Choosing a Phone Plan with Inequalities
Two mobile operators offer monthly plans. Operator A charges a fixed 12 leva per month plus 0.08 leva per minute of calls. Operator B charges a fixed 18 leva per month plus 0.05 leva per minute. A student talks on the phone m minutes per month, where m is a whole number. The student's family budget for the phone is 30 leva per month. The student currently uses about 200 minutes and expects usage to grow. A friend says Operator B is always better because the per-minute rate is lower.
1. Write the monthly cost of each plan as an expression in m, and find the number of minutes for which the two plans cost the same.
Read the worked answer
Cost A: 12 + 0.08m. Cost B: 18 + 0.05m.
Equal when 12 + 0.08m = 18 + 0.05m, so 0.08m − 0.05m = 18 − 12, giving 0.03m = 6, so m = 200 minutes. At m = 200, both cost 12 + 16 = 28 leva (check B: 18 + 10 = 28).
So 200 minutes is the break-even point.
2. For the current 200 minutes and for expected usage of 300 minutes, which plan is cheaper? Solve the inequality to state exactly when each plan is better.
Read the worked answer
Compare: A is cheaper when 12 + 0.08m < 18 + 0.05m, i.e. 0.03m < 6, i.e. m < 200. B is cheaper when m > 200. At m = 200 they are equal.
At 200 minutes: equal, 28 leva each. At 300 minutes: A = 12 + 24 = 36 leva; B = 18 + 15 = 33 leva, so B is cheaper. The friend's claim is only partly right: B is better only above 200 minutes, not always.
3. The budget is 30 leva. For each plan, find the maximum whole number of minutes affordable, and comment on which plan suits growing usage under the budget.
Read the worked answer
Plan A: 12 + 0.08m ≤ 30, so 0.08m ≤ 18, m ≤ 225. Maximum whole minutes = 225. Plan B: 18 + 0.05m ≤ 30, so 0.05m ≤ 12, m ≤ 240. Maximum whole minutes = 240.
Under a 30 leva cap, B allows more minutes (240 vs 225), so for growing usage B is the safer choice as long as usage stays above 200 minutes. Below 200 minutes A is cheaper, but the student is already at 200 and rising, so B fits the trend. The judgement is that the cheaper per-minute rate matters only past the break-even point.