Question

How do you convert $0.15$ repeating as 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 it to get the result.

Complete step-by-step solution:
Given decimal $0.15$.
Let $x=0.15$
Given that $15$ is being repeated. In $15$ we have $2$ digits. So, we will multiply with ${10}^2$ on both sides of the above equation, then we will get
$x\times {10}^2=0.15\times {10}^2$
We know that ${10}^2=100$. Substituting this value in the above equation, then we will have
$x\times 100=0.15\times 100$
Given that $15$ is being repeated, so we can write
$100x=15.15$
Subtracting the value of $x$ from the above equation, then we will get
$\begin{array}{rl}& 100x-x=15.15-0.15\\ & \Rightarrow 99x=15\end{array}$
Dividing the above equation with $99$ on both sides, then we will get
${\displaystyle \frac{99x}{99}}={\displaystyle \frac{15}{99}}$
Cancelling the $99$ which is in both numerator and denominator in the left-hand side, then we will get
$x={\displaystyle \frac{15}{99}}$
We can write $15=5\times 3$ and $99=3\times 33$. Substituting these values in the above equation, then we will get
$x={\displaystyle \frac{5\times 3}{33\times 3}}$
Cancelling the $3$ which is in both numerator and denominator, then we will have
$x={\displaystyle \frac{5}{33}}$
At the beginning we have assumed that $x=0.15$. From this we can write
$0.15={\displaystyle \frac{5}{33}}$
Hence the fractional form of the given decimal $0.15$ is ${\displaystyle \frac{5}{33}}$.

Note: In this problem they have particularly mentioned that the digits are repeated in the given decimal value. So, we have followed the above-mentioned method. If they have not mentioned that the digits are repeated, then it can be easier to convert the given decimal value into fraction. We have two digits after the decimal so we will divide the given number with ${10}^2$ and remove the decimal point. Then we will get $0.15={\displaystyle \frac{15}{100}}$.