Package of problems

## Suggestion

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?

How do you subtract two fractions with different denominators?

How many pairs of fractions could you select?

If you have two algebraic fractions, $\frac{a}{b}$ and $\frac{c}{d}$, how could you apply your approach to numerical fractions, like those on the cards, to add or subtract the algebraic ones?

If you cannot 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, does this mean that it’s impossible?

If you can find a pair of fractions, can you find another? Can you describe the conditions on the values of $a$, $b$, $c$ and $d$ that will, or will not, give a pair of fractions which have a difference that is a unit fraction?