To join a dance lesson, you paid a one-time membership fee of 5 000, then 500 for every session. If they gave you one free session after every ten sessions and you only have attended a number of paid sessions that is a multiple of 10, determine the function that represents the average cost per session and identify if the function is a rational function.

To solve this problem, we will first establish the total cost and the total number of sessions attended, and then compute the function for the average cost per session.

### Step 1: Define the total cost for attending \( n \) sessions.

- You pay a one-time membership fee of 5,000.
- Each session costs 500, but after every 10 paid sessions, you receive one free session. This means for every 10 paid sessions, you attend 11 sessions in total (10 paid + 1 free).

Let \( n \) represent the number of paid sessions. Since you receive 1 free session after every 10 paid sessions, the total number of sessions you actually attend is:

\[
\text{Total sessions attended} = n + \frac{n}{10} = n \times \left(1 + \frac{1}{10}\right) = \frac{11n}{10}
\]

### Step 2: Compute the total cost.

The total cost includes the one-time membership fee and the cost of the \( n \) paid sessions:

\[
\text{Total cost} = 5000 + 500n
\]

### Step 3: Derive the average cost per session.

The average cost per session is given by dividing the total cost by the total number of sessions attended:

\[
\text{Average cost per session} = \frac{\text{Total cost}}{\text{Total sessions attended}} = \frac{5000 + 500n}{\frac{11n}{10}} = \frac{(5000 + 500n) \times 10}{11n}
\]

Simplifying this expression:

\[
\text{Average cost per session} = \frac{50000 + 5000n}{11n}
\]

### Step 4: Is the function a rational function?

Yes, the average cost per session is a rational function. A rational function is defined as a ratio of two polynomials. In this case:

\[
\text{Average cost per session} = \frac{50000 + 5000n}{11n}
\]

Both the numerator and the denominator are polynomials, so this is a rational function.

### Final Answer:

The function that represents the average cost per session is:

\[
f(n) = \frac{50000 + 5000n}{11n}
\]

It is a rational function.

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Do you want more details or have any questions? Here are some related questions:

1. What is the average cost per session if you have attended 20 paid sessions?
2. How does the average cost per session change as you attend more sessions?
3. How would the function change if the free session was given after every 5 paid sessions instead of 10?
4. What is the average cost per session if there was no membership fee and only session costs applied?
5. How can you graph the function and interpret its behavior?

**Tip**: For rational functions, check for asymptotes or limits to understand long-term behavior.

Discover how to calculate the average cost per session when joining a dance lesson, with a membership fee and free sessions after every ten. This solution explains the concept of rational functions and breaks down the cost function for students in Grades 10-12.

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