Question

maria scored 72, 97, and 82 on her first three math tests. she wants to have a mean score of 82 for the quarter. how many points does she need on her last test to earn an 82 for the quarter?
○ 62.75
○ 77
○ 82
○ 83.25

Explanation:

Step1: Recall the formula for the mean

The formula for the mean (average) of a set of numbers \(x_1,x_2,\cdots,x_n\) is \(\bar{x}=\frac{x_1 + x_2+\cdots+x_n}{n}\). Here, \(n = 4\) (since there are 4 tests in total), and \(\bar{x}=82\). Let the score on the fourth test be \(x\).

Step2: Set up the equation using the mean formula

We know that \(\frac{72 + 97+82+x}{4}=82\). First, simplify the numerator of the left - hand side: \(72 + 97+82+x=(72 + 97+82)+x=251+x\). So our equation becomes \(\frac{251 + x}{4}=82\).

Step3: Solve the equation for \(x\)

Multiply both sides of the equation \(\frac{251 + x}{4}=82\) by 4: \(251+x=82\times4\). Calculate \(82\times4 = 328\). Then, subtract 251 from both sides: \(x=328 - 251\).

Step4: Calculate the value of \(x\)

\(x=328-251 = 77\).

Answer:

77