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Carlos has a box of coins that he uses when playing poker with friends. The box currently contains 46 coins consisting of pennies dimes and quarters. The number of pennies is equal to the number of dimes and the total value is $4.09. How many of each denomination of coin does he have?

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facundo3141592

We must write and solve a system of equations to see how many of each type of coin Carlos has.

The answer is:

  • There are 19 pennies.
  • There are 19 dimes.
  • There are 8 quarters.

We start by defining the variables:

  • x = number of pennies in the box.
  • y = number of dimes in the box.
  • z = number of quarters in the box.

We know there are 46 coins in total, thus:

x + y + z= 46

The number of pennies and dimes is the same, thus:

x = y

The total value of the coins in the box is $4.09, then:

x*$0.01 + y*$0.10 + z*$0.25 = $4.09

So we can write the system of equations:

x + y + z= 46

x = y

x*$0.01 + y*$0.10 + z*$0.25 = $4.09

First, we can replace x by y in the first and third equations (by using the second one) to get:

2y + z =  46

y*$0.11 + z*$0.25 = $4.09

Now we can isolate z in the first equation to get:

z = 46 - 2y

And replace that in the other equation:

y*$0.11 + (46 - 2y)*$0.25 = $4.09

Now we can solve the equation for y:

y*$0.11 + (46 - 2y)*$0.25 = $4.09

y*$0.11 + $11.50 - y*$0.50 = $4.09

y*($0.11 - $0.50) = $4.09 - $11.50

y*(-$0.39) = -$7.41

y = (-$7.41)/(-$0.39) = 19

if y = 19, then x = 19

By using the first equation:

x + y + z = 46

19 + 19 + z = 46

z = 46 - 19 - 19  = 8

We can conclude that:

  • There are 19 pennies.
  • There are 19 dimes.
  • There are 8 quarters.

If you want to learn more about systems of equations, you can read:

https://brainly.com/question/13997560

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