Question
Your test scores in one class are 83 and 89. What possible scores can you earn on your next test to have a test average between 86 and 90, inclusive? Your score on the third test must be between and inclusive. (Use ascending order.)
Answers
GPT 3.5
To calculate the average score, we sum all the scores and divide by the number of total tests. Let's call the score on the third test $x$.
The sum of the three scores is $83 + 89 + x$. The average is $\frac{83 + 89 + x}{3}$. To have an average between 86 and 90, inclusive, we need:
$$86 \leq \frac{83 + 89 + x}{3} \leq 90$$
Multiplying through by 3:
$$258 \leq 83 + 89 + x \leq 270$$
Adding $-83-89$ to all terms of the inequality:
$$86 \leq x \leq 98$$
So, the possible scores you can earn on your next test to have a test average between 86 and 90, inclusive, are in the range of $\boxed{86}$ to $\boxed{98}$.
The sum of the three scores is $83 + 89 + x$. The average is $\frac{83 + 89 + x}{3}$. To have an average between 86 and 90, inclusive, we need:
$$86 \leq \frac{83 + 89 + x}{3} \leq 90$$
Multiplying through by 3:
$$258 \leq 83 + 89 + x \leq 270$$
Adding $-83-89$ to all terms of the inequality:
$$86 \leq x \leq 98$$
So, the possible scores you can earn on your next test to have a test average between 86 and 90, inclusive, are in the range of $\boxed{86}$ to $\boxed{98}$.