n a class in which the final course grade depends entirely on the average of four equally weighted 100-point tests, Lisa has scored 94
, 85
, and 78
on the first three. What range of scores on the fourth test will give Lisa a B for the semester (an average between 80
and 89
, inclusive)? Assume that all test scores have a non-negative value.

Express your answer in interval notation.

To determine the range of scores that Lisa needs on her fourth test to achieve a B for the semester, we can follow these steps:

1. **Identify the average grade requirement for a B**:
   A B grade corresponds to an average score between 80 and 89, inclusive.

2. **Set up the equation for the average**:
   Let \( x \) represent Lisa's score on the fourth test. The average of her four test scores is given by:
   \[
   \text{Average} = \frac{94 + 85 + 78 + x}{4}
   \]
   We need this average to be between 80 and 89:
   \[
   80 \leq \frac{94 + 85 + 78 + x}{4} \leq 89
   \]

3. **Solve the inequality**:

   - First, multiply all parts of the inequality by 4 to eliminate the denominator:
     \[
     320 \leq 94 + 85 + 78 + x \leq 356
     \]

   - Simplify by summing the constants on the right-hand side:
     \[
     320 \leq 257 + x \leq 356
     \]

   - Finally, subtract 257 from all parts of the inequality:
     \[
     63 \leq x \leq 99
     \]

Thus, Lisa needs to score between **63 and 99** on her fourth test to get a B for the semester.

In interval notation, the answer is:

\[
[63, 99]
\]

Would you like more details or have any questions? Here are some related questions you might consider:

1. What would be the required score on the fourth test for Lisa to achieve an A (average of 90 or above)?
2. How would the interval change if one of the earlier test scores was lower or higher?
3. What if the tests were not equally weighted? How would that affect the calculation?
4. Can Lisa achieve a B if her fourth test score is exactly 62 or 100? Why or why not?
5. How would the calculation change if the grading scale had different cutoff points for a B?

**Tip**: When working with averages, it’s helpful to understand how each individual score impacts the final outcome, especially when weights are equal or differ.

Find out the range of scores Lisa needs on her fourth test to achieve a B grade for the semester. This math problem involves calculating averages and solving inequalities based on given test scores. Suitable for Grades 7-9.

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