Question

How do you convert 0.26 (6 repeating) to a fraction?

Answer 1

Hint: We will write the given number by assigning a variable to it so that we have it in the form of an equation. Then we will form two equations by multiplying the original equation by powers of 10. Then we will subtract one equation from the other and obtain the whole number on the right hand side and also as a coefficient of the variable. After that we will obtain the fraction form of the given decimal number.

Complete step by step answer:
The given number is $0.2\overline{6}$. Let us write the number by assigning a variable to it. So, we have an equation as follows,
$x=0.2666\ldots $
Now, we will multiply the above equation by a power of 10. This power of 10 will be equal to the number of non-recurring digits after the decimal point. We have one digit which is non-recurring after the decimal point. So, multiplying the above equation by 10, we get
$10x=2.666\ldots \text{ -(i)}$
Next, we will form another equation by multiplying it with a power of 10 that is greater than the previous power of 10 by 1. So, we will multiply the above equation by ${{10}^{2}}$ which is 100. So, we get the following equation,
$100x=26.666\ldots \text{ -(ii)}$
We can see that in both equation (i) and equation (ii), the recurring part in the decimal number on the right hand side is the same. Now, let us subtract equation (i) from equation (ii). We get the following equation,
$90x=24$
Therefore, we obtain the fraction as
$x=\dfrac{24}{90}$
Hence, we have $0.2\overline{6}=\dfrac{24}{90}$.

Note:
We should be familiar with the concept of recurring and non-recurring decimal numbers. We can use this method to convert any recurring decimal number into its fraction form. The recurring decimal numbers and terminating decimal numbers are rational numbers. There are decimal numbers which are not rational numbers, that is, they cannot be written as fractions.