Choosing Between Two Mobile Plans
A phone retailer offers two plans. Plan A charges a flat $18 per month plus $0.06 per gigabyte used. Plan B charges a flat $6 per month plus $0.14 per gigabyte used. A customer tracks her data use over six months: 40, 55, 30, 70, 45, and 60 gigabytes. She wants a single plan for the next year and expects her monthly use to stay within the same range. The retailer also offers a one-time $25 credit if she signs a twelve-month contract, applied once at the start. She asks which plan costs less over the year and how sensitive that answer is to her usage.
1. Model the annual cost of each plan as a function of monthly gigabytes g, including the contract credit, and determine the usage level at which the two plans cost the same.
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Let g be monthly gigabytes.
Plan A annual cost: 12(18 + 0.06g) − 25 = 216 + 0.72g − 25 = 191 + 0.72g. Plan B annual cost: 12(6 + 0.14g) − 25 = 72 + 1.68g − 25 = 47 + 1.68g.
Set equal: 191 + 0.72g = 47 + 1.68g, so 144 = 0.96g, giving g = 150 gigabytes per month.
Interpretation: below 150 GB/month Plan B is cheaper; above 150 GB/month Plan A is cheaper. Her observed range (30–70 GB) is far below the crossover, so Plan B wins comfortably throughout her expected usage.
2. Using her six months of data, estimate her likely annual cost under each plan and state how confident the recommendation is.
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Mean monthly use = (40 + 55 + 30 + 70 + 45 + 60)/6 = 300/6 = 50 GB.
Plan B: 47 + 1.68(50) = 47 + 84 = $131. Plan A: 191 + 0.72(50) = 191 + 36 = $227.
Plan B saves about $96 per year at her average use. Even at her highest month (70 GB), Plan B costs 47 + 1.68(70) = $164.60 while Plan A costs 191 + 0.72(70) = $241.40, so Plan B still wins. She would need to average roughly three times her current usage (150 GB) before Plan A becomes worthwhile, so the recommendation is robust to normal variation in her habits.
3. The retailer changes the credit to $100 for Plan A only. Re-evaluate the crossover and comment on whether the earlier conclusion still holds.
Read the worked answer
New Plan A annual cost: 216 + 0.72g − 100 = 116 + 0.72g. Set equal to Plan B: 116 + 0.72g = 47 + 1.68g, so 69 = 0.96g, giving g = 71.9 GB per month.
The crossover drops from 150 GB to about 72 GB. Her average of 50 GB still favours Plan B, but the margin is now small: at 50 GB, Plan A costs 116 + 36 = $152 versus Plan B's $131, a $21 gap instead of $96. Her 70 GB month is now close to the crossover, so the recommendation becomes sensitive to whether her usage rises. The key judgement is that a large upfront credit shifts the break-even point sharply, so comparing plans requires the crossover, not just a single month's bill.