By solving a system of equations we conclude that Collen has 6 quarters, 4 dimes, and 2 nickels.
<h3>How many of each coin does Colleen have in her pocket?</h3>
First, let's define the variables:
- x = number of quarters.
- y = number of nickels.
- y = number of dimes.
There are 12 coins, so:
x + y + z = 12
She has 2 dollars in total, so:
x*0.25 + y*0.05 + z*0.10 = 2
And she has 3 times as many quarters as nickels.
x = 3y
Then we can write the system of equations (or matrix, depending on how you like to see it):
x + y + z = 12
x*0.25 + y*0.05 + z*0.10 = 2
x - 3y = 0
To use the elimination method we just need to add/subtract these equations to remove variables.
If we subtract the third equation to the first one:
(x + y + z) - (x - 3y) = 12 - 0
4y + z = 12
Now we have two equations:
x*0.25 + y*0.05 + z*0.10 = 2
4y + z = 12
If we multiply the first one by 4:
x + y*0.20 + z*0.40 = 8
Subtracting the third eq:
x + y*0.20 + z*0.40 - (x - 3y) = 8 - 0
y*3.20 + z*0.40 = 8
Now the remaining equations are:
4y + z = 12
y*3.20 + z*0.40 = 8
On the top one, we can rewrite: z = 12 - 4y
Replacing that in the other equation:
y*3.20 + (12 - 4y)*0.40 = 8
Now we eliminated the variables x and z, so we can solve this for y:
y*1.60 + 4.8 = 8
y*1.60 = 3.2
y = 3.2/1.6 = 2
Now that we know the value of y, the values of x and z are:
x = 3y = 3*2 = 6
z = 12 - 4y = 12 - 4*2 = 4
Then we conclude that Collen has 6 quarters, 4 dimes, and 2 nickels.
If you want to learn more about systems of equations:
brainly.com/question/13729904
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