Answer to the February 17, 2003 Problem of the Week

The Stamp Problem

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In the United States, first-class postage is now 37 cents. Before that, it was 34 cents and before that 33 cents. I recently had a supply of all three kinds of stamps , and I stuck some of them on a package to total $3.90.

How many of each kind did I use?

Solution to the Problem:

I used two 33 cent stamps, three 34 cent stamps, and six 37 cent stamps.

Rick Jones sent in the following explanation:

Since twelve 33c stamps would total $3.96, the total number of stamps must be fewer than 12. And since ten 37c stamps would total only $3.70, it follows that there must also have been more than 10 stamps. Thus the total number of stamps had to have been 11.

Let the number of 33c, 34c and 37c stamps be represented by a, b and c, respectively. Then we have

33a + 34b + 37c = 390 and a + b + c = 11

If we multiply the right-hand equation by 33 and subtract it from the first, we obtain

b + 4c = 27

Clearly, since b is positive, c cannot be greater than 6. But if c is 5, then b must be 7, whence b + c = 12, which cannot be. Thus c must be 6, b must be 3 and a must be 2 and this solution is unique. To test it...

2(33) + 3(34) + 6(37) = 66 + 102 + 222 = 390