Question

Computing Grades Going into the final exam, which will count as two tests, Brooke has test scores of 80,83,71,61 , and \(95 .\) What score does Brooke need on the final in order to have an average score of \(80 ?\)

Short answer

Brooke needs a score of 85 on the final exam.

Step-by-step solution

Determine the Number of Tests

Brooke has taken 5 tests so far and the final exam will count as 2 additional tests. So, the total number of tests, including the final as two tests, is 7.

Calculate the Current Total Score

Add the scores of the 5 tests: 80 + 83 + 71 + 61 + 95 = 390

Set Up the Equation for the Average

The desired average is 80. Let the score on the final exam be denoted as \(x\). Given that the final counts as two tests, the equation for the total score needed is: \frac{390 + 2x}{7} = 80

Solve for the Final Exam Score

First, multiply both sides of the equation by 7 to get rid of the fraction: 390 + 2x = 560. Then, subtract 390 from both sides: 2x = 170. Finally, divide by 2: x = 85.

Key concepts

test score

A test score is the numerical result a student receives after completing a test. It is generally expressed in percentage terms but can be in other forms like a fraction or points. For example, if a student answers 80 out of 100 questions correctly, the test score is 80%.
Test scores are used to measure a student's understanding and capabilities in various subjects. Brooke's scenario involves her needing to calculate her average score based on previous test results and a final exam score. Understanding test scores is the first step to solving such problems.

average

An average is a single value that represents the central point of a set of numbers. To compute an average, you sum up all the values and divide by the total number of values.
In Brooke's case, the average score she desires is 80. This means that the sum of all her test scores, including the final exam (counted twice), needs to be divided by the total number of tests to equal 80. The formula for an average is:
\[ \text{Average} = \frac{\text{Sum of all values}}{\text{Number of values}} \]
Breaking this down:

• Brooke has taken 5 tests
• The final exam will count as 2 tests, making it a total of 7 tests
• The total score needed for an average of 80 is then calculated using the formula\[ 80 = \frac{390 + 2x}{7} \]

equation solving

Equation solving involves finding the value of an unknown variable that makes the equation true. In Brooke's case, we need to solve for the score she needs on the final exam to achieve the desired average.
Given the equation \[ \frac{390 + 2x}{7} = 80 \]
we can solve it using the following steps:

• Multiply both sides of the equation by 7 to eliminate the fraction:
• \[ 390 + 2x = 560 \]
• Subtract 390 from both sides to isolate the term with the variable:
• \[ 2x = 170 \]
• Divide by 2 to solve for \( x \):
• \[ x = 85 \]

So, Brooke needs a score of 85 on the final exam, which counts as two tests, to reach an average score of 80.

final exam score

The final exam score is crucial in the calculation of the overall average. Since the final exam counts as two tests in Brooke's scenario, it's essential to double this score in the calculations.
When solving the problem, we set the final exam score as \( x \). Because it counts twice, we express it as \( 2x \) in the equation. This changes how we set up and solve for the average.
In summary, to meet the average of 80, Brooke needs a score of 85 on her final exam. This score will be counted twice, thereby adding a weighted significance to her overall average. Understanding how to factor in the final exam score accurately ensures Brooke correctly determines her needed performance level on the final exam.