$\frac{p}{q}$ form of the number $0.123 ¯ ¯¯¯¯¯¯ ¯ 456$ , where p and q are integers and $q \neq 0$ , is ____

\frac{3333}{99000}

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\frac{41111}{999000}

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\frac{41111}{333000}

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\frac{12333}{99000}

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Solution

The correct option is C $\frac{41111}{333000}$
Given number is $0.123 ¯ ¯¯¯¯¯¯ ¯ 456$
Let, $x = 0.123 ¯ ¯¯¯¯¯¯ ¯ 456$
Multiply both sides of $x$ by 1000
$\Rightarrow 1000 x = 123.456456......$ ---(1)
Now, multiply both sides of $x$ by 1000000
$\Rightarrow 1000000 x = 123456.456456.......$ ---(2)

Subtracting (1) from (2), we get
$1000000 x - 1000 x = (123456.456456...) - (123.456456..)$
$\Rightarrow$ $999000 x = 123333$
$\Rightarrow x = \frac{123333}{999000} = \frac{41111}{333000}$

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Question 2
A number of the form $\frac{p}{q}$ is said to be a rational number, if
(a) p and q are integers.
(b) p and q are integers and q ≠ 0
(c) p and q are integers and p ≠ 0
(d) p and q are integers and p ≠ 0 also q ≠ 0.

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pq form of the number 0.123¯¯¯¯¯¯¯¯456, where p and q are integers and q≠0, is ____

$\frac{p}{q}$ form of the number $0.123 ¯ ¯¯¯¯¯¯ ¯ 456$ , where p and q are integers and $q \neq 0$ , is ____