Proving Equivalence of y is Rational, y/3 is Rational, 2y+5 is Rational

[PowerStrike]

Well I have to prove the following statements are equivalent:
a. y is a rational number
b. y/3 is a rational number
c. 2y+5 is a rational number
So a -> b -> c -> a

I'm not quite sure how you are suppose to prove something is rational however.
I started like this:

* y = q/r Where q & r are integers, r is not 0, no common factors other than 1 [Therefore rational]

* y/3
q/r/3 ... (q/r)(1/3)= q/3r ... Which is rational because everything is integer math?

* 2y+5 ... 2(q/r) + (5/1) ... same reason

This doesn't seem to work like the ones I've done before with proving something is even (2n) or odd (2n+1). Is there some step I'm not getting?

 

[TD]

I don't see why they can't have common factors (other than 1), 8/6 is a rational number too. I think it's sufficient that it can be written as p/q where q is non-zero and p and q are integers. It should be too hard then, it actually seems trivial.

If y = p/q (with p and q as above), then y/3 is p/(3q) and 3q is still an integer since integer*integer is another integer, hence y/3 is rational too. Etcetera, I assume.

 

[0rthodontist]

For 2y+5 you should condense it into a single fraction with nonzero integer numerator and denominator.

 

[VietDao29]

For 2y+5 you should condense it into a single fraction with nonzero integer numerator and denominator.

Why can't a fraction with a numerator of 0 be a rational number. As long as the denominator is not 0, we have y = 0 / r = 0 (a rational number).
Yes, I agree with the nonzero integer denominator part, but not numerator!
Am I missing something?

 

[0rthodontist]

Oh yeah, right. I meant denominator. I kind of added the nonzero part at the end without thinking.