A woman paid a $\$9.75$ debt using only dimes and quarters. In how many ways is this possible?

Question

I am trying to solve this in my numbers theory class. I have no issues finding the gcd of $5$, but I have been having issues with working backwards to find $x$ and $y$ so I can find $t$. To tell the truth, I have an issue with completing the process using the Euclidean Algorithm to find $x$ and $y$. If I can get help finding $x$ and $y$, I am confident that I can do the rest.

Answer 1

So basically your problem is to find number of possible solutions of the equation

$9.75 = 0.1x + 0.25y$ where $x$ is number of dimes and $y$ is number of quarters.

Multiplying by 100 and dividing by $5$ (or multiplying by $20$) on both sides, we have

$$195 = 2x + 5y$$

$$x = \frac{5(39-y)}{2}$$

So here we see that for x to be an integer, $y$ should be odd.

Hence possible values of $y$ are $1,3,5,7,\ldots,39$. Correspondingly calculate $x.$

So there are $20$ solutions or $20$ possible ways to solve your problem.