Answer

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**Hint:**In this question, we are given a decimal number and there is a repetition of 4 in the given number, the repetition is denoted by the bar on the digits 4, so the given number is irrational. In the given question we have to convert the number $0.2\overline 4 $ to a fraction, this is done by multiplying the given number with a power of 10 such that the power is equal to the number of digits that are repeated and then subtracting the obtained equation with the original one, we can reach to the correct answer.

**Complete step-by-step solution:**

The given number is $0.2\overline 4 $

So first we will multiply it by 10 to get only recurring digits after the decimal place –

$0.2\overline 4 \times 10 = 2.\overline 4 $

It can be rewritten as $2.444.....$ , let $2.\overline 4 = R$

In the given question, the number of repeating digits is 1 (4). So we will multiply $2.\overline 4 $ with ${10^1}$ , and then we subtract the result with original equation, as follows –

$

{10^1} \times R = {10^1} \times 2.444.... \\

\Rightarrow 10R = 24.444.... \\

$

Now subtracting $R$ from \[10R\] we get –

$

10R - R = 24.444.... - 2.444.... \\

9R = 22 \\

\Rightarrow R = \dfrac{{22}}{9} \\

\Rightarrow 2.\overline 4 = \dfrac{{22}}{9} \\

$

We know that –

\[

2.\overline 4 = 10 \times 0.2\overline 4 \\

\Rightarrow 0.2\overline 4 = \dfrac{{22}}{{90}} \\

\Rightarrow 0.2\overline 4 = \dfrac{{11}}{{45}} \\

\]

Hence, the recurring decimal \[0.2\overline 4 \] is written in the fraction form as \[\dfrac{{11}}{{45}}\] .

**Note:**\[0.2\overline 4 \] is an irrational number. Irrational numbers are those numbers that have repeating and non-terminating decimal expansion. So, we can convert the irrational numbers into a fraction using the method shown above. After converting it into fraction, we see that they both are a multiple of 2, so we write it in simplified form by canceling out the common factors.

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