Question

How do you convert $0.12345$ ($345$ repeating) to a fraction?

Answer 1

Hint: In the above question, the decimal number $0.12345$ is given to us, in which the digits $345$ are being repeated. So the actual representation of the decimal number will be \[0.12\overline{345}\]. For converting it to a fraction, we can let it equal to some variable $n$ so that we can write the equation $n=0.12\overline{345}$. On multiplying this equation by $100$, we will get the equation $100n=12.\overline{345}$ and on multiplying it by $100000$, we will get \[100000n=12345.\overline{345}\]. On subtracting these two equations obtained after multiplication, we will get the fractional form of the given number.

Complete step by step solution:
The number given in the above question is $0.12345$. Since $345$ is being repeated, we can write the given number as \[0.12\overline{345}\]. Let us equate it to some variable, say $n$. Therefore, we can write the equation
$\Rightarrow n=0.12\overline{345}.......\left( i \right)$
Multiplying both sides of the above equation (i) by $100$, we get
$\begin{align}
  & \Rightarrow 100n=0.12\overline{345}\times 100 \\
 & \Rightarrow 100n=12.\overline{345}........\left( ii \right) \\
\end{align}$
Now, we multiply both sides of the equation (i) by $100000$ to get the equation
\[\begin{align}
  & \Rightarrow 100000n=0.12\overline{345}\times 100000 \\
 & \Rightarrow 100000n=0.12345\overline{345}\times 100000 \\
 & \Rightarrow 100000n=12345.\overline{345}........\left( iii \right) \\
\end{align}\]
Subtracting the equation (iii) from the equation (ii) we get
\[\begin{align}
  & \Rightarrow 100000n-100n=12345.\overline{345}-12.\overline{345} \\
 & \Rightarrow 99900n=12333 \\
\end{align}\]
Finally, dividing both the sides of the above equation by \[99900\], we get
$\begin{align}
  & \Rightarrow \dfrac{99900n}{99900}=\dfrac{12333}{99900} \\
 & \Rightarrow n=\dfrac{12333}{99900} \\
\end{align}$
Simplifying the fraction on the RHS, we get
$\Rightarrow n=\dfrac{4111}{33300}$
Hence, the decimal number given in the above question is converted to the fraction as $\dfrac{4111}{33300}$.

Note: Do not ignore the statement given in the bracket in the above expression, which states that the digits $345$ are being repeated. So do not take the number as $0.12345$. Also, do not forget to simplify the obtained fraction $\dfrac{12333}{99900}$ by cancelling the numerator and the denominator by $3$.