Comparing Two Subscription Plans with Break-Even
A small tutoring business offers two plans. Plan A charges a flat 40 per month plus 12 per session. Plan B charges 100 per month plus 6 per session. A parent currently books 8 sessions per month and is on Plan A, paying 40 + 96 = 136. The business owner wants to know which plan is cheaper for the parent at various usage levels, and whether redesigning the plans could increase revenue without losing customers. A rival offers a flat 150 unlimited plan, which the owner is watching closely.
1. Find the break-even number of sessions between Plan A and Plan B and interpret it for the parent booking 8 sessions.
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Set costs equal: 40 + 12s = 100 + 6s, so 6s = 60, s = 10. For s < 10, Plan A is cheaper (e.g. s = 8: A = 136, B = 148). For s > 10, Plan B is cheaper (e.g. s = 12: A = 184, B = 172). At s = 10 both cost 160. The parent at 8 sessions is correctly on the cheaper plan, saving 12 per month versus Plan B. But the margin is narrow: one extra session per month would narrow the gap to 6, and two extra sessions would flip the cheaper plan. The owner should expect this parent to switch to B if usage rises, which reduces revenue per session.
2. Compare both plans with the rival's flat 150 unlimited plan and advise the owner on a redesign that protects revenue.
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Rival: 150 flat. Plan A equals 150 when 40 + 12s = 150, s ≈ 9.17. Plan B equals 150 when 100 + 6s = 150, s ≈ 8.33. So for s above about 8.3, the rival is cheaper than Plan B; for s above about 9.2, the rival is cheaper than Plan A. The parent at 8 sessions pays 136 on A, still below 150, but only just. Advice: the owner's plans are vulnerable for heavy users. A redesign could cap the per-session charge above a threshold, e.g. Plan A' = 40 + 12·min(s, 9), giving 148 at 9 sessions and 148 thereafter, undercutting the rival at 150 while preserving margin for light users. The judgement call is that matching the rival's flat rate exactly would remove the incentive to book fewer sessions, so a capped tier is preferable to a straight price cut.