Answer
We have $30$ dimes and $12$ quarters.
Work Step by Step
In this exercise, we need to set up a system of equations: one equation adds up the number of coins while the other equation adds up the amount of money we have for the coins.
First, let's define our variables:
Let $x$ = the number of dimes
Let $y$ = the number of quarters
Let $0.10x$ = the total value of the dimes
Let $0.25y$ = the total value of the quarters
Now, we can set up our two equations:
$x + y = 42$
$0.10x + 0.25y = 6$
Let's solve the first equation for $x$ so we can use that expression to plug in for $x$ in the second equation:
$x = 42 - y$
Plug this expression into the second equation:
$0.10(42 - y) + 0.25y = 6$
Use distributive property:
$4.2 - 0.10y + 0.25y = 6$
Combine like terms:
$4.2 + 0.15y = 6$
Subtract $4.2$ from each side of the equation:
$0.15y = 1.8$
Divide each side by $0.15$ to solve for $y$:
$y = 12$
Now that we have the value for $y$, we can plug this into the first equation to find $x$:
$x + 12 = 42$
Subtract $12$ from each side to solve for $x$:
$x = 30$
Therefore, we have $30$ dimes and $12$ quarters.
