WWW.C E M C .U WAT E R LO O.C A | T h e C E N T R E fo r E D U C AT I O N i n M AT H E M AT I C S a n d CO M P U T I N G
Problem of the Week
Problem A and Solution
Bakery Blast
Problem
Aiden goes to the bakery to buy a cake for his dad’s birthday. The cake costs $8.75. He uses a
$10 bill to pay for the cake and gets change in return. In Canada there are 5¢ coins (nickels),
10¢ coins (dimes), 25¢ coins (quarters), $1 coins (loonies) and $2 coins (toonies), and 100¢ is
equal to $1.
A) Suppose the bakery does not have any nickels, but has all other coins. How many coin
combinations can you find that make the correct change?
B) Suppose the bakery does not have any quarters, but has all other coins. How many coin
combinations can you find that make the correct change?
Solution
First we need to determine how much change Aiden receives. We can subtract
the cost of the cake from the $10 using the decimal values, or we can convert
everything to cents. So,
10.00
1000
−8.75 or −875
1.25
125
Alternatively, we can add the change necessary to get from $8.75 to $10.00.
If we add 25¢ to $8.75 we get $9.00. If we add $1.00 to $9.00, we get $10.00.
Using any of these techniques, we see that Aiden receives $1.25 change.
Next we need to determine how many different combinations produce $1.25.
WWW.C E M C .U WAT E R LO O.C A | T h e C E N T R E fo r E D U C AT I O N i n M AT H E M AT I C S a n d CO M P U T I N G
A) Since the amount is less than $2, it is not possible to include a $2 coin in the
change. There are 4 possible combinations of coins that Aiden could receive
that do not include any nickels (5¢).
Combination $1 25¢ 10¢
1
1
1
2
5
3
3
5
4
1
10
B) If the bakery does not have any quarters (25¢), then there are 16
combinations.
Combination $1 10¢ 5¢
1
1
2
1
2
1
1
3
3
1
5
4
12 1
5
11 3
6
10 5
7
9
7
8
8
9
9
7 11
10
6 13
11
5 15
12
4 17
13
3 19
14
2 21
15
1 23
16
25
Note that in order to give correct change for $1.25, we need either quarters or
nickels, since the values, in cents, of all the other coins end with a zero.
WWW.C E M C .U WAT E R LO O.C A | T h e C E N T R E fo r E D U C AT I O N i n M AT H E M AT I C S a n d CO M P U T I N G
Teacher’s Notes
To determine the fewest number of coins required to make correct change many
people would use a technique known as the Greedy Algorithm. With this
approach you choose the highest denomination coin that is less than the amount
of change required. Then you continue choosing coins the highest denomination
that is less than the amount of change still required.
For example, if you need to make change for $4.85, you choose a $2 coin first.
Now you still have $2.85 worth of change to make. Then you choose a second $2
coin. You are left with 85¢ in change. Next you choose a quarter, which leaves
you 60¢, then another quarter leaving 35¢, then another quarter, leaving 10¢.
Finally you choose a dime to cover the remaining change.
In total you have used 6 coins: 2 toonies, 3 quarters, and 1 dime.
The Greedy Algorithm works well for minimizing the number of coins required to
make change with Canadian currency, but it will not always work. Suppose we
introduced a new coin that is worth $1.50, and we wanted to make change for
$4.85. Now, we could do that with 3 $1.50 coins, 1 quarter, and 1 dime for a
total of 5 coins. The Greedy Algorithm might be used in many situations where
we want to minimize or maximize some process. We have to be careful not to
assume that this technique will always provide the best result.