Magic with Fractions and Decimals

1. Start with

$x = 0.3333\ldots$

Here, a single digit 3 is repeating, so multiply both sides by 10.

$10x = 3.3333\ldots$ (The 3’s never end.)

Subtract the first equation from the second equation.

$10x - x = 3.3333\ldots - 0.3333\ldots$

$9x = 3$ (Just like magic!)

$x = \dfrac{3}{9} = \dfrac{1}{3}$

and we are done!

2. Start with

$x = 0.272727\ldots$

Here, two digits 27 are repeating. So, multiply both sides by 100.

$100x = 27.272727\ldots$ (The 27’s never end.)

Subtract the first equation from the second equation.

$100x - x = 27.272727\ldots - 0.272727\ldots$

$99x = 27$

$x = \dfrac{27}{99} = \dfrac{3}{11}$

So,

$0.272727\ldots = \dfrac{3}{11}$

Do you have another way of solving this? Please post a reply below.

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