Comparing Mobile Phone Plans: Algebra and Financial Mathematics
A student is comparing two mobile phone plans. Plan A has a monthly access fee of $30 and includes 5 GB of data. Additional data costs $10 per GB (or part thereof). Plan B has a monthly access fee of $50 and includes 10 GB of data. Additional data costs $8 per GB (or part thereof). The student uses an average of 7 GB per month, but usage varies: some months 4 GB, some months 12 GB. The student signs a 12-month contract. The student wants to minimize total cost over the year. They estimate that in 3 months they will use 4 GB, in 6 months they will use 7 GB, and in 3 months they will use 12 GB. Calculate the total annual cost for each plan under these usage assumptions. Also, determine the break-even monthly data usage (in GB) where the two plans cost the same, assuming data is charged per GB (not per part GB) for simplicity.
1. Calculate the total annual cost for Plan A and Plan B under the given usage pattern. Show your working.
Read the worked answer
Plan A: Monthly fee $30. Included 5 GB. For 4 GB: no extra, cost $30. For 7 GB: extra 2 GB × $10 = $20, total $50. For 12 GB: extra 7 GB × $10 = $70, total $100. Annual cost = 3×$30 + 6×$50 + 3×$100 = $90 + $300 + $300 = $690. Plan B: Monthly fee $50. Included 10 GB. For 4 GB: no extra, $50. For 7 GB: no extra, $50. For 12 GB: extra 2 GB × $8 = $16, total $66. Annual cost = 3×$50 + 6×$50 + 3×$66 = $150 + $300 + $198 = $648. Plan B is cheaper by $42.
2. Determine the break-even monthly data usage (in GB) where the two plans cost the same, assuming data is charged per GB (not per part GB). Let x be the data usage in GB. Consider cases: x ≤ 5, 5 < x ≤ 10, and x > 10.
Read the worked answer
Case 1: x ≤ 5. Plan A cost = $30. Plan B cost = $50. No break-even since 30 ≠ 50. Case 2: 5 < x ≤ 10. Plan A cost = 30 + 10(x-5) = 10x - 20. Plan B cost = $50. Set 10x - 20 = 50 → 10x = 70 → x = 7. This is within the interval, so break-even at 7 GB. Case 3: x > 10. Plan A cost = 30 + 10(x-5) = 10x - 20. Plan B cost = 50 + 8(x-10) = 8x - 30. Set 10x - 20 = 8x - 30 → 2x = -10 → x = -5, not in interval. So no break-even for x > 10. Thus the only break-even point is at 7 GB, where both plans cost $50.