Problem Solving Strategy: Counting Coins

Q: How many ways can you make 25 cents if you can use quarters, dimes, nickels or pennies?
Start with the largest value:

1 Quarter      1 way

2 Dimes, 1 Nickel   You can stop here since it implies 2 ways.
2 Dimes, 0 Nickel ( which implies 5 pennies)

1 Dime, 3 Nickels , which implies 4 ways.
1 Dime, 2 Nickels ( 5 pennies)
1 Dime, 1 Nickel   (10 pennies)
1 Dime, 0 Nickel ( 15 pennies)

0 Dime, 5 N, which implies 6 ways.
: , 4 N  (5 pennies)
: , 3 N  (10 pennies)
: , 2 N  (15 pennies)
: , 1 N  (20 pennies)
: , 0 N (25 pennies)

So altogether 13 ways.

Practice questions: (answers below)

Q 1 : How many ways can you make a. 15cents, b. 20cents, c. 30 cents if you can use quarters, dimes, nickels or pennies?

Q 2 : How many different combination of coins could a person have if she has exactly 21 cents?

Q 3 :Using nickels, dimes, quarters and/or half-dollars, how many ways can you make 75 cents?

Q 4: 20 coins of quarters and nickels add up to 4 dollars. How many nickels are there? 

Q5:  What is the least number of US coins to make changes possible from 1 to  99 cents inclusive? (half dollar is allowed)










Answers: 

#1: a. 6 ways ; b. 9 ways; c 18 ways ;
#2: 9 ways
#3: 22  ways
#4: 5 nickels and 15 quarters   
#5: 9 coins (1 half dollar, 1 quarter, 2 dimes, 1 nickel and 4 pennies)