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Can you express every recurring decimal as a fraction?

Tiny Nines

Age 14 to 16
Challenge Level

Why do this problem

There are fascinating patterns to be found in recurring decimals. This problem explores the relationship between fraction and decimal representations. It's a great opportunity to practise converting fractions to decimals with and without a calculator.

Possible approach

This problem could be explored alongside Repetitiously

Show students the video in the problem, or ask them to find the decimal representations of $\frac19$, $\frac1{99}$, and $\frac1{999}$ for themselves. Invite them to predict what $\frac1{9999}$ will be as a decimal.

Challenge students to convince themselves, and convince you, that the decimal representations really do go on forever. They may do this by performing a division calculation by hand and considering the remainders, or converting the recurring decimals back into a fraction (as demonstrated in Repetitiously). Then they could explore other related fractions such as those suggested in the problem:

  • $\frac13$, $\frac{1}{33}$, $\frac{1}{333}$...
  • $\frac1{11}$, $\frac1{111}$, $\frac1{1111}$...
  • $\frac{23}{99}$, $\frac{37}{99}$, $\frac{52}{99}$, $\frac{n}{99}$

Key questions

How can a fraction be turned into a decimal representation?
Without using a calculator?

If you know that $\frac19=0.\dot{1}$, how can you work out $\frac13$ as a decimal?
If you know that $\frac1{99}=0.0\dot{1}$, how can you work out $\frac1{11}$ as a decimal?

Possible support

If they haven't already done so, students could start by exploring Terminating or Not.

Possible extension

Challenge students to prove that the patterns they have noticed will continue.