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: 80+4n=t
Which statement about the total yearly cost of the memberships is true?
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature twice per question.
(1 point)
Responses
The total yearly cost at Healthy Gym is about $80 lower than the total yearly cost at Fitness Gym.
The total yearly cost at Healthy Gym is about $80 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 $240 at each gym for 40 visits.
The total yearly cost is $240 at each gym for 40 visits.
The total yearly cost is the same at both gyms when members make 25 visits to each one.
Healthy Gym: 6n=t
Fitness Gym: 80+4n=t
Which statement about the total yearly cost of the memberships is true?
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before moving on to the next question. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature twice per question.
(1 point)
Responses
The total yearly cost at Healthy Gym is about $80 lower than the total yearly cost at Fitness Gym.
The total yearly cost at Healthy Gym is about $80 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 $240 at each gym for 40 visits.
The total yearly cost is $240 at each gym for 40 visits.
The total yearly cost is the same at both gyms when members make 25 visits to each one.
Answers
GPT-4o mini
To assess the statements about the total yearly cost of memberships at Healthy Gym and Fitness Gym, we will analyze the equations for each gym and compare the costs for various numbers of visits.
1. **Healthy Gym's Total Cost Equation**:
\[
t = 6n
\]
This represents a cost of $6 per visit.
2. **Fitness Gym's Total Cost Equation**:
\[
t = 80 + 4n
\]
This represents a one-time fee of $80 plus $4 per visit.
Now we can evaluate the total yearly cost at both gyms for the suggested number of visits:
### Cost for 40 visits:
1. **Healthy Gym**:
\[
t = 6 \times 40 = 240
\]
2. **Fitness Gym**:
\[
t = 80 + 4 \times 40 = 80 + 160 = 240
\]
So for 40 visits, both gyms charge $240, making the statement **"The total yearly cost is $240 at each gym for 40 visits."** true.
### Check for 25 visits:
1. **Healthy Gym**:
\[
t = 6 \times 25 = 150
\]
2. **Fitness Gym**:
\[
t = 80 + 4 \times 25 = 80 + 100 = 180
\]
For 25 visits, Healthy Gym costs $150 and Fitness Gym costs $180, which means the costs are not the same.
### Comparison at arbitrary visits:
- When \( n = 0 \):
- Healthy Gym: \( t = 0 \)
- Fitness Gym: \( t = 80 \) (Fitness Gym has a higher cost due to the base fee)
As \( n \) increases, Healthy Gym's costs will always be lower than Fitness Gym's once \( n \) is above a certain number, because the per-visit cost of Healthy Gym ($6) is higher than that of Fitness Gym ($4) after a certain point.
### Conclusion:
1. **Healthy Gym is cheaper except at very low visits.**
2. The statement regarding 40 visits where both charge $240 is **correct**: **"The total yearly cost is $240 at each gym for 40 visits."**
So, the correct statement from the options is:
- **The total yearly cost is $240 at each gym for 40 visits.**
1. **Healthy Gym's Total Cost Equation**:
\[
t = 6n
\]
This represents a cost of $6 per visit.
2. **Fitness Gym's Total Cost Equation**:
\[
t = 80 + 4n
\]
This represents a one-time fee of $80 plus $4 per visit.
Now we can evaluate the total yearly cost at both gyms for the suggested number of visits:
### Cost for 40 visits:
1. **Healthy Gym**:
\[
t = 6 \times 40 = 240
\]
2. **Fitness Gym**:
\[
t = 80 + 4 \times 40 = 80 + 160 = 240
\]
So for 40 visits, both gyms charge $240, making the statement **"The total yearly cost is $240 at each gym for 40 visits."** true.
### Check for 25 visits:
1. **Healthy Gym**:
\[
t = 6 \times 25 = 150
\]
2. **Fitness Gym**:
\[
t = 80 + 4 \times 25 = 80 + 100 = 180
\]
For 25 visits, Healthy Gym costs $150 and Fitness Gym costs $180, which means the costs are not the same.
### Comparison at arbitrary visits:
- When \( n = 0 \):
- Healthy Gym: \( t = 0 \)
- Fitness Gym: \( t = 80 \) (Fitness Gym has a higher cost due to the base fee)
As \( n \) increases, Healthy Gym's costs will always be lower than Fitness Gym's once \( n \) is above a certain number, because the per-visit cost of Healthy Gym ($6) is higher than that of Fitness Gym ($4) after a certain point.
### Conclusion:
1. **Healthy Gym is cheaper except at very low visits.**
2. The statement regarding 40 visits where both charge $240 is **correct**: **"The total yearly cost is $240 at each gym for 40 visits."**
So, the correct statement from the options is:
- **The total yearly cost is $240 at each gym for 40 visits.**