Before taking his last test in a class, the arithmetic mean of Brian's test scores is 91. He has determined that if he scores 98 on his last test, the arithmetic mean of all his test scores will be exactly 92. How many tests, including the last test, does Brian take for this class?

To find the total number of tests Brian takes, we can use the information about his current average and how the average changes with the addition of the last test score.

Let's denote:
- \( n \) as the number of tests Brian has taken before the last test.
- \( S \) as the sum of all Brian's test scores before the last test.

Given:
- The current arithmetic mean of Brian's scores is 91.
- If he scores 98 on his last test, the new arithmetic mean becomes 92.

First, calculate the sum of Brian's scores before the last test using the mean:

\[
S = 91n
\]

After taking the last test, which is his \( n+1 \)-th test, his total score becomes \( S + 98 \). The new mean of all his test scores will be:

\[
\frac{S + 98}{n + 1} = 92
\]

Now substitute \( S = 91n \) into this equation:

\[
\frac{91n + 98}{n + 1} = 92
\]

To solve for \( n \), first clear the fraction by multiplying both sides by \( n + 1 \):

\[
91n + 98 = 92(n + 1)
\]

Expand and simplify:

\[
91n + 98 = 92n + 92
\]

Subtract \( 91n \) and 92 from both sides:

\[
98 - 92 = 92n - 91n
\]

\[
6 = n
\]

So, Brian has taken \( n = 6 \) tests before the last one. Including the last test, the total number of tests Brian takes is:

\[
n + 1 = 6 + 1 = 7
\]

**Therefore, Brian takes a total of 7 tests for this class.**

Would you like more details, or do you have any questions?

Here are five related questions you might find interesting:
1. How would the solution change if the final score was different from 98?
2. What if Brian wanted his average to be 93 instead of 92? How many tests would he need to take?
3. How would the mean change if Brian had taken one more test with a lower score?
4. What is the impact of each additional test score on the arithmetic mean?
5. How do you calculate the mean if the scores have different weights?

**Tip:** When working with averages, remember that adding more data points can change the mean depending on how each new data point compares to the current average.

Explore how Brian determines the number of tests required to achieve a specific arithmetic mean with his test scores. Learn the step-by-step solution involving arithmetic mean formulas and linear equations. Suitable for students in Grades 9-12.

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