November 2009
Problem of the Month

Tax Problem
by Sidney Kravitz
in GAMES Magazine July 2009

• Only one page is needed to print the tax laws of the land of Kerplunkin.
   
• The federal tax is 20 percent, and the only deductions allowed are the state tax to be paid the same year and the municipal tax to be paid the same year.
• The state tax is 10 percent, and the only deductions allowed are the federal tax to be paid the same year and the municipal tax to be paid the same year.
• The municipal tax is 5 percent, and the only deductions allowed are the federal tax to be paid the same year and the state tax to be paid the same year.
   
• If taxes are rounded off to the nearest penny, what are the federal, state, and municipal taxes due on an income of $100,000?

Solution to the Problem:

The federal tax is $17,683.56, the state tax is $7,859.36, and the municipal tax is $3,722.85.

Let f, s, and m equal the amounts of federal, state, and municipal taxes to be paid. Then:

f = 1/5 * (100000 - s - m)
s = 1/10 * (100000 - f - m)
m = 1/20 * (100000 - f - s)

Multiplying both sides of the equations by 5, 10, and 20, respectively, yields:

5f = 100000 - s - m
10s = 100000 - f - m
20m = 100000 - f - s

Subtracting the second equation from the first gives:
5f - 10s = -s + f which can be rewritten 4f = 9s or s = 4f/9.

Subtracting the third equation from the first gives:
5f - 20m = -m + f or 4f = 19m or m = 4f/19.

Substituting 4f/9 for s and 4f/19 for m in the first equation produces:

5f = 100000 - 4f/9 - 4f/19 which simplifies to 967f = 17100000, or f = 17,683.56 (approximately).

Since we have already shown that s = 4f/9 and m = 4f/19, we can easily calculate the values of s and m.

Correctly solved by:

1. Gusti Oggenfuss	Montet (Broye), SWITZERLAND
2. Anna Szentirmai	John Handley High School, Winchester, Virginia

Send any comments or questions to: David Pleacher