Austin scored 85 on the calculus midterm. If the final exam counts twice as much as the midterm exam, then for what range of scores on the final would Austin get an average between 85 and 96? Both tests have a maximum of 100 points.

Austin should score between 85 and 100 in the final exam to have an average between 85 and 95, since reaching 96 is mathematically impossible.

Step-by-step explanation:

The problem has to be solved in two parts, the first to obtain an average of 85 and the second to obtain an average of 96.

In the first part with an average of 85 would be as follows.

Let x be the final exam grade.

Weighted average = (85 + 2 * X) / 3

85 = 85/3 + (2/3) * X

Rearranging

X = (3/2) * (85- (85/3))

Resolving

X = 85, therefore you must take 85 in the final test to get an average of 85.

The second part with an average of 96:

96 = 85/3 + (2/3) * X

Rearranging

X = (3/2) * (96- (85/3))

Resolving

X = 101.5, therefore you must take 101.5 in the final test to get an average of 96, therefore it is impossible to have an average of 96, because the highest score is up to 100.

Taking 100 your average would be:

Weighted average = (85 + 2 * 100) / 3 = 95

Then Austin should score between 85 and 100 in the final exam to have an average between 85 and 95, since reaching 96 is mathematically impossible.

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