25 Coins in a Dollar

Date: 05/07/2001 at 20:05:37
From: Jessica Johnson
Subject: Money

Carlos has 25 U.S. coins whose total is $1.00. What are the coins?

Money isn't my best math topic. I've tried a lot of different coins, 
but I'm either too high or too low. I just can't get the right amount.

Date: 05/08/2001 at 15:03:11
From: Doctor Rick
Subject: Re: Money

Hi, Jessica.

It can be done with guess-and-check, if we pick our guesses wisely.

For instance, let's start by picking a number of pennies. What if 
there is just one penny? If you try it a few times, you'll see that 
whatever the numbers of other coins (nickels, dimes, quarters or half 
dollars), the total value ends in a 1 or a 6. (It doesn't matter how 
many coins you have.) For instance,

  1 quarter + 3 dimes + 4 nickels = 1*25 + 3*10 + 4*5 = 75 cents
  75 cents + 1 penny = 76 cents

There's a reason for this: all coins except pennies are multiples of 5 
cents. Any amount you make by adding multiples of 5 is also a multiple 
of 5 (as 75 is in my example). Multiples of 5 end in either 0 or 5, so 
when you add 1 penny, you get a number that ends in 1 or 6.

We can't make 100 cents if there is only one penny, because 100 
doesn't end in 1 or 6. What numbers of pennies could we have?

Start by supposing you have the largest number of pennies possible. 
How many non-pennies must there be, so that the total number of coins 
is 25? How much are the pennies worth? How much must the non-pennies 
be worth? Can you find this many non-pennies that are worth exactly 
this much?

If not, try the next smaller number of pennies. You can keep going 
this way if you need to. You'll be trying a lot fewer possibilities 
than if we hadn't made these observations.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/   

Date: 05/08/2001 at 15:11:15
From: Doctor Twe
Subject: Re: money

Hi - thanks for writing to Dr. Math.

Actually, there are 5 different ways to make $1.00 using 25 coins.

We know that the solution will involve pennies, because 25 coins that 
are worth $0.05 (a nickel) or more would have to be worth at least

     25 * $0.05 = $1.25

which is more than a dollar.

We also know that the number of pennies has to be a multiple of 5 
(i.e. 5, 10, 15, ...) because otherwise we'd have a few cents left 
over. We know it can't be 25 pennies (that would be 25 coins right 
there, and only totals $0.25) or more.

So that leaves four options for the pennies: 5 pennies, 10 pennies, 
15 pennies, or 20 pennies. Let's consider each case:

5 PENNIES:
That's 5 coins totaling $0.05. That means we need the other 20 coins 
to total $0.95. Since 20 nickels are worth $1.00 (and, of course, 
using larger coins would make the total even larger), we can't get 
$0.95 using 20 coins that aren't pennies.

10 PENNIES:
That's 10 coins totaling $0.10. That means we need the other 15 coins 
to total $0.90. On the average, those coins must be worth:

     $0.90 / 15 = $0.06

Since this is more than a nickel, it might be possible. Can you come 
up with a combination of 15 coins (nickels, dimes, quarters or half-
dollars only) worth $0.90?

15 PENNIES:
That's 15 coins totaling $0.15. That means we need the other 10 coins 
to total $0.85. On the average, those coins must be worth:

     $0.85 / 10 = $0.085   (8.5 cents)

Again, since this is more than a nickel, it might be possible. Can you 
come up with a combination of 10 coins (nickels, dimes, quarters or 
half-dollars only) worth $0.85?

20 PENNIES:
That's 20 coins totaling $0.20. That means we need the other 5 coins 
to total $0.80. On the average, those coins must be worth:

     $0.80 / 5 = $0.16

Once more, it might be possible. Can you come up with a combination of 
5 coins (nickels, dimes, quarters or half-dollars only) worth $0.80?

A couple of hints:

1. There are two ways each to do it with 20 pennies and 15 pennies, 
   and there is one way to do it with 10 pennies.

2. Try solving the 20-pennies combinations first. Since there are 
   fewer coins left (only 5 instead of 10 or 15), they should be 
   easier to find. You may find a system that you can then use to find 
   the other solutions.

3. You can use similar reasoning to break these cases down into 
   sub-cases. For example, for the 10-pennies solutions, we are 
   looking for 5 coins worth $0.80. There could be either 0 or 1 
   half-dollar... There could be either 0, 1, 2, or 3 quarters... 
   and so on.

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum
  http://mathforum.org/dr.math/