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Question id: 159. This question is similar to one that appeared in a GCSE Higher paper (specimen) for 2017. The use of a calculator is allowed.

Express as a single fraction and simplify your answer.

$$\frac{p-1}{q-1}-\frac pq$$Using your answer to part (a), prove that if \(p\) and \(q\) are positive integers and \(p \lt q\), then

$$\frac{p-1}{q-1}-\frac pq\lt 0$$
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