Katherine earned 84, 92, 84, 75, and 70 on her first 5 tests.  What is the minimum grade Katherine needs to earn on the next test to have a mean of 84?

To find the minimum grade Katherine needs to earn on her next test to have a mean of 84, we can follow these steps:

### Step 1: Calculate the sum of the current test scores.
Katherine's current scores are: 84, 92, 84, 75, and 70.

\[
\text{Sum of the first 5 scores} = 84 + 92 + 84 + 75 + 70 = 405
\]

### Step 2: Express the mean of 6 tests including the unknown score.
Let \( x \) be the score Katherine needs on her next test to achieve a mean of 84.

The mean of 6 tests is given by:

\[
\text{Mean} = \frac{\text{Sum of all 6 scores}}{6} = 84
\]

\[
\frac{405 + x}{6} = 84
\]

### Step 3: Solve for \( x \).
Multiply both sides by 6 to clear the fraction:

\[
405 + x = 504
\]

Subtract 405 from both sides:

\[
x = 504 - 405 = 99
\]

### Conclusion:
Katherine needs to score a **minimum of 99** on her next test to have an average of 84.

Would you like to explore any other details or have any questions?

Here are 5 related questions:
1. What would the mean be if Katherine scores 80 on her next test?
2. What is the median of Katherine's first 5 test scores?
3. If Katherine scores 95 on her next test, what will her new average be?
4. How many additional tests with scores above 84 would she need to take to raise her average to 85?
5. What is the range of Katherine’s first 5 test scores?

**Tip:** When aiming for a specific mean, you can always reverse-engineer the total sum required and subtract the sum of known scores to find the necessary score for the remaining tests.

Find out the minimum grade Katherine needs on her next test to achieve an average of 84. Follow a step-by-step solution involving arithmetic mean and equations. Suitable for Grades 7-9.

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