Two Gym Memberships and a Partial Reimbursement
A community gym posts two membership offers. Gym A charges a $45 enrollment fee plus $22 per month. Gym B charges a $120 enrollment fee plus $16 per month. Maria is choosing between them for a stay of m months, where m is a whole number of months, and she does not plan to quit early. Separately, Maria's employer reimburses 60% of monthly membership fees for any employee who completes at least 12 months, but the employer never reimburses enrollment fees. Maria wants to know when the larger monthly fee at Gym A is actually worth paying, and she tends to compare plans by looking only at the monthly rate.
1. Write an expression for the total cost of each gym over m months and find the value of m where the two costs are equal. Explain what that value means given that m must be a whole number.
Read the worked answer
Cost A = 45 + 22m; Cost B = 120 + 16m.
Set equal: 45 + 22m = 120 + 16m → 6m = 75 → m = 12.5 months.
Because m is a whole number, membership never costs exactly the same; the crossing happens between month 12 and month 13:
- m = 12: A = 45 + 264 = $309; B = 120 + 192 = $312 → A cheaper by $3
- m = 13: A = 45 + 286 = $331; B = 120 + 208 = $328 → B cheaper by $3
Judgement point: 12.5 is not an acceptable answer to "which gym is cheaper at 12 months" - the break-even must be rounded up to conclude that B only wins from month 13 onward.
2. Maria stays 18 months and qualifies for the employer reimbursement. Which gym costs her less out of pocket, and by how much?
Read the worked answer
Only monthly fees are reimbursed, so enrollment fees are paid in full and the effective monthly cost is 40% of the sticker rate.
Gym A: monthly fees = 18 × $22 = $396; reimbursement = 0.60 × $396 = $237.60. Out of pocket = $45 + $396 − $237.60 = $203.40.
Gym B: monthly fees = 18 × $16 = $288; reimbursement = 0.60 × $288 = $172.80. Out of pocket = $120 + $288 − $172.80 = $235.20.
Gym A is cheaper by $235.20 − $203.40 = $31.80.
The judgement call: reimbursement shrinks the gap created by the monthly fees but cannot erase the $75 difference in enrollment fees, so A's higher rate is still the better deal at 18 months.
3. Maria argues, "Gym B must be cheaper because $16 is less than $22." Using the reimbursement rules, find the shortest whole-month stay for which Gym B is actually cheaper overall, and explain why the answer moves so far from the no-reimbursement case.
Read the worked answer
Net cost with reimbursement: A = 45 + 0.40(22)m = 45 + 8.8m; B = 120 + 0.40(16)m = 120 + 6.4m.
Set equal: 45 + 8.8m = 120 + 6.4m → 2.4m = 75 → m = 31.25, so the first whole month where B wins is m = 32.
Check: at m = 32, A = 45 + 281.60 = $326.60; B = 120 + 204.80 = $324.80 → B cheaper by $1.80.
Why it moves: reimbursement cuts both monthly rates to $8.80 and $6.40, shrinking the monthly advantage from $6 to $2.40. The fixed $75 enrollment gap now takes 75 ÷ 2.40 ≈ 31.25 months to repay instead of 12.5. Maria's comparison ignores the enrollment fee entirely, which is exactly the quantity that dominates at short stays.