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How do you convert $0.312(312{\text{ repeating)}}$ to a fraction?

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Last updated date: 19th Jun 2024
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Answer
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Hint:Fraction is the number when represented in the form of the numerator upon the denominator. Here we will take the given number and will suppose using some variable. Variables are terms expressed using the small alphabets such as x, y, z,…

Complete step by step solution:
Take the given number: $0.312312....$
Let us suppose that
$x = 0.312\overline {312} $ ….. (A)
(it is given that it is repeating)
Repeating means the number repeated again and again. It is also known as recurring numbers.
Take the equation (A) and multiply both the sides of the equation with the number
Then,
$1000x = 312.312\overline {312} $
The above equation can be re-written as –
$1000x = 312 + 0.312\overline {312} $
From equation (A), the above equation can be written as –
$1000x = 312 + x$
Take a variable on the left hand side of the equation. When you move any term from one side to another then the sign of the term also changes. Positive term changes to the negative term and vice- versa.
$ \Rightarrow 1000x - x = 312$
Simplify the above equation:
$ \Rightarrow 999x = 312$
Term multiplicative on one side, if moved to the opposite side then it goes to the denominator.
$ \Rightarrow x = \dfrac{{312}}{{999}}$
Find the factors of the above term
$ \Rightarrow x = \dfrac{{104 \times 3}}{{333 \times 3}}$
Common multiple from the numerator and the denominator cancel each other.
$ \Rightarrow x = \dfrac{{104}}{{333}}$
This is the required solution.

Note: Always remember that when you multiply any term on one side of the equation, it should be multiplied on both the sides of the equation. Equations should be converted in the form of equivalent. To convert decimal into fraction, place the decimal number over its place value. For example, for $0.6$ the six is in the tenths place so that we place $6$ over $10$ to create the equivalent fraction i.e. $\dfrac{6}{{10}}$