50 Coins = One Dollar

Date: 02/24/97 at 03:22:06
From: wai
Subject: 50 coins to make up one dollar.

Dr. Math,

My teacher has given me a question that I can not find the answer to.
The question is to make up one dollar by using 50 coins.

Can you help me?

Chiho
(grade 3)

Date: 03/02/97 at 23:56:30
From: Doctor Barney
Subject: Re: 50 coins to make up one dollar.

Yes, I think I can help.  This was a fun problem.  I found two 
answers.  The trick is to keep track of how many pennies there are. 
 
I know that the number of pennies must be a multiple of 5, like 15, 
20, 25, 45,  numbers like that. The reason I know this is that since 
the other coins are all multiples of 5 cents, if I had a number like 
47 pennies the closest I could get to one dollar would be $1.02 or 
97 cents. Try adding some of the other coins to 47 cents and see how 
close you can get to one dollar.  

First I wanted to know the greatest number of pennies I could use.  
If I used 50 pennies, I could not use any other coins, so that was too 
many.  

Next I tried 45 pennies. With 45 pennies I could use 5 more coins, 
and those 5 coins would have to add up to 55 cents. Now 55 cents 
seemed like a lot of money for only 5 coins, so I decided to start 
with one of the coins being a quarter. With 45 pennies and one 
quarter I had 70 cents, so to make one dollar I would need 4 more 
coins which would add up to 30 cents. As it turned out, I did find 
one answer using 45 pennies and one quarter. Can you find 4 more 
coins to add to 45 pennies and one quarter to make one dollar all 
together?  (Hint: the other 4 coins are not pennies or quarters.)

So then I had one answer to the question AND I knew that the greatest 
number of pennies I could have was 45.  

Next I wanted to know what was the smallest number of pennies I could 
use. To start with, I guessed 30. Now, if I only had 30 pennies I 
would need to use 20 other coins, and even if they were all nickels 
that would be too much money! (If any of them were dimes or quarters 
that would be even more.) So the smallest number of pennies I could 
use was greater than 30.

Then I tried 35 pennies. With 35 pennies I would need to use 15 
other coins, and even if they were all nickels I would still have 
more than a dollar, right?

So finally I tried 40 pennies. With 40 pennies I needed to find 10 
other coins which would add up to 60 cents, so that the total of all 
the coins was exactly one dollar. It turns out that I was able to 
find a second answer by using 40 pennies. I bet you can too. Why 
not give it a try?  Write back again if you get stuck.  

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