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This problem follows on from Terminating or Not and accompanies Tiny Nines
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.
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):
Can you describe a method that will allow you to express any recurring decimal as a fraction?