Question

A piano instructor charges students a one-time fee for sheet music and an hourly rate for lessons. The graph below shows the total cost for a student who has taken x lessons.

Total Cost of Piano Lessons

A graph has lessons on the x-axis, and dollars on the y-axis. Points are at (1, 50), (2, 80), (3, 110), (4, 140), (5, 170).

To increase her number of students, the instructor put a coupon in the local paper offering to waive the one-time fee for new students. What is the value of the coupon?
$14
$20
$30
$50

Answer 1

To find the value of the coupon, we need to determine the one-time fee the piano instructor charges for sheet music.

From the graph provided, we have the following data points for the total cost after different numbers of lessons:

- (1, 50): Cost after 1 lesson is $50
- (2, 80): Cost after 2 lessons is $80
- (3, 110): Cost after 3 lessons is $110
- (4, 140): Cost after 4 lessons is $140
- (5, 170): Cost after 5 lessons is $170

Let's assume the one-time fee for sheet music is \( f \) and the hourly rate for lessons is \( r \). Therefore, we can represent the total cost after \( x \) lessons as:

\[
\text{Total Cost} = f + r \cdot x
\]

From the values given:

1. For \( x = 1 \):
\[
f + r \cdot 1 = 50 \quad \text{(Equation 1)}
\]

2. For \( x = 2 \):
\[
f + r \cdot 2 = 80 \quad \text{(Equation 2)}
\]

Now, we can subtract Equation 1 from Equation 2 to eliminate \( f \):

\[
(f + r \cdot 2) - (f + r \cdot 1) = 80 - 50
\]
\[
r = 30
\]

Now that we have found the hourly rate \( r = 30 \), we can substitute it back into Equation 1 to find the one-time fee \( f \):

\[
f + 30 \cdot 1 = 50
\]
\[
f + 30 = 50
\]
\[
f = 50 - 30 = 20
\]

Thus, the one-time fee for sheet music is \( f = 20 \).

Since the coupon waives this one-time fee for new students, the value of the coupon is:

\[
\boxed{20}
\]