SOLUTION: How MANY WAYS can you use 21 coins to make $1? i have tried 2 WAYS so fAR.... ONE WAY......2 QUARTERS, 3 DIMES,1 NICKEL,15 PENNIES ANOTHER WAY.......7 DIMES, 4 NICKELS,10 P

Question 561946: How MANY WAYS can you use 21 coins to make $1? i have tried 2 WAYS so fAR....
ONE WAY......2 QUARTERS, 3 DIMES,1 NICKEL,15 PENNIES
ANOTHER WAY.......7 DIMES, 4 NICKELS,10 PENNIES
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
With a systematic analysis I find 4 solutions and can be pretty sure that I have all the solutions.
Let's define variables
= number of quarters
= number of dimes
= number of nickels
= number of pennies
There are 21 coins, so
The value adds to $1.00, so -->
So must be a multiple of 5, meaning that it must be 0, 5, 10, 15, or 20.
It cannot be 0, because the 21 coins would add up to at least $1.05 (21 nickels).
It cannot be 20 pennies, because the remaining coin would have to be worth $0.80.
FIVE PENNIES
Wirh 5 pennies the remaining 16 coins add up to 95 cents, so we have
--> and

Combining both equations we get .
The only integer solution is ,
Substituting in
we get -->
First solution: no quarters, 3 dimes, 13 nickels, 5 pennies
TEN PENNIES
With 10 pennies the remaining 11 coins add up to 90 cents, so we have
--> and

Combining both equations we get .
The integer solutions are , with
and , with
For , with ,
means -->
For , with ,
means -->
Two more solutions:
1 quarter, 3 dimes, 7 nickels, 10 pennies
no quarters, 7 dimes, 4 nickels, 10 pennies
FIFTEEN PENNIES
With 15 pennies the remaining 6 coins add up to 85 cents, so we have
--> and

Combining both equations we get .
The integer solutions are , with ,
, with ,
and , with
However, only the first solution works with
and gives us -->
One more solutions:
2 quarters, 3 dimes, 1 nickel, 15 pennies
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