Bus Fare Problem

Date: 02/12/98 at 23:36:05
From: Heather Zardus
Subject: Math problem

A group of students went on a bus tour. Each one paid his or her exact 
fare with 5 coins. In totaling, the bus driver obtained $21.83.  How 
many pennies did he receive?

I tried to find a prime number that divides into that number but 
did not find one.  I don't even know how to begin this problem because 
I feel that there is too much information missing.  

Please help! Thank you.

Date: 02/19/98 at 18:08:48
From: Doctor Schwenoha
Subject: Re: Math problem

You were on the right track but probably gave up before you got high 
enough.  

I tried your idea of dividing by prime numbers and got $21.83 divided 
by 37 equals $.59. I couldn't find 5 coins which added up to $.59 so I 
kept going with the prime numbers and got to $21.83 divided by 59 
equals $.37. You can get $.37 with five coins by using one quarter, 
two nickels, and two pennies. So, 59 passengers gave the driver two 
pennies each and that means that the driver received 118 pennies.

The information was there for us to explore and the thing that happens 
with a lot of my students is the same thing that happened to you.  
Trust your instincts and take your idea higher and higher. If you 
weren't sure whether or not you needed a prime number, then you could 
have just made a chart of all the possibilities starting with 1 and 
going up one at a time.  

You could save a lot of time with process by thinking about numbers 
that would end in the "3" of the $21.83. One times three (or three 
times one) could work; seven times nine (or nine times seven) could 
work. The number we're looking for must end in a 1, 3, 7, or 9, so 
we could skip all the other numbers in our table. The numbers we 
need to check (divide $21.83 by as possible number of passengers) are 
1, 3, 7, 9, 11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 37, 39, etc.  
Do you see it?  

If you believe the problem is "solvable" then you just have to keep 
trying the numbers until you find the one that works.  

-Doctor Schwenoha,  The Math Forum
Check out our web site http://mathforum.org/dr.math/