In a class in which the final course grade depends entirely on the average of four equally weighted 100-point tests, Harold has scored 94
94
, 85
85
, and 80
80
on the first three. What range of scores on the fourth test will give Harold a C for the semester (an average between 70
70
 and 79
79
, inclusive)? Assume that all test scores have a non-negative value.

Express your answer in interval notation.

To determine what range of scores on the fourth test will give Harold a final average between 70 and 79, we need to first set up the equation for the average of his four test scores.

The formula for the average of four scores is:

\[
\text{Average} = \frac{\text{Score}_1 + \text{Score}_2 + \text{Score}_3 + \text{Score}_4}{4}
\]

Harold's first three scores are 94, 85, and 80. Let \( x \) represent the score on his fourth test. The average can then be expressed as:

\[
\text{Average} = \frac{94 + 85 + 80 + x}{4}
\]

We are given that Harold needs an average between 70 and 79, inclusive. Therefore, we can set up the following inequality:

\[
70 \leq \frac{94 + 85 + 80 + x}{4} \leq 79
\]

### Step 1: Multiply the entire inequality by 4 to eliminate the denominator:

\[
70 \times 4 \leq 94 + 85 + 80 + x \leq 79 \times 4
\]

This simplifies to:

\[
280 \leq 259 + x \leq 316
\]

### Step 2: Subtract 259 from all parts of the inequality:

\[
280 - 259 \leq x \leq 316 - 259
\]

\[
21 \leq x \leq 57
\]

Thus, the range of scores on the fourth test that will give Harold a C for the semester is \( [21, 57] \) in interval notation.

Would you like more details or have any questions?

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Here are 5 related questions:
1. How would the solution change if Harold wanted a B (average between 80 and 89)?
2. What if the first three scores were different, such as 90, 78, and 85?
3. How does the concept of weighted averages apply to situations where tests have unequal weights?
4. Can the fourth test score be negative in any scenario in this problem?
5. What happens to the score range if Harold had already scored 70 in the previous tests?

**Tip:** When working with averages, always ensure that you carefully manage both the total score and the number of items being averaged.

In a class in which the final course grade depends entirely on the average of four equally weighted 100-point tests, Harold has scored 94, 85, and 80 on the first three. What range of scores on the fourth test will give Harold a C for the semester (an average between 70 and 79, inclusive)?

Learn how to calculate the range of Harold's fourth test score to maintain a C average in a class where all test scores are equally weighted. Using algebraic concepts like averages and inequalities, this problem helps students practice solving real-world math problems. Suitable for Grades 6-8.

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