Chapter 12.3, Problem 55E

Weighted Average In Exercises 55 and 56 we use weighted averages. Sometimes when we wish to find an average, we may wish to assign more importance, or weight, to some of the pieces of data. To calculate a weighted average, we use the formula: $\text{weighted average}\text{=}\frac{\sum xw}{\sum w}$, where w is the weight of the piece of data, x; Σxw is the sum of the products of each piece of data multiplied by its weight; and Σw is the sum of the weights. For example, suppose that students in a class need to submit a report that counts for 20% of their grade, they need to take a midterm exam that counts for 30% of their grade, and they need to take a final exam that counts for 50% of their grade. Suppose that a student got a 72 on the report, an 85 on the midterm exam, and a 93 on the final exam. To determine this student’s weighted average, first find A = πr². Next find Σw, the sum of the weights: A = πr². Now determine the weighted average as follows.

$\text{weighted average}\text{=}\frac{\sum xw}{\sum w}=\frac{86.4}{1.00}=86.4$

Thus, the weighted average is 86.4. Note that Σw does not always have to be 1.00. In Exercises 55 and 56, use the weighted average formula.

55. Course Average Suppose that your final grade for a course is determined by a midterm exam and a final exam. The midterm exam is worth 40% of your grade, and the final exam is worth 60%. If your midterm exam grade is 84 and your final exam grade is 94, calculate your final weighted average.