Pocket Change 3

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

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14 comments

  1. An example would be 5 dimes and 10 nickels

  2. Mrs. Elliott's 4th graders

    10 pennies, 1 nickel, 1 dime, 3 quarters
    1 dime, 13 nickels, 1 quarter
    9 dimes, 1 nickel, 5 pennies

  3. Another example us 1 quarter, 13 nickels, 1 dime

  4. Quarters:1
    Nickles:12
    Dimes:4

  5. Quarters:2
    Nickles:5
    Dimes:3
    Pennies:5

  6. Isabella Mendoza

    9 dimes 1 nickel and 5 pennies

  7. 1 quarter 3 dimes 8 nickels 5 pennies = $1.00

  8. 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

  9. I know one answer it is 5 dimes and 10 nickels

  10. 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.

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