To determine Michael's score on the fifth test, we need to use his test average and the scores of his first four tests. Here's a detailed step-by-step solution:
1. Identify the known values:
- Michael's test average [tex]\(\bar{x}\)[/tex] is 83.4.
- The scores of the first four tests are 82, 86, 91, and 74.
2. Calculate the sum of the first four test scores:
[tex]\[
82 + 86 + 91 + 74 = 333
\][/tex]
3. Determine the total sum of the scores required to achieve an average of 83.4 over 5 tests:
The formula for the average is given by:
[tex]\[
\bar{x} = \frac{x_1 + x_2 + \ldots + x_n}{n}
\][/tex]
Here, [tex]\(n = 5\)[/tex] (since there are 5 tests in total) and [tex]\(\bar{x} = 83.4\)[/tex].
So, the total sum required is:
[tex]\[
83.4 \times 5 = 417.0
\][/tex]
4. Calculate the score needed on the fifth test:
Let the score on the fifth test be [tex]\(x_5\)[/tex]. The total sum of all test scores should be equal to 417.0.
Using the equation:
[tex]\[
82 + 86 + 91 + 74 + x_5 = 417.0
\][/tex]
Since we've already summed the first four test scores to get 333, we can solve for [tex]\(x_5\)[/tex]:
[tex]\[
333 + x_5 = 417.0
\][/tex]
Solving for [tex]\(x_5\)[/tex]:
[tex]\[
x_5 = 417.0 - 333 = 84.0
\][/tex]
Therefore, Michael's score on the fifth test must be 84.
