### Why do this problem?

This
problem introduces the idea that a decimal can 'go on forever'.
This idea is subtle and interesting and confusing.

### Possible approach

"I'm thinking of two numbers, I add $6$ to the smaller one to
get the larger one, OR I
could multiply the smaller one by $4$ to get to the larger one.
What are my numbers?"

Continue with more examples to wake everyone up and establish
that a difference and a ratio define two unknowns (but not
necessarily in those words).

Include examples that lead to simple fractions, e.g. $d=5$
$r=3$, and insist on fraction, not decimal answers.

Put $0.222222\ldots$ and $2.22222 \ldots$ on the board, and
ask how to get from the smaller to the larger in two ways. Work
through the processes for finding these "unknowns", alert to all
opportunities for students to talk about the meaning of these
recurring decimals.

Ask students to choose and work on their own pairs of related
recurring decimals, from those in the problem, or later, to make up
their own. It's easy for students to verify their final fraction on
a calculator.

### Key questions

- What does $0.22222222 \ldots$actually mean? How many decimal
places are there?
- Would multiplying by $10$ give a decimal that ended? Why
not?
- Can I do $2.2222222 \ldots$ divided by $0.222222222 \ldots$ on
a calculator?

### Possible extension

- What is the fraction equivalent of $0.999999 \ldots$?
$0.4999999 \ldots$?
- How would you find the fraction for $0.225225225 \ldots$ or
$0.222522252225 \ldots$
- Do you think that all recurring decimals will correspond to a
fraction?
- Can you work out which fractions will correspond to a recurring
decimal and which fractions will not?

### Possible support

Experimentation with a calculator for small numbers can help
students to get into the problem.

Students could be asked to catalogue decimal equivalents of
many common fractions, classifying the decimals as terminating,
recurring and "no obvious repeats". This data set can be used to
check work later, or to suggest recurring decimals to convert back.
Encourage students to classify and describe families of decimals
with clear recurring patterns (e.g. ninths and elevenths).

Encourage students to spot patterns and then to make a conjecture
about the result when dividing two recurring fractions.

$\frac{2.2}{0.22}, \frac{2.22}{0.222},\frac{2.222}{0.2222},
\frac{2.2222}{0.22222},$etc.

Can they extend this to the second part of the question?