In a class in which the final course grade depends entirely on the average of four equally weighted 100-point tests, Mark has scored 76, 94, and 77 on the first three. What range of scores on the fourth test will give Mark 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.

Mark's final grade is determined by the average of four equally weighted 100-point tests. To get a C for the semester, Mark's average score must be between 70 and 79, inclusive. 

Let \( x \) represent Mark's score on the fourth test.

### Step 1: Set up the average equation
Mark's average score for the four tests will be:
\[
\text{Average} = \frac{76 + 94 + 77 + x}{4}
\]
For Mark to earn a C, the average must satisfy the inequality:
\[
70 \leq \frac{76 + 94 + 77 + x}{4} \leq 79
\]

### Step 2: Solve the inequality
First, simplify the sum of the three scores:
\[
76 + 94 + 77 = 247
\]
So the inequality becomes:
\[
70 \leq \frac{247 + x}{4} \leq 79
\]
Now, multiply all parts of the inequality by 4 to eliminate the denominator:
\[
70 \times 4 \leq 247 + x \leq 79 \times 4
\]
\[
280 \leq 247 + x \leq 316
\]

### Step 3: Isolate \( x \)
Subtract 247 from all parts of the inequality:
\[
280 - 247 \leq x \leq 316 - 247
\]
\[
33 \leq x \leq 69
\]

### Step 4: Conclusion
Mark's score on the fourth test must be between 33 and 69, inclusive, for him to earn a C for the semester. Therefore, the range of scores is:

\[
x \in [33, 69]
\]

Would you like more details or have any questions?

Here are 5 related questions you might consider:
1. What happens if Mark scores below 33 on the fourth test?
2. What score does Mark need on the fourth test to get an average of exactly 75?
3. If the weight of the tests were not equal, how would this change the solution?
4. What range of scores would give Mark a B (average between 80 and 89)?
5. How does the number of tests affect the range for a desired average?

**Tip**: When solving inequalities involving averages, always check if multiplying or dividing by a positive number will simplify the equation without changing the inequality.

Determine the range of scores Mark needs on the fourth test to maintain a C grade (average between 70 and 79) in a class where the final grade depends on the average of four equally weighted tests. Explore inequalities and basic algebra concepts. Suitable for Grades 9-10 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation