Are these expressions the same c = 1cc + 1/4c why or why not

Yes 1c = c.

Technically, the "1" in front of the "c" is called an "implied 1" and, so it's simply written as c.

Here's, an example that should make this clear:
    Say c = 4
    Then (1)(c) = (1)(4) = 4 = c

1c + 1/4 c

What is a whole plus 1/4? 1 1/4 or 5/4
Same thing here:
c + 1/4 c or 1c + 1/4 c = 1 1/4c or 5/4c

The answer to that is 6