Courses
Courses for Kids
Free study material
Offline Centres
More
Store

# Express $1.\overline{81}$ in the form of $\dfrac{p}{q}$ where p and q are integers and q is not equal to zero.

Last updated date: 24th Jul 2024
Total views: 348.6k
Views today: 4.48k
Verified
348.6k+ views
Hint: In this type of question we have to use the concept of conversion of decimal numbers into fraction. Here, we consider the given decimal number equal to some variable and call it as equation 1. Then we multiplied both sides by 100 and called it as equation 2. When we perform multiplication by 100, the number of zeros is equal to the number of digits after the decimal point. Then by performing subtraction of equation 1 from equation 2 we can obtain the result.

Here, we have to express $1.\overline{81}$ in the form of $\dfrac{p}{q}$ where p and q are integers and q is not equal to zero.
$\Rightarrow x=1.\overline{81}=1.818181\cdots \cdots \text{ ---}\left( 1 \right)$
\begin{align} & \Rightarrow 100\times x=100\times 1.\overline{81} \\ & \Rightarrow 100\times x=181.8181\cdots \cdots \text{ ---}\left( 2 \right) \\ \end{align}
\begin{align} & \Rightarrow 99x=180 \\ & \Rightarrow x=\dfrac{180}{99} \\ \end{align}
$\Rightarrow x=\dfrac{20}{11}$
Hence, $1.\overline{81}$ can be expressed as $\dfrac{20}{11}$ in the form of $\dfrac{p}{q}$ where p and q are integers and q is not equal to zero.
Note: In this type of question students have to take care when they perform multiplication by 100. Students have to note that the number of zeros over 1 is equal to the number of digits after the decimal point. Also students have to note that in $1.\overline{81}$ the number 81 gets repeated infinitely many times hence we can write $1.\overline{81}$ as 1.818181……