Simplify $: 0. ¯ ¯ ¯ 4 + 0. ¯ ¯¯¯¯ ¯ 61 + 0. ¯ ¯¯¯¯ ¯ 11 - 0. ¯ ¯¯¯¯ ¯ 36$

0. ¯ ¯¯¯¯ ¯ 83

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0. ¯ ¯¯¯¯ ¯ 87

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0. ¯ ¯¯¯¯ ¯ 80

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0. ¯ ¯¯¯¯ ¯ 85

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Solution

The correct option is A ( $0. ¯ ¯¯¯¯ ¯ 80$ )
Given $: 0. ¯ ¯ ¯ 4 + 0. ¯ ¯¯¯¯ ¯ 61 + 0. ¯ ¯¯¯¯ ¯ 11 - 0. ¯ ¯¯¯¯ ¯ 36$
Let $y = ¯ ¯¯¯¯¯ ¯ 0.4$
$\Rightarrow y = 0.44444...$ .......(i)
Multiplying $10$ on both sides, we get
$\Rightarrow 10 y = 4.44444....$ ..... .(ii)
Subtracting equation (ii) and equation(i), we get
$\Rightarrow 10 y - y = 4.44444..... - 0.4444.....$
$\Rightarrow 9 y = 4$
$∴ y = ¯ ¯¯¯¯¯ ¯ 0.4 = \frac{4}{9}$
Similar way, $0. ¯ ¯¯¯¯ ¯ 61 = \frac{61}{100}$
Thus, given expression can be written as
$: 0. ¯ ¯ ¯ 4 + 0. ¯ ¯¯¯¯ ¯ 61 + 0. ¯ ¯¯¯¯ ¯ 11 - 0. ¯ ¯¯¯¯ ¯ 36 = \frac{4}{9} + \frac{61}{99} + \frac{11}{99} - \frac{36}{99} = \frac{4}{9} + \frac{72}{99} - \frac{36}{99}$
$= \frac{4}{9} + \frac{36}{99}$
$= \frac{44}{99} + \frac{36}{99}$
$= \frac{80}{99}$
$= 0. ¯ ¯¯¯¯ ¯ 80$

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