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### 50 Coins = One Dollar

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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
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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
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/
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