Question

How do you convert $0.219(19$repeating) to a fraction?

Answer 1

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 answer:
Take the given number: $0.219(19$repeating)
Let us suppose that
$x = 0.219\overline {19} $ ….. (A)
 (it is given that it is repeating)
Repeating means the number repeated again and again. It is also known as re-occurring numbers.
Take the equation (A) and multiply both the sides of the equation with the number
Then,
$10x = 2.1919\overline {19} $ ….. (B)
Take the equation (A) and multiply both the sides of the equation with the number
Then,
$1000x = 219.1919\overline {19} $ ….. (C)
Now, subtract equation (B) from equation (A), the above equation can be written as –
$1000x - 10x = 219.19\overline {19} - 2..19\overline {19} $
Simplify the above equation finding the subtraction on both the sides of the equation -
$ \Rightarrow 999x = 217.00000$
Term multiplicative on one side, if moved to the opposite side then it goes to the denominator.
$ \Rightarrow x = \dfrac{{217}}{{999}}$
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. $\dfrac{6}{{10}}$