The following eight unit fractions are given: \[\dfrac12\quad\dfrac13\quad\dfrac14\quad\dfrac15\quad\dfrac16\quad\dfrac17\quad\dfrac18\quad\dfrac19\]

What is the probability that a pair of distinct fractions, chosen at random from these, have a difference that is also a unit fraction?

What can you say about the pairs of fractions that have a difference that is a unit fraction?

Can you generalise your findings?

The fractions in this problem are all unit fractions. Can you find a pair of fractions, \(\frac{a}{b}\) and \(\frac{c}{d}\) where \(a\neq 1\) and \(c\neq 1\), which have a difference that is a unit fraction?

Can you expand on your answer to the previous question – under what conditions will a pair of fractions, \(\frac{a}{b}\) and \(\frac{c}{d}\) where \(a\neq 1\) and \(c\neq 1\), have a difference that is a unit fraction?

You may like to print and cut out cards with these fractions, so that they may be physically arranged in pairs.