Question

How do you convert repeating decimals into fractions?

Answer 1

Hint: In this question, we need to find the method to convert repeating decimals into fractions.
Rational numbers, when written as decimals, are either they are terminating, repeating decimals. Converting terminating decimals into fractions is straight forward. However, when the decimals are repeating, things are a little more difficult. Repeating decimals occur very frequently both when doing simple arithmetic and when solving competition problems. So, being able to convert them into fractions is a valuable skill.

Complete step by step solution:
Let us consider the given problem.
Here we are asked to determine a process to convert repeating decimals into fractions.
So we explain the process in some steps along with the example, so that it becomes easier to understand.
We follow the following five steps to convert repeating decimals into fractions.
In the first step, we consider the variable x equal to the repeating decimal we are trying to convert to a fraction.
For example, let us take $x = 0.7777777777$
In the second step, examine the repeating decimal to find out the repeating digits.
In $x = 0.7777777777$, the repeating digit is 7.
In the step three, place the repeating digit or digits to the left of the decimal point.
Now to place the repeating digit 7, to the left of the decimal point, we need to move the decimal point one place to the right.
i.e. moving a decimal point one place to the right is done by multiplying the decimal number by 10.
When we multiply one side by a number, we have to multiply the other side by the same number to keep the equation balanced.
Hence we have, $10 \times x = 10 \times (0.7777777777)$
$ \Rightarrow 10x = 7.777777777$
In the fourth step, place the repeating digit to the right of the decimal point.
If we observe carefully in the example, the repeating digit is already to the right, so there is nothing else to do.
$x = 0.7777777777$
In the fifth step, we use the equations obtained in step 3 and step 4, we subtract both sides of these equations. When we subtract make sure that the difference is positive for both sides.
We have two equations,
$10x = 7.777777777$
$x = 0.7777777777$
Subtracting we get,
$ \Rightarrow 10x - x = 7.777777777 - 0.7777777777$
$ \Rightarrow 9x = 7$
Dividing both sides by 9 we get,
$ \Rightarrow \dfrac{{9x}}{9} = \dfrac{7}{9}$
$ \Rightarrow x = \dfrac{7}{9}$
Hence if we follow the above steps, we can convert a repeating decimal into fractions.

Note :
Students may forget some points while converting repeating decimals into fractions.
Some of them are mentioned below to avoid mistakes.
(a) Forgetting to put the decimal point right before the repeating digit.
(b) While subtracting two equations, forgetting to subtract the smaller one from the bigger one.
(c) Not keeping good track of the number of places the decimal point was moved.