Question

Madison collected a bunch of dimes and quarters for her school fundraiser. she collected a total of 122 coins and a total of $21.50. How many dimes and how many quarters did she collect?

Answer 1

Madison collected 60 dimes and 62 quarters for her school fundraiser by setting up a system of equations and solving for the number of each coin type.

Step-by-step explanation:

Problem Solving: Counting Coins

Madison collected a total of 122 coins consisting of dimes and quarters, which together amount to $21.50. To figure out how many dimes (10 cents) and quarters (25 cents) she collected, we should set up a system of equations. Let d represent the number of dimes and q represent the number of quarters.

The first equation reflects the total number of coins: d + q = 122.

The second equation reflects the total value of the coins in cents: 10d + 25q = 2150 (since $21.50 equals 2150 cents).

Now, we solve this system of equations. Multiplying the first equation by 10 to align with the dime values in the second equation, we get:

10d + 10q = 1220

Then, subtract this from the second equation to eliminate d:

15q = 2150 - 1220

15q = 930

q = 62

Now that we know there are 62 quarters, we substitute q in the first equation to find d:

d + 62 = 122

d = 122 - 62

d = 60

So, Madison collected 60 dimes and 62 quarters for her school fundraiser.

Answer 2

Let:

x = number of dimes

y = number of quarters

She collected a total of 122 coins, so:

`x+y=122`

and a total of 21.50, so:

`0.1x+0.25y=21.50`

Let:

`\begin{gathered} x+y=122_{\text{ }}(1) \\ 0.1x+0.25y=21.50_{\text{ }}(2) \end{gathered}`

From (1):

`x=122-y_{\text{ }}(3)`

Replace (3) into (2):

`\begin{gathered} 0.1(122-y)+0.25y=21.50 \\ 12.2-0.1y+0.25y=21.50 \\ 0.15y=9.3 \\ y=(9.3)/(0.15) \\ y=62 \end{gathered}`

Replace the value of y into (3):

`\begin{gathered} x=122-62 \\ x=60 \end{gathered}`

She collected 60 dimes and 62 quarters