
Express \[1.\overline{81}\] in the form of \[\dfrac{p}{q}\] where p and q are integers and q is not equal to zero.
Answer
410.4k+ 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.
Complete step by step answer:
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.
Let us suppose,
\[\Rightarrow x=1.\overline{81}=1.818181\cdots \cdots \text{ ---}\left( 1 \right)\]
Now, multiplying both sides by 100 we get,
\[\begin{align}
& \Rightarrow 100\times x=100\times 1.\overline{81} \\
& \Rightarrow 100\times x=181.8181\cdots \cdots \text{ ---}\left( 2 \right) \\
\end{align}\]
By subtracting equation (1) from equation (2), we can write,
\[\begin{align}
& \Rightarrow 99x=180 \\
& \Rightarrow x=\dfrac{180}{99} \\
\end{align}\]
Dividing both numerator and denominator by the highest common divisor that is 9, we get
\[\Rightarrow x=\dfrac{20}{11}\]
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……
Complete step by step answer:
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.
Let us suppose,
\[\Rightarrow x=1.\overline{81}=1.818181\cdots \cdots \text{ ---}\left( 1 \right)\]
Now, multiplying both sides by 100 we get,
\[\begin{align}
& \Rightarrow 100\times x=100\times 1.\overline{81} \\
& \Rightarrow 100\times x=181.8181\cdots \cdots \text{ ---}\left( 2 \right) \\
\end{align}\]
By subtracting equation (1) from equation (2), we can write,
\[\begin{align}
& \Rightarrow 99x=180 \\
& \Rightarrow x=\dfrac{180}{99} \\
\end{align}\]
Dividing both numerator and denominator by the highest common divisor that is 9, we get
\[\Rightarrow x=\dfrac{20}{11}\]
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……
Recently Updated Pages
What percentage of the area in India is covered by class 10 social science CBSE

The area of a 6m wide road outside a garden in all class 10 maths CBSE

What is the electric flux through a cube of side 1 class 10 physics CBSE

If one root of x2 x k 0 maybe the square of the other class 10 maths CBSE

The radius and height of a cylinder are in the ratio class 10 maths CBSE

An almirah is sold for 5400 Rs after allowing a discount class 10 maths CBSE

Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

How do you graph the function fx 4x class 9 maths CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Difference Between Plant Cell and Animal Cell

What is pollution? How many types of pollution? Define it

What is the color of ferrous sulphate crystals? How does this color change after heating? Name the products formed on strongly heating ferrous sulphate crystals. What type of chemical reaction occurs in this type of change.
