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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?