Unattributed TMUA Mock Set A Paper 2 · Question 6

Consider the following claim: The sum of two rational numbers is a rational number. Here is an attempted proof of this claim:

I. According to the definition of rational numbers, $\dfrac ab$ and $\dfrac cd$ are rational numbers.
II. Expanding the form $\dfrac ab+\dfrac cd$ , using the rational number definition, we get $\dfrac{ad+bc}{bd}$ .
III. Since both $(ad+bc)$ and $bd$ are integers, and $bd$ is non-zero, according to the definition of rational numbers, $\dfrac{ad+bc}{bd}$ is a rational number.

Which of the following best describes this proof?

AThe proof is completely correct, and the claim is true

BThe proof is completely correct, but there are counterexamples to the claim

CThe proof is wrong, and the first error occurs on line I

DThe proof is wrong, and the first error occurs on line II

EThe proof is wrong, and the first error occurs on line III

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