Question

An amusement park that you are excited to try is $28 per visit.

You have the option to purchase a monthly membership for $90 and then pay $15 for each visit.

How many visits would be approximately the break even point where both options would cost about the same?

Answer 1

To find the break-even point, we need to set up equations to represent the total cost for each option based on the number of visits.

1. **Cost for single visits**:
\[
\text{Cost}_{\text{single}} = 28 \cdot x
\]
where \( x \) is the number of visits.

2. **Cost for the monthly membership**:
\[
\text{Cost}_{\text{membership}} = 90 + 15 \cdot x
\]

To find the break-even point, we set the two cost equations equal to each other:
\[
28x = 90 + 15x
\]

Now, we will solve for \( x \):
1. Subtract \( 15x \) from both sides:
\[
28x - 15x = 90
\]
\[
13x = 90
\]

2. Divide both sides by 13:
\[
x = \frac{90}{13} \approx 6.92
\]

Since the number of visits must be a whole number, we can round \( 6.92 \) to either 6 or 7 visits.

- **Calculating costs for 6 visits**:
- Single visits: \( 28 \cdot 6 = 168 \)
- Membership: \( 90 + 15 \cdot 6 = 90 + 90 = 180 \)

- **Calculating costs for 7 visits**:
- Single visits: \( 28 \cdot 7 = 196 \)
- Membership: \( 90 + 15 \cdot 7 = 90 + 105 = 195 \)

Thus, at 6 visits, the membership is more expensive, while at 7 visits, the membership becomes less expensive.

Therefore, the approximate break-even point is at **7 visits**. Beyond this point, the monthly membership becomes the more cost-effective option.