Here is something that you could do almost without thinking.
The definition of a transitive relation applied to R: for any three numbers x, y, z, if and , then . One has to prove this. Let's fix some numbers x, y and z and assume that and . This means there are rational numbers a / b and c / d such that x - y = a / b and y - z = c / d. One has to prove that x - z is a rational number.
Here you actually turn on the brain: x - z = (x - y) + (y - z). Is this number rational?