Two Gym Memberships: Break-Even, Whole-Month Billing, and a Budget Cap
A student is choosing a gym membership and building a budget model. FitNow charges a $25 joining fee plus $18 per month. CityStrength charges a $70 joining fee plus $12 per month. The student plans to stay in the area for 9 months but could leave after 5, and has $200 set aside for fitness. A classmate writes, 'CityStrength is the better deal because after the joining fee it costs less each month.' The student graphs both cost functions, C_A(m) = 25 + 18m and C_B(m) = 70 + 12m, for 0 ≤ m ≤ 12, where m is a whole number of months. The graph shows the lines crossing between m = 7 and m = 8. The student must decide which plan to buy, justify the decision with the model, and explain to the classmate why the per-month rate alone does not settle the choice.
1. Solve algebraically for the month in which the plans cost the same, and interpret that solution given that memberships are billed in whole months.
Read the worked answer
Set the cost functions equal: 25 + 18m = 70 + 12m. Subtract 12m from both sides: 25 + 6m = 70. Subtract 25: 6m = 45, so m = 7.5.
Check: 25 + 18(7.5) = 25 + 135 = $160, and 70 + 12(7.5) = 70 + 90 = $160. The slopes differ by $6 per month, and the joining fees differ by $45; 45 ÷ 6 = 7.5, so CityStrength's lower monthly rate takes 7.5 months to repay its $45 larger upfront fee.
Because billing is in whole months, 7.5 is not a purchasable answer. At m = 7: FitNow = 25 + 126 = $151; CityStrength = 70 + 84 = $154. FitNow is cheaper by $3. At m = 8: FitNow = 25 + 144 = $169; CityStrength = 70 + 96 = $166. CityStrength is cheaper by $3. The crossover happens during the eighth month, which is exactly why the graph crosses between 7 and 8.
2. Recommend a plan for (a) the 9-month stay and (b) the possibility of leaving after 5 months, both under the $200 budget. Justify each with totals.
Read the worked answer
(a) Nine months: FitNow = 25 + 18(9) = 25 + 162 = $187. CityStrength = 70 + 12(9) = 70 + 108 = $178. Both are under $200, and CityStrength saves $9, so the model favors CityStrength if the student stays 9 months.
(b) Five months: FitNow = 25 + 18(5) = $115. CityStrength = 70 + 12(5) = $130. FitNow saves $15, and CityStrength's lower rate has not had time to repay the fee.
The decision therefore hinges on duration, not on which plan is 'better' in general. Since the student may leave after 5 months, the honest recommendation is conditional: choose FitNow unless the 9-month stay is close to certain, in which case CityStrength wins by a thin $9. A $9 edge over nine months is small enough that non-cost factors (location, equipment) could reasonably decide it. The $200 cap is not binding in either scenario, so it does not change the ranking.
3. Critique the classmate's statement that CityStrength is the better deal because its monthly rate is lower. Write a corrected one-sentence version of the claim.
Read the worked answer
The statement confuses slope with total cost. CityStrength's $12 monthly rate is the slope of its cost line, but total cost is slope times months plus the intercept, and CityStrength's intercept is $45 higher. A lower slope only wins after the break-even point, here 7.5 months. For any horizon under 7.5 months, the lower-fee plan is cheaper despite the higher rate; at exactly 7 or 8 months the gap is only $3. The claim is also unfalsifiable as written because 'better deal' has no time horizon attached, and the classmate's phrase 'after the joining fee' misleadingly treats the fee as if it disappears once paid.
Corrected: 'CityStrength has the lower monthly rate, so it becomes the cheaper plan once the student stays at least 8 months; for shorter stays, FitNow's lower joining fee wins.'