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The \[\dfrac{p}{q}\] form of \[0.5\overline 6 \] is
A. \[\dfrac{{56}}{{100}}\]
B. \[\dfrac{{56}}{{100}}\]
C. \[\dfrac{{59}}{{90}}\]
D. \[\dfrac{{51}}{{90}}\]

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Last updated date: 27th Feb 2024
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IVSAT 2024
Answer
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Hint: Here we have been given the non-terminating decimal number and we need to convert it into the rational number. For that, we will write the given non-terminating decimal number in expanded form. Then we will multiply the expanded decimal form by 10 and again will multiply the expanded decimal form by 100. Then we will subtract these two values. From there, we will get the required answer.

Complete step by step solution:
Let \[x\] be the given rational form of the non terminating decimal number. So,
\[x = 0.5\overline 6 \]
As we can see that this decimal number is a non terminating decimal number as the digits after the decimal point are repeating decimals.
Now, we will write the given non-terminating decimal number in expanded form.
\[ \Rightarrow x = 0.56666....\] …….……… \[\left( 1 \right)\]
Now, we will multiply 10 on both sides of the equation \[\left( 1 \right)\].
\[ \Rightarrow 10 \times x = 10 \times 0.56666....\]
On further simplification, we get
\[ \Rightarrow 10x = 5.6666....\] …………. \[\left( 2 \right)\]
Now, again we will multiply 100 on both sides of the equation \[\left( 1 \right)\].
\[ \Rightarrow 100 \times x = 100 \times 0.56666....\]
On further simplification, we get
\[ \Rightarrow 100x = 56.6666....\] …………. \[\left( 3 \right)\]
Now, we will subtract equation \[\left( 2 \right)\] from equation \[\left( 3 \right)\].
\[100x - 10x = 56.6666.... - 5.6666....\]
Subtracting the like terms, we get
\[ \Rightarrow 90x = 51\]
Now, we will divide both sides by 90. Therefore, we get
\[ \Rightarrow \dfrac{{90x}}{{90}} = \dfrac{{51}}{{90}}\]
On further simplification, we get
\[ \Rightarrow x = \dfrac{{51}}{{90}}\]
Therefore, the required \[\dfrac{p}{q}\] form of \[0.5\overline 6 \] is equal to \[\dfrac{{51}}{{90}}\]

Hence, the correct answer is option D.

Note:
Here we can say that the obtained number is a rational number because it is expressed in the form of a fraction and the denominator is not equal to 0. A decimal number is said to be non terminating if the digits after the decimal number are infinite or repeat itself. Similarly, a decimal number is said to be terminating if the digits after the decimal point are finite or do not repeat itself.