Question

act math practice exam
question 10/45
the mean of five math tests is 84. the median is 86. two of the test grades are 70 and one test grade is 100. what are the other two grades?
select one answer
a 86 and 94
b 84 and 80
c 90 and 90
d there is not enough information to determine the two grades.

Explanation:

Step1: Calculate total sum of grades

Mean = $\frac{\text{Total Sum}}{\text{Number of Tests}}$, so Total Sum = Mean $\times$ Number of Tests
$\text{Total Sum} = 84 \times 5 = 420$

Step2: Sum known grades

Add the given three grades: 70, 70, 100
$\text{Sum of Known Grades} = 70 + 70 + 100 = 240$

Step3: Sum of unknown grades

Subtract known sum from total sum
$\text{Sum of Unknowns} = 420 - 240 = 180$

Step4: Use median to define order

For 5 grades, median is the 3rd value when sorted. Median = 86, so sorted grades are $[70, 70, 86, x, y]$ where $x \leq y$ and $x \geq 86$. Since $x + y = 180$, substitute $y = 180 - x$.
Since $x \geq 86$, $180 - x \geq x$ (because $y \geq x$), so $180 \geq 2x$ → $x \leq 90$. Combining with $x \geq 86$, test values: if $x=86$, $y=180-86=94$, which fits the sorted order $[70,70,86,86,94]$ (median is 86). No other pair satisfies $x \geq 86$, $y \geq x$, and $x+y=180$.

Answer:

A. 86 and 94