3.  Express the following in the form \frac{p}{q} , where p and q are integers and q ≠ 0.

(ii)     0.4\bar{7}

Answers (1)

We can write 0.4\overline7 as

\Rightarrow 0.4\overline7 = \frac{4}{10}+ \frac{0.777..}{10}             -(i)

Now,

Let x = 0.\overline7= 0.7777....             -(ii)

Now, multiply by 10 on both sides

10x= 7.7777...

\Rightarrow 10x = 7 + x \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \(using \ (ii))

\Rightarrow 9x = 7

\Rightarrow x = \frac{7}{9}
Now, put the value of x in equation (i). we will get 

\Rightarrow 0.4\overline7 = \frac{4}{10}+ \frac{7}{10\times 9}= \frac{4}{10}+ \frac{7}{90} = \frac{36+7}{90} = \frac{43}{90}

Therefore,  \frac{p}{q}  form of  0.4\overline7  is  \frac{43}{90}

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