Two Gyms and a Promotion
Two gyms in the same shopping centre advertise these prices:
- FlexFit: $60 joining fee, then $12 per week. No contract.
- BodyWorks: $150 joining fee, then $8 per week. No contract.
Both bill by the week, let you stop at the end of any week, and include the same classes. Tessa will train for a fixed block of weeks and then move overseas, so only the total cost over that block matters to her.
FlexFit has just introduced a promotion: the $60 joining fee is waived for anyone who signs a 30-week commitment. The fine print says that if the 30 weeks are not completed, the joining fee becomes payable after all. Tessa is choosing between 18 weeks, 25 weeks and 30 weeks of training, depending on her work contract.
1. Ignoring the promotion, find the number of weeks for which the two gyms cost the same, and explain what this means for a member who trains for a whole number of weeks.
Read the worked answer
Let w be the number of weeks. FlexFit: C = 60 + 12w BodyWorks: C = 150 + 8w
Set them equal: 60 + 12w = 150 + 8w 4w = 90 w = 22.5
Because weeks are whole numbers, the two never cost exactly the same. Check either side:
- w = 22: FlexFit = 60 + 264 = $324; BodyWorks = 150 + 176 = $326 (FlexFit cheaper by $2)
- w = 23: FlexFit = 60 + 276 = $336; BodyWorks = 150 + 184 = $334 (BodyWorks cheaper by $2)
Meaning: FlexFit is cheaper for 22 weeks or fewer; BodyWorks is cheaper from 23 weeks onward. A member planning around 22–23 weeks should look at the exact dates rather than assume.
2. Tessa first plans to train for 18 weeks. Which gym should she join, and does the 30-week promotion change her choice?
Read the worked answer
At 18 weeks (promotion not available, since 18 < 30):
- FlexFit: 60 + 12 × 18 = 60 + 216 = $276
- BodyWorks: 150 + 8 × 18 = 150 + 144 = $294
FlexFit is cheaper by $294 − $276 = $18.
The promotion cannot help her: it requires a 30-week commitment and she is only training for 18 weeks. The joining fee must be paid, so the answer is FlexFit at $276. This matches the crossover found earlier - 18 weeks is below 22.5, so the gym with the lower weekly rate has not yet recovered its higher joining fee.
3. Tessa now plans 30 weeks and signs the FlexFit promotion, but her contract ends after 25 weeks. Compare the three possible outcomes and judge whether signing the promotion was a good decision.
Read the worked answer
Scenario 1 - 30 weeks, promotion completed:
- FlexFit with waiver: 0 + 12 × 30 = $360
- FlexFit standard: 60 + 360 = $420
- BodyWorks: 150 + 8 × 30 = $390 Promotion wins, saving $30 against BodyWorks.
Scenario 2 - she leaves at 25 weeks, so the waiver is lost:
- FlexFit: 60 + 12 × 25 = 60 + 300 = $360
- BodyWorks: 150 + 8 × 25 = 150 + 200 = $350 FlexFit is now $10 worse.
Judgement: the promotion only removes the joining fee, it does not change the weekly rate, so the crossover is still 22.5 weeks. The promotion is worth $60 only if the commitment is actually completed. Since her contract was uncertain, the safer choice was FlexFit without the commitment (or BodyWorks above 22.5 weeks). The lesson: compare the guaranteed cost, not the advertised headline.