Your test scores in one class are 81 and 87. What possible scores can you earn on your next test to have a test average between 84 and 99, inclusive?
Let us assume that the score of the next test will be denoted as $x.$ We are given that $\frac{81+87+x}{3}\geq84,$ which when simplified gives $200\geq x.$ We are also given that $\frac{81+87+x}{3}\leq99,$ which when simplified gives $x\leq123.$ Thus, all values of $x$ from $\boxed{200}$ to $\boxed{123}$ (inclusive) will give us a test average between 84 and 99.
To find the possible scores you can earn on your next test, you need to consider the current test scores and the desired test average.
Let's start by calculating the total possible score range. Since the average is between 84 and 99 inclusive, we can determine the total possible score range by multiplying the desired average (84) by the number of tests (3) and subtracting the sum of the current test scores (81 + 87).
Total possible score range = (desired average * number of tests) - sum of current test scores
Total possible score range = (84 * 3) - (81 + 87)
Total possible score range = 252 - 168
Total possible score range = 84
Now, to find the possible scores on your next test, subtract the range (84) from the maximum score (100).
Maximum possible score on next test = Maximum score - Total possible score range
Maximum possible score on next test = 100 - 84
Maximum possible score on next test = 16
So, the possible scores you can earn on your next test to have a test average between 84 and 99, inclusive, range from 0 to 16.
To determine the possible scores on your next test, you need to find the range of scores that will give you a test average between 84 and 99, inclusive.
Step 1: Calculate the minimum average:
To find the minimum average of 84, you can sum up the scores you have (81 + 87) and multiply the result by 3 (since you have taken two tests):
(81 + 87) * 3 = 504
Step 2: Calculate the maximum average:
To find the maximum average of 99, you can add the highest possible score to the total sum of the two previous test scores, then multiply the result by 3:
(81 + 87 + highest possible score) * 3 = 897
Step 3: Determine the possible scores:
To find the possible scores on the next test, you need to find all values within the range of 504 (minimum average) and 897 (maximum average), inclusive. So any score between 504 and 897, inclusive, will give you a test average between 84 and 99.
Therefore, the possible scores you can earn on your next test to have a test average between 84 and 99, inclusive, are any scores between 504 and 897, inclusive.