Question

Express each of the following decimals number in the p/q from 0.36 bar

Answer 1

Express each of the following decimal number(s) in the ${\sf{\dfrac{p}{q}}}$ form!

• ( I thought that bar is on both 3 and 6 and can be expressed as ${\pmb{\sf{0.\bar{36}}}}$ )

${\pmb{\sf{:\implies 0.\bar{36}}}}$

${\sf{:\implies 0.\bar{36} \: = x}}$

${\sf{:\implies x \: = 0.\bar{36}}}$

${\sf{:\implies \therefore \: x \: = 0.363636 ... - - - \: Eq^n \: (1)}}$

Let us multiply eqⁿ (1) with 100!

• (We are multiplying this eqⁿ by 100 because it it changing 2 times as 363636 not 666...etc as according to non-terminating or repeating digits.)

${\sf{:\implies 100 \times x = 0.363636 ... \times 100}}$

${\sf{:\implies 100x = 36.3636 ... - - - \: Eq^n \: (2)}}$

Now let us subtract eqⁿ (2) & eqⁿ (1)!

${\sf{:\implies 100x = 36.3636 ... - - - \: Eq^n \: (2)}}$

${\sf{:\implies x \: = 0.363636 ... - - - \: Eq^n \: (1)}}$

${\sf{:\implies \therefore \: 100x - x = 36.3636 ... - 0.363636 ...}}$

${\sf{:\implies 99x = 36}}$

${\sf{:\implies x = \dfrac{36}{99}}}$

• Henceforth, expressed!