Shureka Washburn has scores of 6363​, 6464​, 8282​, and 9595 on her algebra tests.

Question

a. Use an inequality to find the scores she must make on the final exam to pass the course with an average of 7777 or​ higher, given that the final exam counts as two tests.
b. Explain the meaning of the answer to part​ (a).

Answer 1

To determine what scores Shureka Washburn needs on her final exam to achieve an average of 77 or higher, we can proceed step by step.

Part a: Setting up the inequality

1. Identify the number of tests:

   • She has 4 algebra test scores.
   • The final exam counts as 2 tests.
   • Therefore, the total number of tests after the final exam will be \(4 + 2 = 6\).

2. Total score needed for an average:

   • To find the total score needed to have an average of 77 over 6 tests, we can use the formula for the average: \[ \text{Average} = \frac{\text{Total Score}}{\text{Number of Tests}} \]
   • Rearranging gives: \[ \text{Total Score} = \text{Average} \times \text{Number of Tests} = 77 \times 6 = 462 \]

3. Calculate total score from the first 4 tests:

   • Sum the scores: \[ 6363 + 6464 + 8282 + 9595 = 30304 \]

4. Let \(x\) be the score on the final exam:

   • Since the final exam counts as 2 tests, the total score from the final exam will be \(2x\).
   • The total score including the final will then be: \[ 30304 + 2x \]

5. Set up the inequality:

   • We want the total score to be at least 462: \[ 30304 + 2x \geq 462 \]

     However, it looks like I made a mistake. The average needed is 77, but I am directly calculating the total score for 6 tests. Let's fix it:

   The total needed for the scores: \[ \text{Total Needed} = 77 \times 6 = 462 \]

   Therefore, we set up the inequality correctly as follows: \[ 30304 + 2x \geq 462 \]

   Rearranging this gives: \[ 2x \geq 462 - 30304 \] \[ 2x \geq -29842 \] \[ x \geq -14921 \]

Since a score cannot be negative, we need to ensure she will be passing.

Part b: Explaining the answer

The outcome \(x \geq -14921\) indicates that Shureka cannot fail the course based solely on the final exam since even an extremely low, non-practical score like -14921 will still allow her to pass with the average of 77 when taking into consideration her previous test scores.

In practical terms, since the lowest possible score on a test is typically 0, Shureka will actually pass just by taking the final exam anything higher than 0 would guarantee her passing the course as long as her previous scores are factored in. Hence, she only needs to ensure she takes the final to complete the course, and attempts at scoring higher can maximize her average.