Choosing a Gym Membership: When Do the Plans Cross?
Two fitness centers advertise different payment plans. Center A charges a $25 joining fee plus $30 per month. Center B charges an $85 joining fee plus $20 per month. A customer plans to join for at least 6 months but no more than 18 months. She has budgeted $400 for the entire membership, including the joining fee, and she wants to choose the plan that costs less over the time she actually stays. She is uncertain whether she will move after 14 months or stay the full 18 months. Both centers require whole-month payments and charge no cancellation fee after the joining fee. The customer writes the equations A(m)=25+30m and B(m)=85+20m, where m is the number of months, and solves them by setting them equal. She gets m=6 and concludes that Center B is always the better choice because it has a lower monthly rate. Her friend says the joining fee should decide the choice. The customer asks you to check the mathematics and the recommendation for her actual time range.
1. Solve the system and interpret m=6 in the context of total cost. Does m=6 mean Center B is always cheaper? Explain using a test month on each side of 6.
Read the worked answer
Set A(m)=B(m): 25+30m = 85+20m. Subtract 20m: 25+10m = 85. Subtract 25: 10m = 60, so m=6. At m=6, A(6)=25+30(6)=25+180=205, and B(6)=85+20(6)=85+120=205. The intersection means that for exactly 6 months the two plans cost the same: $205.
It does not mean B is always cheaper. For m=3, A(3)=25+90=115 and B(3)=85+60=145, so A is cheaper by $30. For m=10, A(10)=25+300=325 and B(10)=85+200=285, so B is cheaper by $40. The joining fee dominates at short stays, while B's lower monthly rate dominates at longer stays. Since the customer plans at least 6 months, B is never more expensive in her range and is cheaper after 6 months; at exactly 6 months they tie. The conclusion that B is always better is only true for m ≥ 6, not for all m.
2. Calculate the total cost at 14 months and at 18 months for each center. Which plan should she choose if she stays 14 months? If she stays 18 months? Compare with the $400 budget.
Read the worked answer
At 14 months: A(14)=25+30(14)=25+420=445; B(14)=85+20(14)=85+280=365. At 18 months: A(18)=25+540=565; B(18)=85+360=445.
If she stays 14 months, Center B costs $365, which is under her $400 budget, while Center A costs $445, which exceeds it. She should choose B. If she stays 18 months, Center B costs $445 and Center A costs $565; both exceed $400. Center B is still the cheaper of the two, but it does not fit her budget as stated. She would need to increase the budget by $45, negotiate a different plan, or shorten the membership.
The recommendation therefore depends on the actual duration: B is the lower-cost plan for both 14 and 18 months, but the $400 constraint is satisfied only in the 14-month case. This is a judgement call, not just a rate comparison.
3. The customer says Center B is always better because its monthly rate is lower. Write a response that uses the joining fee and the intersection point to correct the reasoning, and state any assumption about m.
Read the worked answer
A lower monthly rate is not enough by itself. Center B saves $10 per month compared with Center A, but it costs $60 more upfront ($85 vs. $25). The $60 upfront difference is recovered after 60/10 = 6 months, which is why the intersection is at m=6. For m < 6, the joining fee dominates and A is cheaper; for m > 6, the monthly savings dominate and B is cheaper. At m=6 they tie at $205.
So B is better only if she stays more than 6 months. Since she plans to stay at least 6 months, B is a reasonable choice, but the statement 'always better' is false for shorter stays. The comparison assumes whole-month payments, no hidden fees, and that the monthly rates and joining fees remain fixed for the entire membership. If any of those assumptions change, the intersection could move.