Question

The value of 0.666… in the form of p/q , where p and q are integers and q ≠ 0 is

Answer 1

$\rm{ Let \:us \:consider \:the\: number \:as\: x}$

$\longrightarrow{\rm{x = 0.666....}}$ -- (1)

Now let us simplify count the number of digits after the decimal point.

There are 3 digits after the decimal point.

So , we will multiply both the sides with a number that has 3 zeros after 1. The only number which has 3 zeros after 1 is 1000.

So , we will multiply both the sides by 1000.

$\longrightarrow{\rm{ 1000x = 666.66666...}}$---(2)

Now let us subtract (2) from (1)

$\longrightarrow{\rm{1000x - x = 666.6666... - 0.6666.....}}$

$\longrightarrow{\rm{ 999x = 666}}$

$\longrightarrow{\rm{ x = \dfrac{999}{666}}}$

$\implies{\large\boxed{\rm{ x = \dfrac{3}{2}}}}$

Answer 2

$\bf\large{\underline{Question:-}}$

The value of 0.666… in the form of p/q , where p and q are integers and q ≠ 0 is.

$\bf\large{\underline{Solution:-}}$

$\tt→ Let\: 0.\bar666.....equ(1)$

Now,

Multiplying both side by 10, we have

$\tt→ 10x= 6.\bar666......equ(2)$

so,

subtracting equation (1) from(2)

$\tt →10x-x=6.\bar666-0.\bar666 \\\tt→ 9x=6 \\\tt→ x=\frac{6}{2}\\\tt→ x=\frac{2}{3}$

Hence,

0.666 will be in the form p/q is = 2/3

.Verification.

L.H.S

$\tt→ \frac{2}{3}=0.\bar666\\\tt→ 0.\bar666=0.\bar666$

L.H.S=R.H.S

Verified