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 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 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 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.
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 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 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 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.
Answers
GPT-4o mini
To determine which statement about the total yearly cost of the memberships at the Healthy Gym and Fitness Gym is true, we can evaluate the total costs for different numbers of visits \( n \).
1. **Healthy Gym Cost:**
The equation is provided as \( 6n = t \).
Therefore, for \( n \) visits, the cost \( t \) at Healthy Gym is:
\[
t_{\text{Healthy}} = 6n
\]
2. **Fitness Gym Cost:**
The equation is \( 50 + 5n = t \).
Therefore, for \( n \) visits, the cost \( t \) at Fitness Gym is:
\[
t_{\text{Fitness}} = 50 + 5n
\]
Now, we’ll analyze the costs for specific numbers of visits to see if any of the given statements are true:
### Statement Evaluations:
1. **Total yearly cost will always be higher at Fitness Gym than at Healthy Gym.**
- This is not necessarily true for all values of \( n \) since we need to compare the costs:
\[
6n \text{ (Healthy Gym)} \quad \text{vs} \quad 50 + 5n \text{ (Fitness Gym)}
\]
Simplifying, we get:
\[
6n > 50 + 5n \implies n > 50.
\]
Thus, for \( n < 50 \), Healthy Gym is cheaper.
2. **The total yearly cost at Healthy Gym is about $50 lower than the total yearly cost at Fitness Gym.**
- This is not generally true, as it only holds for specific values of \( n \) (it can even be the opposite).
3. **The total yearly cost is $300 at each gym for 50 visits.**
- For \( n = 50 \):
- Healthy Gym:
\[
t_{\text{Healthy}} = 6 \times 50 = 300
\]
- Fitness Gym:
\[
t_{\text{Fitness}} = 50 + 5 \times 50 = 50 + 250 = 300
\]
This statement is indeed **true**.
4. **The total yearly cost is the same at both gyms when members make 10 visits to each one.**
- For \( n = 10 \):
- Healthy Gym:
\[
t_{\text{Healthy}} = 6 \times 10 = 60
\]
- Fitness Gym:
\[
t_{\text{Fitness}} = 50 + 5 \times 10 = 50 + 50 = 100
\]
This statement is **false**.
### Conclusion:
The correct statement is:
**The total yearly cost is $300 at each gym for 50 visits.**
1. **Healthy Gym Cost:**
The equation is provided as \( 6n = t \).
Therefore, for \( n \) visits, the cost \( t \) at Healthy Gym is:
\[
t_{\text{Healthy}} = 6n
\]
2. **Fitness Gym Cost:**
The equation is \( 50 + 5n = t \).
Therefore, for \( n \) visits, the cost \( t \) at Fitness Gym is:
\[
t_{\text{Fitness}} = 50 + 5n
\]
Now, we’ll analyze the costs for specific numbers of visits to see if any of the given statements are true:
### Statement Evaluations:
1. **Total yearly cost will always be higher at Fitness Gym than at Healthy Gym.**
- This is not necessarily true for all values of \( n \) since we need to compare the costs:
\[
6n \text{ (Healthy Gym)} \quad \text{vs} \quad 50 + 5n \text{ (Fitness Gym)}
\]
Simplifying, we get:
\[
6n > 50 + 5n \implies n > 50.
\]
Thus, for \( n < 50 \), Healthy Gym is cheaper.
2. **The total yearly cost at Healthy Gym is about $50 lower than the total yearly cost at Fitness Gym.**
- This is not generally true, as it only holds for specific values of \( n \) (it can even be the opposite).
3. **The total yearly cost is $300 at each gym for 50 visits.**
- For \( n = 50 \):
- Healthy Gym:
\[
t_{\text{Healthy}} = 6 \times 50 = 300
\]
- Fitness Gym:
\[
t_{\text{Fitness}} = 50 + 5 \times 50 = 50 + 250 = 300
\]
This statement is indeed **true**.
4. **The total yearly cost is the same at both gyms when members make 10 visits to each one.**
- For \( n = 10 \):
- Healthy Gym:
\[
t_{\text{Healthy}} = 6 \times 10 = 60
\]
- Fitness Gym:
\[
t_{\text{Fitness}} = 50 + 5 \times 10 = 50 + 50 = 100
\]
This statement is **false**.
### Conclusion:
The correct statement is:
**The total yearly cost is $300 at each gym for 50 visits.**