To determine the range of scores Alexandra needs on her last test to achieve an average between 80 and 85, we can follow these steps:
1. Identify the known test scores:
Alexandra's current test scores are 89, 77, and 81.
2. Calculate the total number of tests:
Including the upcoming last test, Alexandra will have completed 4 tests in total.
3. Set the target average range:
Alexandra wants her average to be between 80 and 85.
4. Calculate the sum of the current test scores:
The sum of 89, 77, and 81 is:
[tex]\[
89 + 77 + 81 = 247
\][/tex]
5. Formulate the mathematical expressions:
- To achieve a minimum average of 80:
[tex]\[
\frac{247 + a}{4} \geq 80
\][/tex]
- To achieve a maximum average of 85:
[tex]\[
\frac{247 + a}{4} \leq 85
\][/tex]
6. Solve for [tex]\( a \)[/tex] in both inequalities:
- For the minimum average of 80:
[tex]\[
\frac{247 + a}{4} \geq 80
\][/tex]
Multiply both sides by 4:
[tex]\[
247 + a \geq 320
\][/tex]
Subtract 247 from both sides:
[tex]\[
a \geq 73
\][/tex]
- For the maximum average of 85:
[tex]\[
\frac{247 + a}{4} \leq 85
\][/tex]
Multiply both sides by 4:
[tex]\[
247 + a \leq 340
\][/tex]
Subtract 247 from both sides:
[tex]\[
a \leq 93
\][/tex]
7. Combine the two conditions:
Alexandra's last test score [tex]\( a \)[/tex] must satisfy:
[tex]\[
73 \leq a \leq 93
\][/tex]
Therefore, the range of scores Alexandra needs on her last test to average between 80 and 85 is:
[tex]\[
73 \leq a \leq 93
\][/tex]
