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The average, or mean, [tex] T [/tex], of three exam grades, [tex] w, y [/tex], and [tex] r [/tex], is given by the following formula:

[tex] T = \frac{w + y + r}{3} [/tex]

(a) Solve the formula for [tex] r [/tex].

(b) Use the formula in part (a) to solve this problem: On your first two exams, your grades are 85% and 89% ([tex] w = 85 [/tex] and [tex] y = 89 [/tex]). What must you get on the third exam to have an average of 90%?



Answer :

GinnyAnswer
Alright, let's tackle the problem step-by-step.

### Part (a): Solve the formula [tex]\( T = \frac{w + y + r}{3} \)[/tex] for [tex]\( r \)[/tex].

1. Start with the given formula:
[tex]\[ T = \frac{w + y + r}{3} \][/tex]

2. To isolate [tex]\( r \)[/tex], first we need to get rid of the denominator, 3. Multiply both sides of the equation by 3:
[tex]\[ 3T = w + y + r \][/tex]

3. Next, isolate [tex]\( r \)[/tex] by subtracting [tex]\( w \)[/tex] and [tex]\( y \)[/tex] from both sides:
[tex]\[ r = 3T - w - y \][/tex]

So, the formula solved for [tex]\( r \)[/tex] is:
[tex]\[ r = 3T - w - y \][/tex]

### Part (b): Use the formula [tex]\( r = 3T - w - y \)[/tex] to find the third exam grade.

Given:
- Desired average [tex]\( T = 90 \% \)[/tex]
- First exam grade [tex]\( w = 85 \% \)[/tex]
- Second exam grade [tex]\( y = 89 \% \)[/tex]

Substitute the known values into the formula:
[tex]\[ r = 3 \cdot 90 - 85 - 89 \][/tex]

Let’s break this down step-by-step:
1. Calculate [tex]\( 3 \cdot 90 \)[/tex]:
[tex]\[ 3 \cdot 90 = 270 \][/tex]

2. Subtract the first exam grade [tex]\( 85 \% \)[/tex] from [tex]\( 270 \)[/tex]:
[tex]\[ 270 - 85 = 185 \][/tex]

3. Subtract the second exam grade [tex]\( 89 \% \)[/tex] from [tex]\( 185 \)[/tex]:
[tex]\[ 185 - 89 = 96 \][/tex]

So, the third exam grade [tex]\( r \)[/tex] you need to achieve to have an average of [tex]\( 90 \% \)[/tex] is:
[tex]\[ r = 96 \% \][/tex]

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