Question

How do you convert $0.78023$($023$ repeating) into a fraction?

Answer 1

Hint: We will consider the fraction to be a variable $x$ then since there is repetition, we will multiply the fraction to remove the $3$ decimal places and then subtract the initial value from the original value of the fraction such that we can eliminate the recurring terms in the decimal and then we will simplify the expression to get the required solution.

Complete step by step solution:
We have the number given to us as:
$\Rightarrow 0.78023$ ($023$ repeating)
We will consider the number to be $x$ therefore, it can be written as:
$\Rightarrow x=0.78023$ ($023$ repeating)
Now to remove the recurring decimal place, we will multiply both the sides of the equation by $1000$. It is to be kept in mind that even after multiplying, the number is still recurring.
$\Rightarrow 1000x=780.23$ ($023$ repeating)
We can see that the recurring decimal places start from the third decimal place in the number therefore, to remove the decimal place, we will subtract the initial value $x$ from the sides of the equation.
$\Rightarrow 1000x-x=780.23-x$
On substituting the value of $x$ from the right-hand side of the equation, we get:
$\Rightarrow 1000x-x=780.23-0.7803$
On simplifying the left-hand side of the equation, we get:
$\Rightarrow 999x=780.23-0.7803$
On simplifying the right-hand side of the equation, we get:
$\Rightarrow 999x=779.45$
Note that in the above step we have removed the recurring decimal places. Now on transferring the term $999$ from the left-hand side to the right-hand side, we get:
$\Rightarrow x=\dfrac{779.45}{999}$
Now we have to eliminate the decimal in the numerator therefore, we will multiply the numerator and denominator by $100$.
On multiplying, we get:
$\Rightarrow x=\dfrac{779.45}{999}\times \dfrac{100}{100}$
On simplifying, we get:
$\Rightarrow x=\dfrac{77945}{99900}$, which is the required fraction.

Note: In the above question, we have a smaller number in the numerator and a larger number in the denominator. These types of fractions are called proper fractions. There also exist improper fractions which are the opposite of it. It is to be remembered that multiplying and dividing a fraction by the same number does not change its value.