Question

Express the following decimal in the form p/q, where p, q are integers and q ≠ 0
0.1 38bar

.

Answer 1

Given Question -

Express the following decimal in the form p/q, where p, q are integers and q ≠ 0 , 0.1 38bar.

Required Answer -

In the portion it is not defined properly that bar is on which number . But bar can be on

• 8
• 38
• 138

So let's do the all three cases

I Case -

If the bar is on 8

${\sf{:\implies 0.13 \bar{8}}}$

Let 0.1388888... be x

${\sf{:\implies 0.138888...=x}}$

Multiplying both side with 100,

${\sf{:\implies 013.8888...=100 x}}$....(i)

Multiplying both side with 10

${\sf{:\implies 0138.888...=1000 x}}$....(ii)

Now subtract (i) from (ii)

As we know that .888... Is non terminating so there will be no change if we write it as zero on subtracting .

${\sf{:\implies (138.8888..)-(13.8888..)=(1000 x)-(100 x)}}$

${\sf{:\implies 125=900 x}}$

• ${\sf{:\implies x = \dfrac{125}{900} = \dfrac{5}{36} }}$

If you will convert it again into decimal then you will get result as 0.13(8)bar only

__________________

II Case -

If the bar is on 38

${\sf{:\implies 0.1 \bar{38}}}$

Let 0.138383838... be x

${\sf{:\implies 0.138383838...=x}}$

Multiplying both side with 10,

${\sf{:\implies 01.38383838...=10x}}$....(i)

Multiplying both side with 100

${\sf{:\implies 0138.383838...=1000x}}$....(ii)

Now subtract (i) from (ii)

As we know that .383838... Is non terminating so there will be no change if we write it as zero on subtracting .

${\sf{:\implies (138.383838..)-(1.383838..)=(1000 x)-(10 x)}}$

${\sf{:\implies 137=990 x}}$

• ${\sf{:\implies x = \dfrac{137}{990} }}$

If you will convert it again into decimal then you will get result as 0.1(38)bar only

__________________

III Case -

If the bar is on 138

${\sf{:\implies 0. \bar{1} \bar{3} \bar{8}}}$

Let 0.138138138138138... be x

${\sf{:\implies 0.138138138138...=x}}$ ....(i)

Multiplying both side with 1000,

${\sf{:\implies 0138.138138138...=1000x}}$....(ii)

Now subtract (i) from (ii)

As we know that .138138138... Is non terminating so there will be no change if we write it as zero on subtracting .

${\sf{:\implies (138.138138138..)-(0.138138138..)=(1000 x)-( x)}}$

${\sf{:\implies 138=999 x}}$

• ${\sf{:\implies x = \dfrac{138}{999} }}$

If you will convert it again into decimal then you will get result as 0.(138)bar only

Answer 2

Given Question -

Express the following decimal in the form p/q, where p, q are integers and q ≠ 0 , 0.1 38bar.

Required Answer -

${\sf{:\implies x = \dfrac{125}{900} = \dfrac{5}{36} }}$