Answer to September 6, 2004 Problem of the Week

When I get a haircut, I normally pay my barber with an exact number of dollar bills (which include the cost of the haircut plus the tip). There are no coins involved in the transaction.

One day, I did not have the exact number of dollar bills, so I gave her a twenty-dollar bill so that she would give me change for the cost of my haircut (which costs less than ten dollars). I would then have enough dollar bills to give her a tip, assuming that she gave me all one-dollar bills.

To my surprise, I discovered that my barber thought I was a senior citizen (people 62 or older get one dollar off the regular price of a haircut) because she gave me one dollar too much in change. I was tipping her 37.5% (based on a regular haircut), but how much did my barber think I had been tipping her (based on a senior citizen haircut)? What is the cost of a senior citizen haircut?

Extra Credit: How much do you think I will tip my barber next time?

Solution to the Problem:

My barber thought I was tipping her 57.1% and the cost of a senior citizen haircut is $7.00 (one dollar off the regular price of $8.00).

Let x = the cost of a regular haircut.
Let x-1 = the cost of a senior citizen haircut.
Then x + .375x = whole number of dollars (cost of haircut plus tip)
So 1.375x = whole number of dollars
And 1 3/8 x = whole number of dollars
Therefore x must be a multiple of 8.
If x = 8, Then the total of the haircut and tip is 11 dollars.
If x = 16, Then the total is 22 dollars, but x must be less than 10 dollars.
So the cost of a haircut is $8.00, and
the cost of a senior citizen haircut is $7.00.
I normally tipped her $3.00 on an eight dollar haircut for 37.5%.
But she thought I was tipping her $4.00 on a seven dollar haircut for 57.1%!

Correctly solved by:

1. James Alarie	University of Michigan -- Flint, Flint, Michigan
2. Josh Feingold	California Institute of Technology Pasadena, California
3. Tristan Collins	Winchester, Virginia
4. Ben Schultz	Winchester, Virginia
5. Jim Arrison	Norristown, Pennsylvania
6. Jeffrey Gaither	Winchester, Virginia
7. Tyler Windham	Columbus, Georgia
8. Joe Jenkins	College of William & Mary Williamsburg, Virginia
9. Walt Arrison	Philadelphia, Pennsylvania
10. John Funk	Ventura, California
11. Julie W.	Dallas, Texas
12. John Harcourt	Columbus, Georgia
13. David Amos	Winchester, Virginia
14. Chris McCormick	Virginia Tech, Blacksburg, Virginia
15. David & Judy Dixon	Bennettsville, South Carolina
16. Erik Hultgren	Tullängsskolan, Örebro, Sweden
17. Jonas Sutinen	Tullängsskolan, Örebro, Sweden
18. Misty Carlisle	Winchester, Virginia

September 6, 2004 Problem of the Month

Correctly solved by:

September 6, 2004
Problem of the Month