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Answer :
Great question! Let's break it down step-by-step to find which equation represents Justine’s scenario.
1. Understanding the Problem:
- Justine pays a one-time registration fee of \[tex]$45.
- She also pays a fee of \$[/tex]9 per lesson.
- The average cost per music lesson is \[tex]$12.
- She has taken \( z \) (let’s use \( z \) instead of \( x \) to match the question's terminology) lessons.
2. Setting Up the Scenario:
- The total cost Justine pays after \( z \) lessons is the sum of the registration fee and the cost for \( z \) lessons.
Total cost = Registration fee + (Fee per lesson × Number of lessons)
So, the total cost \( T \) is:
\[
T = 45 + 9z
\]
3. Average Cost per Lesson:
- The average cost per lesson is the total cost divided by the number of lessons.
\[
\text{Average cost per lesson} = \frac{T}{z} = \frac{45 + 9z}{z}
\]
4. Given Information:
- We know the average cost per lesson is \$[/tex]12.
Therefore, we set up the equation:
[tex]\[
12 = \frac{45 + 9z}{z}
\][/tex]
Now, we match this with the given options:
A. [tex]\( 12 = \frac{54}{2} \)[/tex]
- This does not relate to our scenario and is incorrect.
B. [tex]\( 12 = 45 + \frac{9}{z} \)[/tex]
- This also doesn’t match our equation and is incorrect.
C. [tex]\( 12 = \frac{45 + 9z}{z} \)[/tex]
- This is exactly what we derived and is correct.
D. [tex]\( 12 = \frac{45z + 9}{z} \)[/tex]
- This is not correct as it doesn’t represent the relationship correctly.
So, the correct answer is:
[tex]\[
\boxed{C}
\][/tex]
1. Understanding the Problem:
- Justine pays a one-time registration fee of \[tex]$45.
- She also pays a fee of \$[/tex]9 per lesson.
- The average cost per music lesson is \[tex]$12.
- She has taken \( z \) (let’s use \( z \) instead of \( x \) to match the question's terminology) lessons.
2. Setting Up the Scenario:
- The total cost Justine pays after \( z \) lessons is the sum of the registration fee and the cost for \( z \) lessons.
Total cost = Registration fee + (Fee per lesson × Number of lessons)
So, the total cost \( T \) is:
\[
T = 45 + 9z
\]
3. Average Cost per Lesson:
- The average cost per lesson is the total cost divided by the number of lessons.
\[
\text{Average cost per lesson} = \frac{T}{z} = \frac{45 + 9z}{z}
\]
4. Given Information:
- We know the average cost per lesson is \$[/tex]12.
Therefore, we set up the equation:
[tex]\[
12 = \frac{45 + 9z}{z}
\][/tex]
Now, we match this with the given options:
A. [tex]\( 12 = \frac{54}{2} \)[/tex]
- This does not relate to our scenario and is incorrect.
B. [tex]\( 12 = 45 + \frac{9}{z} \)[/tex]
- This also doesn’t match our equation and is incorrect.
C. [tex]\( 12 = \frac{45 + 9z}{z} \)[/tex]
- This is exactly what we derived and is correct.
D. [tex]\( 12 = \frac{45z + 9}{z} \)[/tex]
- This is not correct as it doesn’t represent the relationship correctly.
So, the correct answer is:
[tex]\[
\boxed{C}
\][/tex]
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