If a and b are two rational numbers, prove that a + b, a - b, ab are rational numbers. If b ≠ 0, show that a/b is also a rational number.

Let a = p/q where q ≠ 0 and b = r/s where s ≠ 0 be the rational numbers then
(i) a + b ==  
where q s ≠ 0, since q ≠ 0 and s ≠ 0.
Also p s + r q is an integer.
Hence a + b is a rational number.
(ii) a - b = =
where q s ≠ 0, since q ≠ 0 and s ≠ 0. 
Also p s - r q is an integer
Hence a - b is a rational number.
(iii) ab =
where q s ≠ 0, since q ≠ 0 and s ≠ 0.
Also pr is an integer.
Hence ab is a rational number.
(iv) Since b ≠ 0, we have r/s ≠ 0 thus r ≠ 0 and s ≠ 0.

where q ≠ 0 and r ≠ 0.
Also ps is an integer
Hence a/b is a rational number.