Samir and Ben are planning a day trip with their friends. Samir wants to go to a waterpark that charges $40 per person for admission, plus $5 per person for tube rentals. Ben wants to go to a rock-climbing gym that charges $150 to rent the facility, plus an additional $20 per person.
Which equation can you use to find p, the number of people who would need to go on the trip for the two options to cost the same?

To find the number of people $ p $ for which the costs of both activities would be the same, we can set up an equation based on the costs of each option:

1. **Waterpark Cost**:
   - The cost per person for admission is $40.
   - The cost per person for tube rentals is $5.
   - The total cost per person for the waterpark is therefore $ 40 + 5 = 45 $.
   - The total cost for $ p $ people at the waterpark would be $ 45p $.

2. **Rock-climbing Gym Cost**:
   - The fixed cost to rent the facility is $150.
   - The cost per person is $20.
   - The total cost for $ p $ people at the gym is $ 150 + 20p $.

Now, set the two costs equal to each other to find the value of $ p $ when both options cost the same:

\[
45p = 150 + 20p
\]

This is the equation you can use to find $ p $.

Would you like me to solve for $ p $? 

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Here are 5 questions based on the problem:
1. How would the equation change if the waterpark had no tube rental fees?
2. What happens to the break-even point if the cost per person at the waterpark increases?
3. How does a fixed cost like the gym rental affect the total cost for large groups?
4. What if the rock-climbing gym charged a higher facility rental fee—how would that affect the number of people?
5. What if the waterpark offered a discount for larger groups? How would you model that?

**Tip**: Always check if there are fixed costs when comparing different pricing options—these often have a big impact for small groups!

Samir and Ben are planning a day trip with their friends. Samir wants to go to a waterpark that charges $40 per person for admission, plus $5 per person for tube rentals. Ben wants to go to a rock-climbing gym that charges $150 to rent the facility, plus an additional $20 per person. Which equation can you use to find p, the number of people who would need to go on the trip for the two options to cost the same?

Learn how to set up and solve a linear equation to compare the total costs of two different day trip options: a waterpark charging $45 per person versus a rock-climbing gym with a $150 fixed fee and $20 per person. This problem involves balancing a linear equation and is suitable for students in Grades 7-8.

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