Question

# Show that $0.2353535\ldots ..=0.\overline{235}$ can be expressed in the form of $\dfrac{p}{q}$, where p and q are integers and $q\ne 0$.

Hint: Assume the given number as x and multiply it by 100, you will get a new equation. Now subtract x from the new equation as $99x=23.53535....-0.2353535....$, then simplify the equation you will get the fraction number.

To convert the above decimal into a fractional number we will assume the number as ‘x’
Therefore, $x=0.2353535\ldots ..=0.\overline{235}$ …………………………… (1)
We will multiply the above equation by 100 therefore we will get,
$\therefore 100x=0.2353535....\times 100$
After multiplication we will get,
$\therefore 100x=23.53535....$ ……………………………… (2)
Now we will subtract equation (1) from equation (2) therefore we will get,
$\therefore 100x-x=23.53535....-0.2353535....$
By taking ‘x’ common we will get,
$\therefore 99x=23.53535....-0.2353535....$
If we subtract 0.2353535…. from 23.53535…. we will get,
$\therefore 99x=23.300000....$
As we know that the zero’s on the right side of the decimal has no values, therefore we can write the above equation as shown below,
$\therefore 99x=23.3$
If we shift the 99 on the right hand side of the equation we will get,
$\therefore x=\dfrac{23.3}{99}$
Now if we observe the above equation then we can say that the decimal is nearly converted into the fraction except the remaining one decimal.
To convert the remaining decimal into fraction completely we will multiply and dived the right hand side of the equation by 10 as there is only one number after the decimal. Therefore we will get,
$\therefore x=\dfrac{23.3\times 10}{99\times 10}$
Multiplying 23.3 by 10 we will get,
$\therefore x=\dfrac{233}{99\times 10}$
Also multiplying 99 by 10 we will get,
$\therefore x=\dfrac{233}{990}$
Now we have to simplify the above equation if possible. As there’s no possibility of further simplification therefore we can write the final answer as,
$\therefore x=\dfrac{233}{990}$
If we compare the above equation with equation (1) we will get,
$0.2353535\ldots ..=0.\overline{235}=\dfrac{233}{990}$

Note: While subtracting $99x=23.53535....-0.2353535....$ do remember the chain of 35….. will continue and has no end therefore the subtraction after the first 5 will become zero.
Like, Don’t represent $99x=23.53535....-0.2353535....$ as $99x=23.5353500-0.2353535....$ you will get the wrong answer.