Which is better?

Question

A store advertised two phone cards.  Which cards would you buy?
Card 1: 3.9 cents a minute with a 37 cent charge for each call.
Card 2: 7.9 cents a minute; no connection charge.

Answer 1

Card 1: for n calls at 37 connection charge and total time of T minutes the cost is 37n+3.9T cents.

Card 2: for n calls at 0c connection charge and total time of T minutes the cost is 7.9T cents.

37n+3.9T=7.9T so 37n=4T and T/n=37/4=9.25 minutes per call is the average call time when the costs of the two cards are the same. If the average call time is less than 9.25 minutes, then card 2 gives better value; if the average call time is greater than 9.25 minutes card 1 is better value. The following inequalities confirm this:

Card 1>card 2: 37n+3.9T>7.9T, 37n>4T, so 4T<37n and T/n<9.25 mins per call on average;

Card 2>card 1: 37n+3.9T<7.9T, 37n<4T, so 4T>37n and T/n>9.25 mins per call on average.