SOLUTION: What are the different combinations of nickels, dimes, and/or quarters that equal exactly 45 cents?

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Question 341148: What are the different combinations of nickels, dimes, and/or quarters that equal exactly 45 cents?
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!

The equation is:

5n + 10d + 25q = 45

It can be divided through by 5
 
n + 2d + 5q = 9

We will consider two cases

Case 1.  Using no quarters, i.e., q=0, then

n + 2d + 5(0) = 9

n + 2d = 9

d = %289-n%29%2F2

n must be an odd number to make the numerator even so it
can be divided evenly by the denominator 2.

1.  n=1 nickel, %289-1%29%2F2=8%2F2=%22%224 dimes
2.  n=3 nickels, %289-3%29%2F2=6%2F2=%22%223 dimes.
3.  n=5 nickels, %289-5%29%2F2=4%2F2=%22%222 dimes.
4.  n=7 nickels, %289-7%29%2F2=2%2F2=%22%221 dime.
5.  n=9 nickels, %289-9%29%2F2=0%2F2=%22%220 dimes.

Case 2. q=1

n + 2d + 5(1) = 9

n + 2d + 5 = 9

n + 2d = 4

d = %284-n%29%2F2

n must be an even number to make the numerator even so it
can be divided evenly by the denominator 2.  (Don't forget
that 0 is an even number!)

6.  n=0 nickels, %284-0%29%2F2=4%2F2=%22%222 dimes
7.  n=2 nickels, %284-2%29%2F2=2%2F2=%22%221 dime.
8.  n=4 nickels, %284-4%29%2F2=0%2F2=%22%220 dimes.

So there are 8 ways to make 45 cents using nickels and/or dimes and/or 
quarters. 

Edwin