Question

Ques 3 : Show that $0.2353535.... = 0.2 \overline{35}$ can be expressed in the form of $\displaystyle \frac{p}{q}$, where p and q are integers and q ≠ 0​

Answer 1

☑️ $\bf{\red{\boxed{ Correct\:Question : }}}$

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Show that $0.2353535.... = 0.2 \overline{35}$ can be expressed in the form of $\displaystyle \frac{p}{q}$, where p and q are integers and q ≠ 0

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$\bf{\red{\boxed{ Solution : }}}$

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$\tt{One\: digit \: after\: the\: decimal\: point\: is\: without\:bar.\: So,\:10x = 0.2 \overline{35}}$

$\tt{\green{ 10x = 0.2 \overline{35}}}$

$\tt{\green{\implies 10x = 2. \overline{35}}}$

$\tt{\green{\implies 10x = 2 + \dfrac{35}{99}}}$

$\tt{\green{\implies 10x = \dfrac{198 + 35}{99}}}$

$\tt{\green{\implies 10x = \dfrac{233}{99}}}$

$\tt{\green{\implies x = \dfrac{233}{99} \div \dfrac{1}{10} }}$

$\tt{\green{\implies x = \dfrac{233}{990}}}$

$\tt{\green{\implies x = 0.2353535}}$

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$\textbf {Hence, it shows that 0.2353535.... = 0.2 \overline{35}}$

Answer 2

Given :-

0.2353535 - - - -

$\tt{ 0.2\overline{35}}$

Required to find :-

• p/q form of $\tt{ 0.2\overline{35}}$

Conditions mentioned :-

• p and q are integers

• q ≠ 0

Solution :-

Given that :-

0.235353535 - - - - -

$\boxed{\boxed{\begin{minipage}{ 6cm } \sf{ Period = 35 } \\ \sf{ Periodicity = 2 } \end{minipage}}}$

So,

Let x = 0.235353535 - - - - $\red{\longrightarrow{\text{Equation 1 }}}$

consider this is as equation 1

Since the periodicity is 2

Multiply equation 1 with 100

Hence,

100 ( x ) = 100 ( 0.235353535 - - - - - )

100x = 23.535353535- - - - - $\red{\longrightarrow{\text{Equation 2 }}}$

Consider this as equation 2

Now,

Subtract equation 1 from equation 2

So,

100x = 23.535353535 - - - - - -

x = 0.235353535 - - - - -

$\rule{200}{1}$

99x = 23.300000000 - - - - - -

$\rule{200}{1}$

This implies ,

99x = 23.3

$\tt{ x = \dfrac{ 23.3 }{99}}$

Since it is also mentioned that p,q are integers

So,

Multiply numerator and denominator with 10

Hence,

$\tt{ x = \dfrac{23.3 \times 10 }{ 99 \times 10 }}$

$\tt{ x = \dfrac{233}{990 }}$

$\large{\rm{\therefore{ 0.2\overline{35} = \dfrac{233}{990} }}}$

Where,

p , q are integers and q ≠ 0

Additional information :-

1. Only a non - terminating recurring decimal can be converted into p/q form .

2. Period is the digits which are repeating in the given decimal expansion

3. Periodicity refers to the number of digits which are repeating in the given decimal expansion