Choosing a Cell Phone Plan with Overage and Taxes
A student is comparing two monthly cell phone plans. Plan A costs $32 per month and includes 4 GB of data; each additional 0.5 GB (or part of 0.5 GB) costs $6. Plan B costs $48 per month and includes 8 GB; each additional 1 GB (or part) costs $10. The student typically uses between 6 and 7.5 GB per month. Both plans add 11% tax and fees to the total monthly charge (base plus overage). The student wants to keep the average monthly cost below $60 and is unsure which plan is safer if usage varies. They also have a one-time $40 activation fee on either plan, which they prefer to ignore when comparing monthly budgets.
1. Calculate the total monthly cost (including tax) for each plan at 6.5 GB of usage. Which plan is cheaper at this usage level?
Read the worked answer
Plan A: Included 4 GB, so overage = 6.5 - 4 = 2.5 GB. Billing is in 0.5 GB blocks: 2.5 / 0.5 = 5 blocks. Overage charge = 5 × $6 = $30. Pre-tax total = $32 + $30 = $62. With 11% tax: $62 × 1.11 = $68.82.
Plan B: Included 8 GB, so 6.5 GB is within the allowance. Pre-tax total = $48. With tax: $48 × 1.11 = $53.28.
At 6.5 GB, Plan B is cheaper by $68.82 - $53.28 = $15.54 per month.
2. Determine the usage level (in GB) at which the two plans cost the same before tax, assuming usage falls in whole 0.5 GB increments above 4 GB. Show the reasoning.
Read the worked answer
Let x be the usage in GB, with 4 < x ≤ 8 so Plan B has no overage.
Plan A pre-tax cost = 32 + 6 × (number of 0.5 GB blocks above 4). The number of blocks is ceil((x - 4) / 0.5).
Plan B pre-tax cost = 48.
Set them equal: 32 + 6n = 48, where n is the number of blocks. So 6n = 16, n = 2.67 blocks. Since blocks are whole, the plans are not exactly equal at any block count. At n = 2 blocks (usage from 5 to 5.5 GB), Plan A = 32 + 12 = $44, which is less than $48. At n = 3 blocks (usage from 5.5 to 6 GB), Plan A = 32 + 18 = $50, which is more than $48.
So the crossover happens between 5 and 6 GB: Plan A is cheaper up to and including 5.5 GB, and Plan B becomes cheaper once usage exceeds 5.5 GB.
3. The student's usage is uncertain between 6 and 7.5 GB. Recommend a plan and justify it using worst-case and best-case costs including tax.
Read the worked answer
Evaluate the endpoints for each plan.
Plan A: At 6 GB, overage = 2 GB = 4 blocks, cost = 32 + 24 = $56; with tax $62.16. At 7.5 GB, overage = 3.5 GB = 7 blocks, cost = 32 + 42 = $74; with tax $82.14.
Plan B: At 6 GB and at 7.5 GB, usage is within 8 GB, so cost = $48; with tax $53.28 in both cases.
Plan B stays at $53.28 across the whole range, which is below the $60 target. Plan A ranges from $62.16 to $82.14, exceeding the $60 target at both ends. Plan B is the safer choice: it meets the budget in the worst case and avoids overage risk entirely within the stated usage range.