Mr. P has two credit cards that reward him with cash based on how much he charges on the credit cards.

The Mastercard Rewards program is simple:
    Mr. P gets back 1% of all that he charges on that card during the year.

The American Express rewards program is a bit more complicated:
    On the first $6,500 of purchases in a year,
        Mr. P gets 1% on supermarket, drug store, and gas purchases, and
        he gets .5% on all other purchases.
    On all purchases over $6,500,
        Mr. P gets 5% on supermarket, drug store, and gas purchases, and
        he gets 1.25% on all other purchases.

Suppose that 20% of Mr. P's yearly charges are for supermarket, drug store, and gas purchases.

(1) What is the "break-even" point on the two cards?   That is, what amount
      must be charged on each card so that the rewards would be equal?

(2) If Mr. P's annual charges are less than this amount,
      which card should he use?

(3) If Mr. P's annual charges are more than this amount,
      which card should he use?

All three questions must be answered correctly to get credit.

Solution to the Problem:

(1) The break-even point is $9,100.
(2) If he charges less than $9,100, he should use his Mastercard.
(3) If he charges more than $9,100, he should use his American Express card.

Let x = the break-even point.
Then the Mastercard Rewards = .01x.
Then the American Express Rewards = .01(.20(6500)) + .005(.80(6500)) +
          .05(.20(x - 6500)) + .0125(.80(x - 6500))

Set them equal and solve:
.01x = .01(.20(6500)) + .005(.80(6500)) + .05(.20(x - 6500)) + .0125(.80(x - 6500))
.01 x = 91
So, x = $9,100.

Correctly solved by:

1. James Alarie Flint, Michigan
2. Steven Burak Drexel University,
Philadelphia, Pennsylvania