Answer
Georgia has two more dimes than quarters in her bag.
Work Step by Step
Let us define some variables to set up the equations to be solved:
$x$ = the number of dimes Georgia has
$y$ = the number of quarters Georgia has
Each dime is worth $10$ cents and each quarter is worth $25$ cents, so
$0.10x$ = the amount of money Georgia has in dimes
$0.25y$ = the amount of money Georgia has in quarters
Let us set up an equation that states the number of dimes and quarters Georgia has is equal to $18$:
$x + y = 18$
We set up another equation that states that the amount of money Georgia has in dimes and the amount of money she has in quarters equals three dollars:
$0.10x + 0.25y = 3$
Let us put the two equations together to form the system:
$x + y = 18$
$0.10x + 0.25y = 3$
Let us try to solve using substitution. We solve the first equation for $x$ in terms of $y$ by subtracting $y$ from both sides of the equation to isolate $x$:
$x = 18 - y$
Let us plug this expression for $x$ into the second equation to solve for $y$:
$0.10(18 - y) + 0.25y = 3$
Use distributive property to simplify:
$0.10(18) - 0.10y + 0.25 y = 3$
$1.8 - 0.10y + 0.25y = 3$
Combine like terms:
$1.8 + 0.15y = 3$
Subtract $1.8$ from each side to isolate constants to the right side of the equation:
$0.15y = 1.2$
Divide each side by $0.15$ to solve for $y$:
$y = 8$
Now that we have the value for $y$, we can use this value to substitute for $y$ in the first equation:
$x + 8 = 18$
Subtract $8$ from both sides of the equation to solve for $x$
$x = 10$
Thus, Georgia has $10$ dimes and $8$ quarters.
This means that Georgia has two more dimes than quarters in her bag.
