Question

You have test scores of \(84,92,\) and \(76 .\) There will be one more test in the marking period. You want your mean test score for the marking period to be 85 or higher. a. Let \(x\) represent your last test score. Write an expression for the mean of your test scores for the marking period. b. Write and solve an inequality to find what you must score on the last test. c. Solve the problem without using algebra. Describe your method. Do your answers agree?

Short answer

To get a mean score of at least 85 for all tests, at least 88 must be scored on the last test.

Step-by-step solution

Expression for the Mean

The mean or the average of a set of numbers is calculated by adding all the numbers in the set and then dividing the sum by the count of numbers. In this case, we have three test scores and the upcoming test score 'x'. So, the mean of all test scores would be: \((84 + 92 + 76 + x) / 4\).

Formulate the Inequality

The problem wants a mean score of 85 or more. So, we set the mean expression from Step 1 equal to or greater than 85 and simplify the inequality: \((84 + 92 + 76 + x) / 4 \geq 85 \). This simplifies to \(252 + x \geq 340\) .

Solve the Inequality

To find the value of 'x', we subtract 252 from both sides, which gives us: \( x \geq 340 - 252 \), simplifying to \( x \geq 88 \). Hence, to have a mean of 85 or more, a score of at least 88 is required in the last test.

Solve without Algebra

For a confirmation, let's solve the problem without using algebra. We want a total of \(85 \times 4 = 340\) points from all 4 tests, as the mean wanted is 85. But we already have \(84 + 92 + 76 = 252\) points. Hence, we need \(340 - 252 = 88\) points on the fourth test to get a mean of 85. This confirms our previous result.