Phone Plans: Break-Even, Promo, and Usage Variability
A student compares two phone plans. Plan A costs $18 per month and includes 100 texts; each additional text costs $0.10. Plan B costs $25 per month with unlimited texts. The student usually sends about 240 texts per month, but in some months-like exam weeks and summer-usage drops to about 120. A one-time promotion gives $10 off the first month of Plan A only; Plan B has no promotion. The student plans to keep the plan for 12 months and wants the lowest total cost, not necessarily the lowest monthly bill. A friend says Plan A must be cheaper because its base price is lower, while the student worries that the per-text charge could erase the savings.
1. At how many texts in a month do the two plans cost the same, ignoring the one-time promotion?
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Let t be the number of texts, with t greater than 100. Plan A costs 18 + 0.10(t − 100). Plan B costs 25. Set them equal: 18 + 0.10(t − 100) = 25. Subtract 18: 0.10(t − 100) = 7. Divide by 0.10: t − 100 = 70, so t = 170. At 170 texts, both plans cost $25. Below 170, Plan A is cheaper; above 170, Plan B is cheaper.
2. Compare the monthly cost at 240 texts and at 120 texts. Which plan is cheaper in each case?
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At 240 texts, Plan A charges for 240 − 100 = 140 extra texts. Extra cost = 140 × $0.10 = $14. Plan A total = $18 + $14 = $32. Plan B = $25, so Plan B saves $7 that month. At 120 texts, Plan A charges for 20 extra texts: 20 × $0.10 = $2. Plan A total = $18 + $2 = $20. Plan B = $25, so Plan A saves $5 that month. The friend’s claim that Plan A is always cheaper is wrong for high usage.
3. Compute the 12-month total cost if the student sends 240 texts every month, including the promotion. Then explain how the recommendation changes if usage is consistently low instead.
Read the worked answer
At 240 texts, Plan A costs $32 per month. Over 12 months before the promotion: 12 × $32 = $384. Apply the one-time $10 discount: $384 − $10 = $374. Plan B costs 12 × $25 = $300. Plan B is cheaper by $74 over the year. If usage were consistently 120 texts, Plan A would cost $20 per month: 12 × $20 = $240, minus $10 = $230. Plan B would still be $300, so Plan A would be cheaper by $70. The promo does not change the break-even usage; it only shifts the first-month comparison. The best choice depends on whether usage stays above or below 170 texts.