Question

The table below shows the cost for a movie theater membership with an initial start-up fee charged in the first month.
Number of months. cost
1 75
2 105
3 135

the cost per mouth is $__
the initial fee is $___

Answer 1

To determine the cost per month and the initial fee from the information given in the table, we can analyze the data.

From the table:

- For 1 month, the cost is $75.
- For 2 months, the cost is $105.
- For 3 months, the cost is $135.

### Initial Fee Calculation:

First, we can deduce the initial fee by setting up an equation. Let's assume:
- \( I \) = Initial fee
- \( C \) = Monthly cost

From the data:
1. For 1 month: \( I + C = 75 \)
2. For 2 months: \( I + 2C = 105 \)
3. For 3 months: \( I + 3C = 135 \)

### Solving the equations:

From the first equation, we can express the cost per month \( C \) in terms of the initial fee \( I \):
\[ C = 75 - I \]

Now, substitute \( C \) in the second equation:
\[ I + 2(75 - I) = 105 \]
\[ I + 150 - 2I = 105 \]
\[ -I + 150 = 105 \]
\[ -I = 105 - 150 \]
\[ -I = -45 \]
\[ I = 45 \]

Now that we have the initial fee, we can find the monthly cost:
\[ C = 75 - I \]
\[ C = 75 - 45 = 30 \]

### Final Results:

- The cost per month is **$30**.
- The initial fee is **$45**.