Two Gym Plans, a Five-Month Semester, and a Break-Even Claim
A student is choosing a gym for a semester that runs from mid-August through the end of December. Gym A charges a $50 enrollment fee plus $22 per month. Gym B charges no enrollment fee and $30 per month. Both gyms bill any partial month as a full month, so the half-month of August counts as a full billing month. The student's total fitness budget for the semester is $175. A friend advises choosing Gym B because "it has no sign-up fee, so it has to be cheaper." The student is also considering staying through the following summer, which would extend the membership to 8 billed months.
1. How many months will the student be billed, and what does each gym cost for the semester?
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Count billed months: August (partial, billed full), September, October, November, December = 5 billed months.
Gym A: 50 + (22 × 5) = 50 + 110 = $160. Gym B: 30 × 5 = $150.
Difference: $10, with Gym B cheaper for this term. Both totals sit inside the $175 budget, so affordability does not decide the question - the cost comparison does.
2. Find the break-even point and evaluate the friend's reasoning.
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Set the costs equal, with m = billed months:
50 + 22m = 30m → 50 = 8m → m = 6.25 months.
So Gym A becomes cheaper once the membership passes 6.25 months (that is, from the 7th month onward).
Check it: at 6 months, A = 50 + 132 = $182 and B = $180 (B still ahead by $2). At 7 months, A = 50 + 154 = $204 and B = $210 (A ahead by $6).
The friend's reasoning is wrong because it compares only the fixed fee and ignores the per-month rate. The enrollment fee is a one-time cost that gets spread thinner the longer the student stays; it is a real advantage to Gym B only for short memberships, here anything under 6.25 months.
3. If the student extends to 8 billed months, which plan should be chosen, and how much does the answer depend on how long they stay?
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8 months: Gym A = 50 + (22 × 8) = 50 + 176 = $226. Gym B = 30 × 8 = $240. Gym A is cheaper by $14.
The decision flips: Gym B wins for stays shorter than 6.25 months, Gym A for longer ones, and the two are within a few dollars of each other in the 6–7 month zone.
Judgement: the student knows the semester length with confidence (5 months) but only hopes for the summer extension. Choosing Gym B costs nothing if the extension falls through, while prepaying or committing to Gym A on the strength of a maybe is the riskier bet. A defensible answer names the assumption - 5 months guaranteed, summer uncertain - rather than picking a plan from a single projection.