Question

Express 0.32 (bar i on 2 only) in the form p/q

Answer 1

$\huge{\tt{\red{\underline{\underline{\blue{\overline{\overline{\green{ANSWER}}}}}}}}}$

$\underline{\large{\mathbb{G}{i}{v}{e}{n}}}$

0.32 ( bar is on 2)

$\rule{400}{2}$

$\mathsf{\large{Required\;to\;find \: : }}$

$\longrightarrow{ Express \; in \; the \; form \; of \; \dfrac{p}{q}}$

$\rule{400}{2}$

Explanation :

Before solving this question we need to know what is meant by period and periodicity .

$\rule{400}{1}$

What is period ?

In simple terms , we can say that the numbers which are repeating or recurring in that expansion .

For example :

Find the period of 3.33333-----

Here, the recurring Digit in that expansion is 3 .

So,

$\red{\Rightarrow{\tt{Period = 3 }}}$

$\rule{400}{1}$

What is periodicity ?

In simple terms , we can say that the no. of digits ( here digits refers to numbers ) which are repeating/recurring in that expansion .

For example :

Find the periodicity of 9.35353535----

Here, the no. of digits which are repeating is 2

So,

$\blue{\Rightarrow{\tt{Periodicity = 2}}}$

In this type of questions periodicity plays an major role in the solution .

The major role .

Depending upon the number of digits recurring we have to choose the number which should be multipled .

For example :

If Periodicity is 2 .

The number which should be multipled is 100 .

If Periodicity is 1 .

The number which should be multipled is 10 .

Depending on the Periodicity the number of zeros will get increased .

Knowing this concept is important to Solve questions like this .

$\rule{400}{2}$

Solution :

Given number :

0.32222---

0.32 ( bar is on 2)

Here,

$\boxed{\tt{period \: = 2}}$

$\boxed{\tt{ periodicity = 1}}$

Since,

The Periodicity is 1 the number which is needed to be multipled is 10 .

Hence,

Let's consider

$\orange{x = 0.3222------}{\longrightarrow{\tt{Equation\; - \;1 }}}$

Multiply with 10 on both sides ..

$\green{10(x) = 10(0.32222-----)}$

consider this as ;

$\green{10x = 3.2222------}{\longrightarrow{\tt{Equation\; - 2\;}}}$

However,

Subtract equation 1 from 2

$\tt{\green{10x = 3.2222------}}$

$\tt{\orange{ \: \: \: \: \: x = 0.3222------}}$

$\rule{90}{2}$

$\tt{9x = 2.9000--- }$

$\tt{\Rightarrow{ x = \dfrac{2.9}{9}}}$

$\tt{Multiply\: Numerator\: and\: denominator \: with \: 10}$

$\tt{\Rightarrow{ x = \dfrac{2.9}{9} \times \dfrac{10}{10}}}$

$\tt{\implies{ x = \dfrac{29}{90}}}$

The p/q form of 0.3222-- is

$\blue{\boxed{\longrightarrow{\therefore{0.322--- \:=\: \dfrac{29}{90}}}}}$

$\rule{400}{2}$

✅ Hence Solved ...