Choosing a Mobile Plan Under Uncertain Usage
Priya is comparing two phone plans. Plan A charges a flat $22 per month and includes 8 GB of data; each extra GB (or part thereof) costs $6. Plan B has no monthly fee, charges $0.09 per MB for the first 500 MB, then $12 per GB (or part thereof) beyond that. Priya's monthly data use varies: in a typical month she uses about 6.4 GB, but in two months of the year she uses around 11.3 GB. She wants the cheaper option across a full year, assuming 10 typical months and 2 heavy months. She also dislikes bills that jump sharply when she slightly exceeds a limit.
1. Compute Priya's annual cost under each plan and recommend one, showing all rounding decisions.
Read the worked answer
Plan A, typical month (6.4 GB): within 8 GB allowance, so cost = $22. Plan A, heavy month (11.3 GB): extra = 11.3 − 8 = 3.3 GB, charged as 4 GB (part GB counts), so 4 × $6 = $24; total = $22 + $24 = $46. Annual A = 10 × 22 + 2 × 46 = 220 + 92 = $312.
Plan B, typical month (6.4 GB = 6400 MB): first 500 MB at $0.09/MB = $45. Remaining 5900 MB = 5.9 GB, charged as 6 GB at $12 = $72. Total = $45 + $72 = $117. Plan B, heavy month (11.3 GB = 11300 MB): first 500 MB = $45; remaining 10800 MB = 10.8 GB → 11 GB × $12 = $132; total = $177. Annual B = 10 × 117 + 2 × 177 = 1170 + 354 = $1524.
Plan A is far cheaper ($312 vs $1524). The rounding rule matters: treating part-GB as free would understate Plan A's heavy months by $6 each and Plan B's totals by $12 per month, but the ranking is unchanged.
2. Priya's usage estimate could be wrong. At what typical-month usage does Plan A stop being cheaper than Plan B within that single month? Explain why this comparison is awkward.
Read the worked answer
Within a single typical month, Plan A costs $22 for up to 8 GB. Plan B already costs $45 for the first 500 MB, so Plan B exceeds $22 before any data beyond 500 MB is used. Solving: Plan B = $45 + $12 × (GB beyond 0.5, rounded up). For Plan B to equal $22 is impossible because the first 500 MB alone costs $45.
So in every month Plan B costs at least $45, while Plan A costs $22 unless usage exceeds 8 GB. Plan A is cheaper whenever monthly usage is at most 8 GB. Beyond 8 GB, Plan A rises by $6 per part-GB while Plan B rises by $12 per part-GB, so Plan A stays cheaper at every usage level. The comparison is awkward because the two plans have different fixed/variable structures: Plan B's high per-MB opening rate means it never undercuts Plan A, so the 'crossover' does not exist rather than being a single number.
3. Priya says she dislikes sharp bill jumps. Compare the marginal cost of exceeding a limit by a small amount on each plan.
Read the worked answer
On Plan A, exceeding 8 GB by even 0.01 GB triggers a whole extra GB charge of $6. So a tiny overrun costs $6 - a step jump. On Plan B, the first 500 MB is charged per MB at $0.09, so a small overrun there costs only fractions of a cent per MB (e.g. 10 MB = $0.90). But crossing from 500 MB to the next GB block costs $12 per part-GB, again a step of up to $12 for a tiny overrun.
Conclusion: both plans use 'part-GB counts as full GB' rounding, so both have step jumps; Plan B's steps are larger ($12 vs $6). Plan A is both cheaper overall and gentler at the margin, so the recommendation stands. The judgement call is recognising that 'per part-GB' rounding, not the headline rate, drives the jump.