Answer

Verified

341.7k+ views

**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.

Recently Updated Pages

The branch of science which deals with nature and natural class 10 physics CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Define absolute refractive index of a medium

Find out what do the algal bloom and redtides sign class 10 biology CBSE

Prove that the function fleft x right xn is continuous class 12 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Draw a diagram of nephron and explain its structur class 11 biology CBSE

What is the color of ferrous sulphate crystals? How does this color change after heating? Name the products formed on strongly heating ferrous sulphate crystals. What type of chemical reaction occurs in this type of change.

Write the difference between soap and detergent class 10 chemistry CBSE

Give 10 examples of unisexual and bisexual flowers

Differentiate between calcination and roasting class 11 chemistry CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

What is the difference between anaerobic aerobic respiration class 10 biology CBSE

a Why did Mendel choose pea plants for his experiments class 10 biology CBSE