Annual versus Monthly Units in a Gym Membership Model
A fitness studio offers two annual plans. Plan A charges AED 120 to join and AED 35 per class. Plan B charges AED 300 to join and AED 20 per class. A member expects to attend 8 to 14 classes per month. A staff member writes A(x) = 120 + 35x and B(x) = 300 + 20x, where x is the number of classes per year, solves the equations to get x = 12, and advertises: 'Plan B is cheaper for anyone attending more than 12 classes per year.' The joining fees are paid once for the year, but the member's attendance estimate is monthly. The member may also cancel after six months; joining fees are not refunded.
1. Identify the modelling error, correct the annual cost functions, and determine which plan is cheaper under the 8–14 monthly attendance range.
Read the worked answer
Let m be monthly classes. Over 12 months, Plan A costs 120 + 35(12m) = 120 + 420m. Plan B costs 300 + 20(12m) = 300 + 240m. Setting equal gives 120 + 420m = 300 + 240m, so 180m = 180 and m = 1. For any m > 1, Plan B is cheaper. At 8 monthly classes, A = 120 + 420 × 8 = AED 3480 and B = 300 + 240 × 8 = AED 2220, so B is much cheaper. At 14 monthly classes, A = 6000 and B = 3660.
The staff error is mixing units: x was defined as classes per year, but the member's 8–14 estimate is monthly. The break-even of 12 classes per year is correct only if x is annual attendance. The advertisement applies that annual break-even to a member whose annual attendance is 96–168 classes, so it badly misstates the comparison.
2. If the member cancels after six months, compare total costs for m = 1 and m = 3 monthly classes under both plans. Does the answer depend on m?
Read the worked answer
For six months, Plan A costs 120 + 35(6m) = 120 + 210m. Plan B costs 300 + 20(6m) = 300 + 120m. Equal when 120 + 210m = 300 + 120m, so 90m = 180 and m = 2.
At m = 1: A = 120 + 210 = AED 330; B = 300 + 120 = AED 420, so A is cheaper. At m = 3: A = 120 + 630 = AED 750; B = 300 + 360 = AED 660, so B is cheaper. At m = 2 both cost AED 540.
Yes, the answer depends on m. A six-month membership at very low attendance can favour Plan A, while the annual advertisement's conclusion does not hold automatically.