Let's solve the problem step by step.

We are given the test scores $77, 83, 52, 74$ and one remaining test score. The tasks involve finding the required final score based on two conditions.

Part (a):

The student needs a mean score of $70$ to earn a C.

1. The formula for the mean is: $\text{Mean} = \frac{\text{Sum of scores}}{\text{Number of tests}}$

2. To find the required final score, let the missing score be $x$. The mean should equal $70$ over 5 tests: $\frac{77 + 83 + 52 + 74 + x}{5} = 70$

3. Calculate the sum of the known scores: $77 + 83 + 52 + 74 = 286$

4. Substituting into the equation: $\frac{286 + x}{5} = 70$

5. Multiply through by $5$: $286 + x = 350$

6. Solve for $x$: $x = 350 - 286 = 64$

The student must score $64$ on the final test to earn a C.

Part (b):

The student needs a mean score of $80$ to earn a B.

1. The mean should equal $80$ over 5 tests: $\frac{286 + x}{5} = 80$

2. Multiply through by $5$: $286 + x = 400$

3. Solve for $x$: $x = 400 - 286 = 114$

The student must score $114$ on the final test to earn a B.

Final Answers:

(a) The student must score 64 to earn a C.
(b) The student must score 114 to earn a B.

Would you like further explanation on these calculations?

Related Questions:

1. How does the mean change with each additional test score?
2. What if the required mean for a C was 75 instead of 70?
3. What is the highest possible average if the final score is 100?
4. How does changing one of the previous test scores impact the final score needed?
5. Can the student still earn a B if the highest possible score on the final is capped at 100?

Tip:

Always double-check your arithmetic when summing or solving equations, as small errors can lead to incorrect conclusions!