Question

Express the following in the form of \[\dfrac{p}{q}\] , where p and q are integers. \[0.555................\]

Answer 1

Hint: We know that a number which is in the form of \[\dfrac{p}{q}\] where p and q are integers, then the number is called a rational number. From the question, we should convert the given number into a rational number. We know that the given number is a terminating repeating decimal. By using the method to convert the terminating repeating decimal to rational number, the conversion takes place.

Complete step-by-step answer:
We are given to convert the given number \[0.555................\] into \[\dfrac{p}{q}\] form, where p and q are integers.
For solving this question let us consider the given equation as equation (1) and assume it as S.
\[S=0.555...............\text{ }...\left( 1 \right)\]
Now for converting the equation (1) into rational form we have to eliminate the decimal in the number.
Let us multiply the equation (1) with 10 and consider it as equation (2).
By multiplying the equation (1) with 10, we get
\[\Rightarrow 10S=5.555.............\text{ }\]
Let us consider the above equation as equation (2), we get
\[\Rightarrow 10S=5.555.............\text{ }......\left( 2 \right)\]
Now let us subtract the equation (1) with equation (2) to eliminate the decimal, we get
\[\Rightarrow 10S-S=5.5555....-0.555555\]
By simplifying the above equation, we get
\[\Rightarrow 9S=5\]
Let us consider the above equation as equation (3), we get
\[\Rightarrow 9S=5.................\left( 3 \right)\]
Now by dividing the equation (3) by 9, we get
\[\Rightarrow S=\dfrac{5}{9}\]
Let us consider the above equation as equation (4), we get
\[\Rightarrow S=\dfrac{5}{9}.............\left( 4 \right)\]
Therefore equation (4) is the solution for a given problem.

Note: Students should be able to identify the rational number whether it is terminating decimal or non-terminating decimal or recurring decimal or non-recurring decimal. If a small mistake is done, then the process to find the \[\dfrac{p}{q}\] will change and the exact form may not get obtained.