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
