Comparing Mobile Phone Plans with Data Overage
A student compares two mobile plans over a 12-month period. Plan X costs $25 per month and includes 15 GB of data; excess data is charged at $10 per GB, billed in whole GB blocks. Plan Y costs $40 per month with 30 GB included, and excess data at $8 per GB in whole GB blocks. The student's monthly data use over the year was: 12, 14, 18, 22, 26, 31, 35, 29, 24, 19, 16, 13 GB. The student also has a $60 handset repayment on Plan X only, spread over 12 months at $5 per month. The student wants the cheapest total for the year but also wants to avoid any month where the bill exceeds $55.
1. Calculate the total annual cost of each plan including handset repayment where applicable.
Read the worked answer
Plan X: base = 25 × 12 = $300. Handset = 5 × 12 = $60. Excess: allowance 15 GB, so months over 15 GB are 18, 22, 26, 31, 35, 29, 24, 19, 16 = 9 months. Excess GB: 3+7+11+16+20+14+9+4+1 = 85 GB. Cost = 85 × 10 = $850. Total = 300 + 60 + 850 = $1210.
Plan Y: base = 40 × 12 = $480. No handset. Excess over 30 GB: months 31 and 35 → 1 + 5 = 6 GB. Cost = 6 × 8 = $48. Total = 480 + 48 = $528.
Plan Y is cheaper by $1210 − $528 = $682.
2. Check the $55 monthly bill ceiling for each plan and identify any breaching months.
Read the worked answer
Plan X monthly bill = 25 + 5 + 10 × excess GB. Breach when 30 + 10e > 55, i.e. e > 2.5, so e ≥ 3 GB excess (usage ≥ 18 GB). Months with excess ≥ 3: 18 (3), 22 (7), 26 (11), 31 (16), 35 (20), 29 (14), 24 (9), 19 (4) → 8 months breach. Worst is 35 GB: 30 + 200 = $230.
Plan Y monthly bill = 40 + 8 × excess GB. Breach when 40 + 8e > 55, i.e. e > 1.875, so e ≥ 2 GB excess (usage ≥ 32 GB). Months: 35 GB only (e = 5) → 40 + 40 = $80. One month breaches.
So Plan Y breaches the ceiling once, Plan X eight times. The judgement is that the cheaper annual plan still breaks the monthly rule, so the student must decide whether the ceiling is hard or a preference.
3. The student could buy a data add-on on Plan X: $15 per month for an extra 10 GB, applied every month. Evaluate whether this makes Plan X competitive.
Read the worked answer
With the add-on, Plan X allowance becomes 25 GB at a flat $15 per month = $180 per year extra.
Excess now over 25 GB: months 26, 31, 35, 29 = 4 months. Excess GB: 1 + 6 + 10 + 4 = 21 GB → $210.
Total Plan X with add-on = 300 + 60 + 180 + 210 = $750.
Still more than Plan Y's $528, by $222. Monthly ceiling: bill = 25 + 5 + 15 + 10e = 45 + 10e > 55 when e ≥ 2, so months 31, 35, 29 breach (3 months) - better than 8 but worse than Plan Y's 1.
The conclusion: the add-on improves Plan X but does not overturn the ranking, and it is only worth it if the student values the handset arrangement or needs the flexibility. This is the judgement being tested - evaluating a third option rather than only comparing the original two.