Mathematics: Comparing Two Phone Plans with a Threshold
A customer is choosing between two monthly phone plans. Plan X charges a flat $30 per month and $0.10 per gigabyte of data used. Plan Y charges a flat $18 per month and $0.25 per gigabyte. The customer typically uses between 60 and 100 gigabytes per month but occasionally uses as little as 40. The customer wants the lower-cost plan for the typical range but does not want to lose money badly in a low-use month. A friend claims Plan Y is always cheaper because its base fee is lower.
1. Write cost expressions for both plans and find the usage level at which they cost the same.
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Let g be gigabytes used. Plan X costs 30 + 0.10g dollars. Plan Y costs 18 + 0.25g dollars.
Set them equal: 30 + 0.10g = 18 + 0.25g. Subtract 18 from both sides: 12 + 0.10g = 0.25g. Subtract 0.10g: 12 = 0.15g. Divide by 0.15: g = 80.
At 80 gigabytes both plans cost 30 + 8 = $38, and 18 + 20 = $38. The break-even point is 80 gigabytes.
2. Evaluate the friend's claim and recommend a plan given the customer's usage pattern.
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The friend's claim is false. Below 80 gigabytes, Plan Y is cheaper; above 80 gigabytes, Plan X is cheaper. Check 60 gigabytes: Plan X costs 30 + 6 = $36; Plan Y costs 18 + 15 = $33, so Y wins. Check 100 gigabytes: Plan X costs 30 + 10 = $40; Plan Y costs 18 + 25 = $43, so X wins.
For typical use of 60 to 100 gigabytes, Plan X is cheaper for the upper part of the range and Plan Y for the lower part. In a 40-gigabyte month, Plan X costs $34 and Plan Y costs $28, a $6 difference favoring Y. The recommendation depends on how often usage exceeds 80 gigabytes; if it usually does, Plan X saves money, and the worst low-use penalty is modest.