Answer
Sam has $7$ dimes and $5$ 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 represented by the coins.
First, define the variables:
Let $x$ = the number of dimes
Let $y$ = the number of quarters
Let $0.10x$ = how much the dimes add up to
Let $0.25y$ = how much the quarters add up to
Set up the two equations:
There are $12$ coins in all so
$x + y = 12$
The total value of the coins is $\$1.95$ so
$0.10x + 0.25y = 1.95$
Hence, the system of equations needed to solve this problem is:
$x+y=12$
$0.10x+0.25y=1.95$
Solve the first equation for $x$ so we can use that expression to plug in for $x$ in the second equation:
$x = 12 - y$
Plug this expression into the second equation:
$0.10(12 - y) + 0.25y = 1.95$
$1.2 - 0.10y + 0.25y = 1.95$
$1.2 + 0.15y = 1.95$
Subtract $1.2$ from each side of the equation:
$0.15y = 0.75$
Divide each side by $0.15$ to solve for $y$:
$y = 5$
Now that the value for $y$ has been determined, plug this value into the first equation to find $x$:
$x + 5 = 12$
$x = 7$
Therefore, Sam has $7$ dimes and $5$ quarters.
