Question

on two examinations, you have grades of 81 and 96. there is an optional final examination, which counts as one grade. you decide to take the final in order to get a course grade of a, meaning a final average of at least 90.
a. what must you get on the final to earn an a in the course?
b. by taking the final, if you do poorly, you might risk the b that you have in the course based on the first two exam grades. if your final average is less than 80, you will lose your b in the course. describe the grades on the final that will cause this to happen.

Explanation:

Step1: Recall average formula

The average of three grades (the two existing grades and the final - exam grade) is calculated as $\text{Average}=\frac{\text{Sum of grades}}{\text{Number of grades}}$. Let the grade on the final exam be $x$. The sum of the three grades is $81 + 96+x$, and the number of grades is 3.

Step2: Set up inequality for part a

We want the average to be at least 90, so $\frac{81 + 96+x}{3}\geq90$. First, simplify the numerator: $81+96 = 177$. The inequality becomes $\frac{177 + x}{3}\geq90$. Multiply both sides of the inequality by 3: $177+x\geq270$. Then subtract 177 from both sides: $x\geq270 - 177$, so $x\geq93$.

Step3: Set up inequality for part b

We want the average to be less than 80, so $\frac{81 + 96+x}{3}<80$. Simplify the numerator: $81 + 96=177$. The inequality is $\frac{177 + x}{3}<80$. Multiply both sides by 3: $177+x<240$. Subtract 177 from both sides: $x<240 - 177$, so $x<63$.

Answer:

a. 93
b. Grades less than 63 on the final will cause you to lose your B in the course.