Linear Diophantine equation

Could someone walk me through the steps on solving this one?:

A man cashes a check for d dollars and c cents at a bank. Assume that the teller by mistake gives the man c dollars and d cents. Assume that the man does not notice the error until he has spent 23 cents. Assume further that he then notices that he has 2d dollars and 2c cents. Assume still further that he asks you what amount the check was for. Assuming that you can except all the assumptions, what is the answer?

Thanks for your help!

Expressing everything in terms of cents, we have
(100c + d) - 23 = 100 * (2d) + 2c
==> 98c - 199d = 23.

To solve the diophantine equation, let's reduce modulo 98:
0c - 3d = 23 (mod 98)
==> -3d = 23 - 98 = -75 (mod 98)
==> d = 25 (mod 98).

Writing d = 25 + 98k for some integer k, we obtain
98c - 199(25 + 98k) = 23
==> c = 199k + 51.

Since c is originally in cents, we need c in [0, 100].
Hence, k = 0.
==> (c, d) = (51, 25).

So, the original check was $25.51.

I hope this helps!