Question

lamont keeps track of his math grades by recording them in a table. he wants to keep an average of 90, so he also lists the amount that each grade is above or below 90. a. complete the table. b. use the numbers in the above/below 90 row to find out whether lamonts average is above or below 90. show your work. c. what grade does lamont need to get on the next test to have an average of exactly 90? explain your answer. you may want to group the positive numbers and the negative numbers.

Explanation:

Step1: Calculate the sum of the current grades' differences from 90

Let the grades be \(g_1 = 83\), \(g_2=94\), \(g_3 = 79\), \(g_4=96\) with differences from 90 as \(d_1=83 - 90=- 7\), \(d_2=94 - 90 = 4\), \(d_3=79 - 90=-11\), \(d_4=96 - 90 = 6\). The sum of differences \(S=d_1 + d_2+d_3 + d_4=-7 + 4-11 + 6=-8\).

Step2: Determine the grade needed on the next test

Let the grade on the next test be \(x\). We want the average of all 5 grades to be 90. The sum of all 5 - grade differences from 90 should be 0. So, \(-8+(x - 90)=0\). Solving for \(x\):
\[

$$\begin{align*} -8+(x - 90)&=0\\ x-90&=8\\ x&=98 \end{align*}$$

\]

Answer:

Lamont needs to get a 98 on the next test to have an average of exactly 90.

For part b:

1. First, find the actual grades from the differences:

• For a difference of \(-7\), the grade is \(90-7 = 83\).
• For a difference of \(4\), the grade is \(90 + 4=94\).
• For a difference of \(-11\) (since \(79-90=-11\)), the grade is \(79\).
• For a difference of \(6\), the grade is \(90 + 6=96\).
• For a difference of \(-3\), the grade is \(90-3 = 87\).

1. Then, calculate the average of these grades:

• The sum of the grades \(S=83 + 94+79+96+87=439\).
• The average \(\bar{x}=\frac{439}{5}=87.8\). Since \(87.8<90\), Lamont's average is below 90.