Question

How do you convert $0.1624$ ($24$ being repeated) to a fraction?

Answer 1

Hint: In the above question, we have been given a decimal number $0.1624$, in which the digits $24$ are being repeated. So the actual decimal number given can be written as $0.16\overline{24}$. For converting it to a fraction, we can equate it to some arbitrary variable, say $n$ to write the equation $n=0.16\overline{24}$. On multiplying both sides of the equation by $100$ and $10000$, we will obtain two equations. On subtracting the two equations and using the algebraic operations, we will obtain the given number in the form of a fraction.

Complete step by step solution:
According to the above question, we are given the decimal number $0.1624$ in which $24$ is being repeated. So we can write the given decimal number as $0.16\overline{24}$. Let us equate it to a variable $n$ so that we can write the equation
$\Rightarrow n=0.16\overline{24}......\left( i \right)$
Multiplying both sides of the above equation by $100$, we get
\[\begin{align}
  & \Rightarrow 100n=0.16\overline{24}\times 100 \\
 & \Rightarrow 100n=16.\overline{24}......\left( ii \right) \\
\end{align}\]
Now, we multiply both the sides of the equation (i) by $10000$ to get
\[\begin{align}
  & \Rightarrow 1000n=0.16\overline{24}\times 10000 \\
 & \Rightarrow 1000n=0.1624.\overline{24}\times 10000 \\
 & \Rightarrow 1000n=1624.\overline{24}......\left( iii \right) \\
\end{align}\]
Subtracting the equation (ii) from the equation (ii) w get
$\begin{align}
  & \Rightarrow 10000n-100n=1624.\overline{24}-16.\overline{24} \\
 & \Rightarrow 9900n=1608 \\
\end{align}$
Finally, dividing both sides of the above equation, we get
$\begin{align}
  & \Rightarrow \dfrac{9900n}{9900}=\dfrac{1608}{9900} \\
 & \Rightarrow n=\dfrac{1608}{9900} \\
\end{align}$
Simplifying the fraction on the RHS, we finally get
$\Rightarrow n=\dfrac{134}{825}$
Hence, the given decimal number is converted to the fraction as $\dfrac{134}{825}$.

Note: While solving the subtraction $1624.\overline{24}-16.\overline{24}$ in the above solution, do not get confused. The bar symbol over a digit means that digit is being repeated infinitely many times. So the digits $\overline{24}$ after the decimal points in the subtraction $1624.\overline{24}-16.\overline{24}$ will get cancelled and will be simplified as $1624-16$.