Directions: You have $1.00 in change in your pocket. You have 15 coins. What coins do you have?

### Hint

Modifications: Provide coin manipulatives to make the problem more concrete.

Questions: Have you drawn out the coin possibilities?

### Answer

Solutions may vary (e.g. 5 dimes and 10 nickels)

Source: Andrew Gael

An example would be 5 dimes and 10 nickels

10 pennies, 1 nickel, 1 dime, 3 quarters

1 dime, 13 nickels, 1 quarter

9 dimes, 1 nickel, 5 pennies

Another example us 1 quarter, 13 nickels, 1 dime

Nickles:5

Dimes:9

Quarters:1

Nickles:12

Dimes:4

Quarters:2

Nickles:5

Dimes:3

Pennies:5

9 dimes 1 nickel and 5 pennies

1 quarter 3 dimes 8 nickels 5 pennies = $1.00

5 dimes and 50 pennies

There are exactly six solutions. Let (pennies, nickels, dimes, quarters) represent the number of each type of coin. For example, (0,10,5,0) represents no pennies, 10 nickels, 5 dimes, and no quarters. The six solutions, therefore, are:

(0,10,5,0)

(0,13,1,1)

(5,1,9,0)

(5,4,5,1)

(5,7,1,2)

(10,1,1,3)

I knew that the number of pennies had to be a multiple of 5. I also knew the number of quarters would greatly affect the total. I combined these two ideas with the method of solving systems of equations. I found more solutions, but they have negative numbers in them, so they obviously don’t apply to this situation, but they did reveal a cool pattern.

First, I started with no pennies and no quarters. This gave the solution (0,10,5,0). Then, I examined the case of no pennies and one quarter. This resulted in the solution (0,13,1,1). Then, I considered no pennies and two quarters. The resulted in (0,16,-3,2), which is not possible. Note however that it does result in 15 coins and $1. The next case was (0,19,-7,3). Notice that each time I add a quarter, the number of nickels goes up by 3 and the number of dimes goes down by 4. This makes sense because adding 3 nickels (15 cents) and subtracting 4 dimes (-40 cents) results in a decrease of 25 cents, which is offset by the increase of the quarter. Notice also that adding 1 quarter and 3 nickels and subtracting 4 dimes results in 4 added coins and 4 removed coins, keeping the number of coins the same.

Displaying the results in a table(ish) form, further illuminates the pattern:

no pennies

0, 10, 5, 0

0, 13, 1, 1

0, 16, -3, 2

0, 19, -7, 3

0, 22, -11, 4

five pennies

5, 1, 9, 0

5, 4, 5, 1

5, 7, 1, 2

5, 10, -3, 3

5, 13, -7, 4

ten pennies

10, -8, 13,0

10, -5, 9, 1

10, -2, 5, 2

10, 1, 1, 3

10, 4, -3, 4

10 pennies

4 dimes

1 half-dollar

I know one answer it is 5 dimes and 10 nickels

here you go all answers 0,10,5,0.

0,13,1,1.

5,1,9,0.

5,4,5,1.

5,7,1,2.

10,1,1,3.

five dimes and ten nickels