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Tiny Nines

What do you notice about these families of recurring decimals?

Repetitiously

Age 14 to 16
Challenge Level

This problem follows on from Terminating or Not and accompanies Tiny Nines

A recurring decimal is a decimal with a digit, or group of digits, that repeats forever.
For example, $\frac13 = 1 \div 3 = 0.33333333...$ with the $3$s repeating forever.
We can write this as $0.\dot{3}$

Imagine I started with the number $x=0.\dot{2}$
How could you write $2.\dot{2}$ in terms of $x$?
Can you find two different ways?

Click below to reveal a hint.

What do you need to multiply $x$ by to get $2.\dot{2}$?
What do you need to add to $x$ to get $2.\dot{2}$?

Can you create an equation, and then solve it to express $x$ as a fraction?

Now let's consider $y=0.2525252525...$, where the digits $2$ and $5$ keep alternating forever.

This can be written as $0.\dot{2}\dot{5}$, with dots over the first and last digit in the repeating pattern.

How could you write $25.\dot{2}\dot{5}$ in terms of $y$, in two different ways?

Can you create an equation, and then solve it to express $y$ as a fraction?

Now try writing the following recurring decimals as fractions (if you get stuck, take a look here):

  • $0.\dot{4}0\dot{5}$
  • $0.8\dot{3}$
  • $0.002\dot{7}$

Can you describe a method that will allow you to express any recurring decimal as a fraction?