Extented Response. How many different ways can you make change for a quarter?

Question

Show all possible ways. Explain in words how you got your answer and why you took the steps you did to solve the problem.

Answer 1

Start with no pennies:
1 quarter
2 dimes, 1 nickel
1 dime, 3 nickels
5 nickels

Now try 5 pennies:
2 dimes, 5 pennies
1 dimes, 2 nickels, 5 pennies
4 nickels, 5 pennies

Now try 10 pennies
1 dime, 1 nickel, 10 pennies
3 nickels, 10 pennies

Then 15 pennies
1 dime, 15 pennies
2 nickels, 15 pennies

And 20 pennies:
1 nickel, 20 pennies

Finally:
25 pennies

Answer 2

Start by getting using 25 cents in dimes and nickels:
2 dimes + 1 nickel
1 dime + 3 nickels
5 nickels

Then get 20 cents in dimes and nickels, plus 5 pennies:
2 dimes + 5 pennies
1 dime + 2 nickels + 5 pennies
4 nickels + 5 pennies

Then get 15 cents in dimes and nickels, plus 10 pennies:
1 dime + 1 nickel + 10 pennies
3 nickels + 10 pennies

Then get 10 cents in dimes and nickels, plus 15 pennies:
1 dime + 15 pennies
2 nickels + 15 pennies

Lastly, you have:
1 nickel + 20 pennies
25 pennies

I get 12 different ways. I am assuming that we are not using 1/2 cents, 2 cent pieces, and 3 cent pieces, which they haven't made since the 1800's.

Answer 3

I think the most systematic way to work this out is to start by assuming no pennies. The solutions are: 2 dimes + 1 nickle, 1 dime + 3 nickles, or 5 nickles (3 solutions)

Now assume exactly 5 pennies. To make 20 cents using no pennies, you can have: 2 dimes, 1 dime + 2 nickles, or 4 nickles (3 more solutions)

Now assume exactly 10 pennies. To make 15 cents using no pennies, you can have: 1 dime + 1 nickle, or 3 nickles (2 more solutions)

Now assume exactly 15 pennies. To make 10 cents using no pennies, you can have: 1 dime or 2 nickles (2 more solutions)

Now assume exactly 20 pennies. To make 5 cents using no pennies, the remaining coin has to be a nickle (1 more solution)

Lastly, you can have 25 pennies. (1 last solution)

Add them up, and you get 12 solutions in all

Answer 4

25 pennies
5 nickels
two dimes one nickel
4 nickels 5 pennies
one dime one nickel 10 pennies.
you know i really don't feel like listing them all. but you get the point

Answer 5

Not to be a jerk but wouldnt it be faster to do your own homework?