Question
Healthy Gym charges members a fee for each visit, while Fitness Gym charges members a one-time yearly charge plus a fee for each visit. The equations below represent the amounts members pay at each gym, where t represents the total yearly cost and n represents the number of visits.
Healthy Gym: 6n=t
Fitness Gym: 50+5n=t
Which statement about the total yearly cost of the memberships is true?
(1 point)
Responses
The total yearly cost at Healthy Gym is about $50 lower than the total yearly cost at Fitness Gym.
The total yearly cost at Healthy Gym is about $50 lower than the total yearly cost at Fitness Gym.
The total yearly cost will always be higher at Fitness Gym than at Healthy Gym.
The total yearly cost will always be higher at Fitness Gym than at Healthy Gym.
The total yearly cost is $300 at each gym for 50 visits.
The total yearly cost is $300 at each gym for 50 visits.
The total yearly cost is the same at both gyms when members make 10 visits to each one.
The total yearly cost is the same at both gyms when members make 10 visits to each one.
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Healthy Gym: 6n=t
Fitness Gym: 50+5n=t
Which statement about the total yearly cost of the memberships is true?
(1 point)
Responses
The total yearly cost at Healthy Gym is about $50 lower than the total yearly cost at Fitness Gym.
The total yearly cost at Healthy Gym is about $50 lower than the total yearly cost at Fitness Gym.
The total yearly cost will always be higher at Fitness Gym than at Healthy Gym.
The total yearly cost will always be higher at Fitness Gym than at Healthy Gym.
The total yearly cost is $300 at each gym for 50 visits.
The total yearly cost is $300 at each gym for 50 visits.
The total yearly cost is the same at both gyms when members make 10 visits to each one.
The total yearly cost is the same at both gyms when members make 10 visits to each one.
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Answers
GPT-4o mini
To analyze the costs at Healthy Gym and Fitness Gym, we can use the given equations:
1. **Healthy Gym:**
\[ t = 6n \]
(This means the total cost \( t \) is $6 per visit.)
2. **Fitness Gym:**
\[ t = 50 + 5n \]
(This means the total cost \( t \) includes a one-time fee of $50 plus $5 for each visit.)
Now, let's evaluate the total yearly costs based on the number of visits \( n \).
### Total Cost for Different Visit Counts
1. **For 10 visits (\( n = 10 \)):**
- Healthy Gym:
\( t = 6 \times 10 = 60 \)
- Fitness Gym:
\( t = 50 + 5 \times 10 = 50 + 50 = 100 \)
- **Comparison:**
Healthy Gym costs $60 and Fitness Gym costs $100.
2. **For 50 visits (\( n = 50 \)):**
- Healthy Gym:
\( t = 6 \times 50 = 300 \)
- Fitness Gym:
\( t = 50 + 5 \times 50 = 50 + 250 = 300 \)
- **Comparison:**
Both gyms cost $300.
### Conclusion
Now we can evaluate the given statements in your options:
1. **"The total yearly cost at Healthy Gym is about $50 lower than the total yearly cost at Fitness Gym."**
- This is true for 10 visits but not for 50 visits.
2. **"The total yearly cost will always be higher at Fitness Gym than at Healthy Gym."**
- Not true; it is only true for certain visit counts.
3. **"The total yearly cost is $300 at each gym for 50 visits."**
- True, they both cost $300 at 50 visits.
4. **"The total yearly cost is the same at both gyms when members make 10 visits to each one."**
- Not true; Healthy Gym costs $60 and Fitness Gym costs $100 at 10 visits.
Based on this breakdown, the correct statement is:
**"The total yearly cost is $300 at each gym for 50 visits."**
1. **Healthy Gym:**
\[ t = 6n \]
(This means the total cost \( t \) is $6 per visit.)
2. **Fitness Gym:**
\[ t = 50 + 5n \]
(This means the total cost \( t \) includes a one-time fee of $50 plus $5 for each visit.)
Now, let's evaluate the total yearly costs based on the number of visits \( n \).
### Total Cost for Different Visit Counts
1. **For 10 visits (\( n = 10 \)):**
- Healthy Gym:
\( t = 6 \times 10 = 60 \)
- Fitness Gym:
\( t = 50 + 5 \times 10 = 50 + 50 = 100 \)
- **Comparison:**
Healthy Gym costs $60 and Fitness Gym costs $100.
2. **For 50 visits (\( n = 50 \)):**
- Healthy Gym:
\( t = 6 \times 50 = 300 \)
- Fitness Gym:
\( t = 50 + 5 \times 50 = 50 + 250 = 300 \)
- **Comparison:**
Both gyms cost $300.
### Conclusion
Now we can evaluate the given statements in your options:
1. **"The total yearly cost at Healthy Gym is about $50 lower than the total yearly cost at Fitness Gym."**
- This is true for 10 visits but not for 50 visits.
2. **"The total yearly cost will always be higher at Fitness Gym than at Healthy Gym."**
- Not true; it is only true for certain visit counts.
3. **"The total yearly cost is $300 at each gym for 50 visits."**
- True, they both cost $300 at 50 visits.
4. **"The total yearly cost is the same at both gyms when members make 10 visits to each one."**
- Not true; Healthy Gym costs $60 and Fitness Gym costs $100 at 10 visits.
Based on this breakdown, the correct statement is:
**"The total yearly cost is $300 at each gym for 50 visits."**