Question

show that 0.3333...= 0.3 can be expressed in the form p/q where p and q are integer and q ≠ 0...​

Answer 1

$\Large{\underbrace{\sf{\orange{Required\:Solution:}}}}$

Let's αssume thαt x = 0.3333.

$\longrightarrow{\sf{x=0.{\overline{3}}\:\:\:\:\:\:....(1)}}$

• Multiply 10 with both the sides becαuse 1 digit is repeαting.

$\longrightarrow{\sf{x\times 10=0.{\overline{3}}\times 10}}$

$\longrightarrow{\sf{10x=3.{\overline{3}}\:\:\:\:\:\:.....(2)}}$

• Subtract equation (1) from equation (2).

$\longrightarrow{\sf{10x-x = 3.{\overline{3}} - 0.{\overline{3}}}}$

$\longrightarrow{\sf{9x = 3}}$

• Transpose 9 to R.H.D.

$\longrightarrow{\sf{x = \dfrac{3}{9}}}$

• Cancel out the numbers.

$\longrightarrow{\sf{x =\dfrac{\cancel{3}}{\cancel{9}}}}$

$\longrightarrow{\sf{x = \dfrac{1}{3}}}$

Therefore, $\large{\boxed{\mathbf{\orange{0.333\:or\:0.{\overline{3}}= \dfrac{1}{3}}}}}$

__________________________

Answer 2

$\huge \sf \red{ \underline{“Solution”}}$

• Since, we don't know what .03 is, let us call it ‘x’and so,

⠀⠀⠀⠀⠀⠀$\sf \to \blue{x =0.3333...}$

• Now, here is, where the trick comes in. look at:-

⠀________________________

Now:-

⠀⠀⠀$\sf \to \pink{ \underline{10x = 10 \times (0.333...) = 3.333...}}$

⠀$\sf \to \green{ \underline{3.3333... = 3 + x. \: since \: x = 0.3333...}}$

⠀⠀⠀⠀⠀⠀⠀$\sf \to \pink{ \underline{10x = 3 + x}}$

• Solving for x , we get:-

⠀________________________

⠀⠀⠀$\sf \to \red{ \underline{9x = 3.i.e. \: x = \frac{1}{3}}}$

⠀________________________