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Can you express every recurring decimal as a fraction?

Tiny Nines

Age 14 to 16
Challenge Level

This problem follows on from Terminating or Not and accompanies Repetitiously

In the video below, watch as three calculations are performed on the calculator.
The fourth calculation is not completed. Can you predict what the result will be?

This video has no sound.

If you can't see the video, click below.

The calculations shown are $$1 \div 9$$ $$1 \div 99$$ $$1 \div 999$$
Work out these answers, and then use them to predict the answer to $1 \div 9999$.

The decimal representations of $\frac19$, $\frac{1}{99}$, $\frac{1}{999}$ and $\frac{1}{9999}$ can be used to help you work out the decimal representations of other families of fractions. 

Can you use what you now know to make predictions about the decimal representations of these and other fractions?

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

Can you show that the recurring decimals in your predictions are equivalent to the fractions that they are supposed to represent?

Calculator used in video: