Choosing a Mobile Plan with Break-Even Reasoning
A family compares two phone plans for a single line. Plan A charges a flat $35 per month and includes 12 GB of data; each extra GB costs $6. Plan B charges $20 per month and includes 4 GB; each extra GB costs $9. The family estimates monthly usage is uncertain, somewhere between 6 GB and 20 GB. They want one plan for the whole year and will not change mid-year. A friend claims Plan A is always cheaper because its extra-GB rate is lower. The family wants to know the usage level where the two plans cost the same, and whether the friend's claim holds across their likely range.
1. Write cost functions for both plans in terms of data used (g GB, for g above each plan's included allowance) and find the usage where total monthly costs are equal.
Read the worked answer
Let g be monthly GB used.
Plan A (included 12 GB): for g ≤ 12, C_A = 35. For g > 12, C_A = 35 + 6(g − 12) = 6g − 37.
Plan B (included 4 GB): for g ≤ 4, C_B = 20. For g > 4, C_B = 20 + 9(g − 4) = 9g − 16.
Set 6g − 37 = 9g − 16: 6g − 37 = 9g − 16 −37 + 16 = 9g − 6g −21 = 3g g = 7.
At g = 7: Plan A = 6(7) − 37 = 5, which is wrong because 7 GB is below Plan A's 12 GB allowance, so Plan A actually costs $35. Plan B at 7 GB = 9(7) − 16 = $47.
The algebra gives g = 7, but this lies outside the valid range for the Plan A formula (which only applies for g > 12). So the linear-equality solution is invalid. Checking the real boundary: at g = 12, Plan A = $35, Plan B = 9(12) − 16 = $92. Plan A is already cheaper. Below 12 GB, Plan A is a flat $35 while Plan B rises from $20. Plan B equals $35 when 9g − 16 = 35, giving g = 5.67 GB. So for g below about 5.7 GB, Plan B is cheaper; above that, Plan A is cheaper.
2. Evaluate the friend's claim that Plan A is always cheaper, using the family's 6–20 GB range.
Read the worked answer
The friend is wrong for the low end of the range but right for most of it.
- At 6 GB: Plan A = $35 (within allowance). Plan B = 9(6) − 16 = $38. Plan A cheaper by $3.
- At 5 GB: Plan A = $35. Plan B = 9(5) − 16 = $29. Plan B cheaper by $6.
- Break-even: 9g − 16 = 35 → g = 5.67 GB.
Within the stated 6–20 GB range, Plan A is cheaper at every point (at 6 GB it wins by $3, and the gap widens as usage grows because Plan A's marginal rate is lower). So within their likely range, the friend's conclusion happens to hold. But the claim "always" is false: below 5.67 GB, Plan B is cheaper.
The key judgement is that the friend's reasoning (lower extra-GB rate) is not the right justification. The correct reasoning is that Plan A's higher fixed cost is offset once usage exceeds about 5.7 GB. A student who simply compares the $6 vs $9 rates has not shown when the fixed-cost difference is recovered.
3. The family's usage is uncertain. Recommend a decision rule and identify the risk of choosing Plan A.
Read the worked answer
Decision rule: choose Plan A if expected usage is confidently above about 5.7 GB; choose Plan B only if usage might drop below that.
Risk analysis for Plan A:
- Maximum downside if usage falls to 4 GB or lower: Plan A = $35, Plan B = $20, so Plan A costs $15 more per month, or $180 over the year.
- Upside if usage is high (say 20 GB): Plan A = 6(20) − 37 = $83; Plan B = 9(20) − 16 = $164. Plan A saves $81 per month.
The downside is capped at $15/month because Plan A's fixed cost does not fall with low usage, while the upside grows without limit as usage rises. Given the family expects 6–20 GB, all of which favours Plan A, Plan A is the sensible choice. The residual risk is only if their estimate is badly wrong at the low end. Recommend Plan A, but note the $15/month exposure if usage collapses below ~5.7 GB.