will have a C grade, thus the interval is [25,61].

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