Answer to the Problem of the Week
for the week of April 25, 2005

Another Barber Problem

A barber had his first customer of the day, who happened to be a friend. When he was done, the barber refused to take the money from the customer. The fellow said, "Look, I know we're friends, but, business is business. I want to pay for my haircut."

The barber said, "Here's what we'll do. You open the cash register. I don't have any idea how much money is in there. But, you match whatever is in there, and then take out 20 bucks."

The customer says, "Okay," and he does that.

The barber says, "Gee, I kind of like this." So, the next customer comes in, he gets his haircut, and the barber says, "You can do the same thing my first customer did. Open the cash register, match what's in there, and then take out 20 bucks."

The second customer does that, and he leaves. The third customer does the same. The fourth customer, after receiving his haircut, opens the cash register, and says, "I can't do it. "

The barber says, "Why not?"

"There's no money in here. Not a cent."

The question is, how much money was in there to start?

 

Solution to the Problem:

There was originally $17.50 in the drawer.
Let x = amount of money in the drawer at the start of the day.
After the first customer, there is 2x - 20 dollars in the drawer.
After the second customer, there is 2 (2x - 20) - 20 dollars left,
or 4x - 60 dollars.

After the third customer, there is 2 (4x - 60) - 20 dollars left,
or 8x - 140 dollars.

Set 8x - 140 = 0 and solve for x:
x = 17.50.

Keith Mealy made the following observations:
"If the till starts with $20, there is an equilibrium point, as each person adds and removes $20.

If the till starts with more than $20, then each person adds (progressively) more than he removes and the till will increase with each customer."

Jeffrey Gaither also remarked that, " The barber might want to be sure he has at least $20.01 in his register if he wants to be at all profitable in the future."

Correctly solved by:

1. Mike Singer Winchester, Virginia
2. William Proffitt Winchester, Virginia
3. Jim Arrison Norristown, Pennsylvania
4. John Funk Ventura, California
5. Mac Dillon Columbus, Georgia
6. Arsalan Heydarian Harrisonburg, Virginia
7. Walt Arrison Philadelphia, Pennsylvania
8. Larry Schwartz Trumbull, Connecticut
9. Angelo Leone Middletown, Connecticut
10. Amir Mohammadzaeh Tehran, Iran
11. David & Judy Dixon Bennettsville, South Carolina
12. Derick Gilkeson Harrisonburg, Virginia
13. Misty Carlisle Winchester, Virginia
14. Keith Mealy Cincinnati, Ohio
15. António Baptista Lisboa - Portugal
16. Alex McIntire Columbus, Georgia
17. Charles Washington Winchester, Virginia
18. Rob Adams Winchester, Virginia
19. Daniel Surber Winchester, Virginia
20. Jeffrey Gaither Winchester, Virginia
21. Ali Wade Columbus, Georgia
22. Arin Smith Winchester, Virginia
23. James Alarie University of Michigan -- Flint,
Flint, Michigan
24. Nathan Seifert Harrisonburg, Virginia
25. Ann Norman Columbus, Georgia
26. Jason Storer Winchester, Virginia
27. Chris Rogers Winchester, Virginia
28. Jackie Henson Columbus, Georgia
29. Nathan Riley Wichita, Kansas
30. Michael Lewis ----------
31. Rebecca Columbus, Georgia
32. Wajih Ansari Harrisonburg, Virginia
33. Sheila Shah Arlington, Virginia
34. Henry Woodward Columbus, Georgia
35. Emily Auerbach Columbus, Georgia
36. ShowSam@aol.com ----------
37. Gilbert Melanson Sierra Vista, Arizona
38. Jeanette Crawford Columbus, Georgia
39. Mary Helen Calhoun Columbus, Georgia