## How Many Dimes?

How Many Dimes?
A woman has $\2.15$ in change in her purse, made up entirely of dimes and quarters. Given that there are more quarters than dimes in her purse, what is the total number of dimes?
Source: NCTM Mathematics Teacher, November 2006

SOLUTION
Let $q$ be the number of quarters and $d$ the number of dimes.
$25q+10d=215$
$10d=215-25q$
$=5(43-5q)$
Simplify
$2d=43-5q\qquad (1)$
Since $2d$ is even, $43-5q$ is even which implies that $q$ is odd.
If $q=9, 9q=225 > 215$
If $q=7, 7q=175 < 215$
Substitute the value of $q$ into Eq.$(1)$
$2d=43-5(7)=8$
$d=4$
If $q=5$, Eq. $(1)$ is also satisfied but with $9$ dimes.
The total number of dimes equals $4$.

Answer: $4$ dimes