Question

How do you convert $0.23\bar 4$ (with $4$ repeating) as a fraction

Answer 1

In this question, we need to convert $0.23\bar 4$ (with $4$ repeating) into fraction. Here, we will consider $0.23\bar 4$ as x. So, to bring the repeating entity immediately after the decimal point, we multiply and divide the given decimal $0.23\bar 4$ by $100$. Then, as there is only $1$ digit being repeated. So, we multiply and divide the decimal by $10$.

Complete step-by-step solution:
In this question, we need to convert $0.23\bar 4$ to a fraction.
Let x be that fraction.
Here, consider the given value as $x = 0.23\bar 4$.
Now, let us multiply and divide $0.23\bar 4$ by $100$, we have,
$x = 0.23\bar 4 \times \dfrac{{100}}{{100}}$
Then, $100x = 0.23\bar 4 \times 100$
$\Rightarrow 100x = 23.\bar 4$
Hence, $100x = 23.4444....$
Let us consider this as the equation $\left( 1 \right)$.
Now, let us multiply and divide $23.\bar 4$ by $10$, we have,
$100x = 23.\bar 4 \times \left( {\dfrac{{10}}{{10}}} \right)$
Then, $1000x = 23.\bar 4 \times 10$
$\Rightarrow 1000x = 234.\bar 4$
Hence, $1000x = 234.\bar 4$
Let us consider this as equation $\left( 2 \right)$.
Now, we will subtract equation $\left( 1 \right)$ from equation $\left( 2 \right)$.
Therefore, we have,
$1000x - 100x = \left( {234.\bar 4 - 23.\bar 4} \right)$
Hence, $900x = \left( {234.4444.... - 23.4444.....} \right)$
$\Rightarrow 900x = 211$
$\Rightarrow x = \dfrac{{211}}{{900}}$
Therefore, $x = \dfrac{{211}}{{900}}$
Hence, the converted value of $0.23\bar 4$ to a fraction is $\left( {\dfrac{{211}}{{900}}} \right)$.

Note: In this question it is important to note that, here we have multiplied and divided $0.23\bar 4$ firstly by $100$ and then by $10$ respectively, then subtracted both the equations to determine the value of x as in this question we have a repetition of a repetition of $4$ in $0.23\bar 4$. The scenario may be different in each question depending on the situation as the decimal may have more number of digits as its repeating entity.