Answer to February 17, 2003 Problem

The Stamp Problem

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


Correctly solved by:

1. Kathleen Altemose Winchester, Virginia
2. William Funk San Antonio, Texas
3. Richard Johnson La Jolla, Texas
4. Tina Zahel Winchester, Virginia
5. Rick Jones Kennett Square, Pennsylvania
6. John Funk Ventura, California
7. Christopher March Virginia Tech,
Blacksburg, Virginia
8. Walt Arrison Philadelphia, Pennsylvania
9. James Alarie University of Michigan -- Flint
Flint, Michigan
10. Steve Muller Clearbrook, Virginia
11. Ben Reames Columbus, Georgia
12. Tristan Collins Winchester, Virginia
13. Ashley Neumann Winchester, Virginia
14. Pam Hedrick Winchester, Virginia