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.
A. The total yearly cost at Healthy Gym is about $80 lower than the total yearly cost at Fitness Gym.
B. The total yearly cost will always be higher at Fitness Gym than at
Healthy Gym.
C. The total yearly cost is the same at both gyms when members make 25 visits to each one.
D. The total yearly cost is $240 at each gym for 40 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.
A. The total yearly cost at Healthy Gym is about $80 lower than the total yearly cost at Fitness Gym.
B. The total yearly cost will always be higher at Fitness Gym than at
Healthy Gym.
C. The total yearly cost is the same at both gyms when members make 25 visits to each one.
D. The total yearly cost is $240 at each gym for 40 visits.
Answers
GPT-4o mini
To determine which statement is true, let's analyze the equations given for both gyms.
**Healthy Gym:**
\[ t = 6n \]
This means that for every visit (n), the cost is $6. Therefore, the total cost for n visits is simply 6 times n.
**Fitness Gym:**
\[ t = 80 + 4n \]
This means that there is a one-time yearly charge of $80, plus $4 for each visit. So, the total cost for n visits is $80 + 4 times n.
Let's explore the statements one by one:
**A. The total yearly cost at Healthy Gym is about $80 lower than the total yearly cost at Fitness Gym.**
Let's calculate the total costs for a few values of n.
For n = 0 (no visits):
- Healthy Gym: t = 6(0) = $0
- Fitness Gym: t = 80 + 4(0) = $80
Difference: $80
For n = 10:
- Healthy Gym: t = 6(10) = $60
- Fitness Gym: t = 80 + 4(10) = $120
Difference: $60
For n = 20:
- Healthy Gym: t = 6(20) = $120
- Fitness Gym: t = 80 + 4(20) = $160
Difference: $40
As we can see, the difference in total costs is not consistently $80; it decreases as the number of visits increases. So, statement A is **not true**.
**B. The total yearly cost will always be higher at Fitness Gym than at Healthy Gym.**
Let's find out when the costs are equal. We set the two equations equal to each other:
\[ 6n = 80 + 4n \]
Solving this:
\[ 6n - 4n = 80 \]
\[ 2n = 80 \]
\[ n = 40 \]
This means at 40 visits, the costs are equal. For values less than 40, Healthy Gym is actually less expensive, hence statement B is **not true**.
**C. The total yearly cost is the same at both gyms when members make 25 visits to each one.**
We calculate for n = 25:
- Healthy Gym: t = 6(25) = $150
- Fitness Gym: t = 80 + 4(25) = 80 + 100 = $180
Since $150 is not equal to $180, statement C is **not true**.
**D. The total yearly cost is $240 at each gym for 40 visits.**
Let's check the total costs for n = 40:
- Healthy Gym: t = 6(40) = $240
- Fitness Gym: t = 80 + 4(40) = 80 + 160 = $240
Both gyms have the same cost of $240 at 40 visits, so statement D is **true**.
Thus, the correct answer is **D. The total yearly cost is $240 at each gym for 40 visits.**
**Healthy Gym:**
\[ t = 6n \]
This means that for every visit (n), the cost is $6. Therefore, the total cost for n visits is simply 6 times n.
**Fitness Gym:**
\[ t = 80 + 4n \]
This means that there is a one-time yearly charge of $80, plus $4 for each visit. So, the total cost for n visits is $80 + 4 times n.
Let's explore the statements one by one:
**A. The total yearly cost at Healthy Gym is about $80 lower than the total yearly cost at Fitness Gym.**
Let's calculate the total costs for a few values of n.
For n = 0 (no visits):
- Healthy Gym: t = 6(0) = $0
- Fitness Gym: t = 80 + 4(0) = $80
Difference: $80
For n = 10:
- Healthy Gym: t = 6(10) = $60
- Fitness Gym: t = 80 + 4(10) = $120
Difference: $60
For n = 20:
- Healthy Gym: t = 6(20) = $120
- Fitness Gym: t = 80 + 4(20) = $160
Difference: $40
As we can see, the difference in total costs is not consistently $80; it decreases as the number of visits increases. So, statement A is **not true**.
**B. The total yearly cost will always be higher at Fitness Gym than at Healthy Gym.**
Let's find out when the costs are equal. We set the two equations equal to each other:
\[ 6n = 80 + 4n \]
Solving this:
\[ 6n - 4n = 80 \]
\[ 2n = 80 \]
\[ n = 40 \]
This means at 40 visits, the costs are equal. For values less than 40, Healthy Gym is actually less expensive, hence statement B is **not true**.
**C. The total yearly cost is the same at both gyms when members make 25 visits to each one.**
We calculate for n = 25:
- Healthy Gym: t = 6(25) = $150
- Fitness Gym: t = 80 + 4(25) = 80 + 100 = $180
Since $150 is not equal to $180, statement C is **not true**.
**D. The total yearly cost is $240 at each gym for 40 visits.**
Let's check the total costs for n = 40:
- Healthy Gym: t = 6(40) = $240
- Fitness Gym: t = 80 + 4(40) = 80 + 160 = $240
Both gyms have the same cost of $240 at 40 visits, so statement D is **true**.
Thus, the correct answer is **D. The total yearly cost is $240 at each gym for 40 visits.**