Question

You are planning to take a group of friends to the movies for your birthday. The local movie theater charges $12.99 per person. When you check the company's website, you find that they also offer a special group rate. The group rate is only $8.99 per person, but also includes a $20 reservation fee in addition.

What is the minimum number of people that would need to attend your party in order for the group rate to be a better deal?
Responses
A 4 people4 people
B 5 people5 people
C 6 people6 people
D 7 people7 people
E 8 people

Answer 1

wrong it was 6 people

Answer 2

To determine the minimum number of people needed for the group rate to be a better deal, we can set up an inequality comparing the total cost of both options.

1. **Cost without Group Rate:**
- Cost per person = $12.99
- Total cost for \( n \) people = \( 12.99n \)

2. **Cost with Group Rate:**
- Cost per person = $8.99
- Reservation fee = $20
- Total cost for \( n \) people = \( 8.99n + 20 \)

Now we need to find \( n \) such that the group rate cost is less than the regular cost:

\[
8.99n + 20 < 12.99n
\]

Subtract \( 8.99n \) from both sides:

\[
20 < 12.99n - 8.99n
\]

This simplifies to:

\[
20 < 3n
\]

Now, divide both sides by 3:

\[
\frac{20}{3} < n
\]

Calculating:

\[
\frac{20}{3} \approx 6.67
\]

Since \( n \) must be a whole number, we round up to the next whole number, which is 7.

Thus, the minimum number of people that need to attend for the group rate to be a better deal is **7**.

The correct response is:
**D 7 people**

Answer 3

Let's re-evaluate the calculation to clarify the situation.

### Cost Comparisons

1. **Cost without Group Rate:**
- Cost per person = $12.99
- Total cost for \( n \) people:
\[
\text{Cost}_{\text{normal}} = 12.99n
\]

2. **Cost with Group Rate:**
- Cost per person = $8.99
- Reservation fee = $20
- Total cost for \( n \) people:
\[
\text{Cost}_{\text{group}} = 8.99n + 20
\]

### Setting Up the Inequality

We want to find \( n \) such that the group rate is cheaper than the normal rate:

\[
8.99n + 20 < 12.99n
\]

Subtract \( 8.99n \) from both sides:

\[
20 < 12.99n - 8.99n
\]

This simplifies to:

\[
20 < 3n
\]

Now, divide by 3:

\[
n > \frac{20}{3} \approx 6.67
\]

Since \( n \) must be a whole number, we take the smallest integer greater than \( 6.67 \), which is **7**.

### Conclusion

The minimum number of people needed for the group rate to be a better deal is **7**, not 6.

However, if we check exactly:

- For \( n = 6 \):
- Cost without group rate: \( 12.99 \times 6 = 77.94 \)
- Cost with group rate: \( 8.99 \times 6 + 20 = 53.94 + 20 = 73.94 \)

At \( n = 6 \), the group rate cost \( 73.94 \) is less than the normal cost \( 77.94 \).

- For \( n = 5 \):
- Cost without group rate: \( 12.99 \times 5 = 64.95 \)
- Cost with group rate: \( 8.99 \times 5 + 20 = 44.95 + 20 = 64.95 \)

At \( n = 5 \), both costs are equal.

So, the group rate is better starting at **6 people**.

Thank you for your patience, and you were indeed correct; it is **6 people**.