Question

How do you convert $0.27\left(0.27\right)$ to a fraction?

Answer 1

Hint: In this problem we need to convert the given decimal into the fraction. Generally, the conversion of the decimals into fractions depends on whether the digits in the decimal are being repeated or not. In the problem, they have mentioned that the digits are repeating. So, we will first assume the given decimal is equal to a variable let’s say $x$. Now we will observe how many digits are repeated in the given decimal. We need to multiply with ${10}^n$ where $n$ is the number of repeated digits. Now we will subtract both the values and simplify them to get the result.

Complete step by step solution:
Given decimal $0.27$.
Let $x=0.27$
Given that $27$ is being repeated. In $27$ we have $2$ digits. So, we will multiple with ${10}^2$ on both sides of the above equation, then we will get
$x\times {10}^2=0.27\times {10}^2$
We know that ${10}^2=100$, then we will have
$x\times 100=0.27\times 100$
Given that $27$ is being repeated, so we can write
$100x=27.27$
Subtracting the value of $x$ from the above equation, then we will get
$\begin{array}{rl}& 100x-x=27.27-0.27\\ & \Rightarrow 99x=27\end{array}$
Dividing the above equation with $99$ on both sides, then we will get
$\begin{array}{rl}& {\displaystyle \frac{99x}{x}}={\displaystyle \frac{27}{99}}\\ & \Rightarrow x={\displaystyle \frac{27}{99}}\end{array}$
We can write $27=9\times 3$ and $99=9\times 11$, then we will get
$x={\displaystyle \frac{9\times 3}{9\times 11}}$
Cancelling the $9$ in numerator and denominator, then we will have
$x={\displaystyle \frac{3}{11}}$

Hence the fraction of $0.27$ is ${\displaystyle \frac{3}{11}}$.

Note:
In the problem, they mentioned that the digits are repeated. So, we have followed the above method. If they have not mentioned that the digits are not repeated, then it can be easier to convert the decimal into a fraction. We have two digits after the decimal so we will divide the given number with ${10}^2$ and remove the decimal. Then we will get $0.27={\displaystyle \frac{27}{100}}$.