Question

How do you convert $0.7326$($326$ being repeated) to 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:
$\Rightarrow 0.7326$ ($326$ repeating)
We will consider the number to be $x$ therefore, it can be written as:
$\Rightarrow x=0.7326$ ($326$ repeating)
Now the number is recurring in $3$ decimal places therefore we will multiply the number by $1000$. It is to be kept in mind that the number is still recurring after the multiplication.
On multiplying, we get:
$\Rightarrow 1000x=732.6326$ ($326$ 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.
$\Rightarrow 1000x-x=732.6326-x$
On substituting the value of from the right-hand side of the equation, we get:
$\Rightarrow 1000x-x=732.6326-0.7326$
On simplifying the left-hand side of the equation, we get:
$\Rightarrow 999x=732.6326-0.7326$
On simplifying the right-hand side of the equation, we get:
$\Rightarrow 999x=731.9$
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:
$\Rightarrow x=\dfrac{731.9}{999}$
Now we have to eliminate the decimal in the numerator therefore, we will multiply the numerator and denominator by $10$.
On multiplying, we get:
$\Rightarrow x=\dfrac{731.9}{999}\times \dfrac{10}{10}$
On simplifying, we get:
$\Rightarrow x=\dfrac{7319}{9990}$, which is the required fraction.

Note: It is to be remembered that whenever a value is added, subtracted, multiplied or divided on both the sides of the equation, the value of the equation does not change. It is to be remembered that when a term which is in multiplication is transferred across the $=$ sign, it has to be written as division. Same rule applies for addition and subtraction.