Question

express the following in p/q form where q is not equal to 0 (i) 0.12 (bar on 12) (ii) 0.123 (bar on 3)​

Answer 1

Answer:

$(i) \: Let \: x = 0. \overline{12}$

$100 \times x = 0.12121212... \times 100$

$100x = 12.121212... \\ \underline{{}^{ ( - )} x = {}^{( - )} 0.121212...} \\ \underline{99x = 12.000000... \: \: \: }$

$x = \frac{12}{99} = \frac{4}{33}$

$\therefore \: 0. \overline{12} = \frac{4}{33}$

To know:-

We are multiplying both sides by 100 because the two numbers are repeating here.

$(ii) \:Let \: x = 0.12 \overline{3}$

$10 \times x = 123333... \times 10$

$10x = 1.233333... \\ \underline{ {}^{ ( - )} x = {}^{ ( - ) } 0.123333...} \\ \underline{9x = 1.110000... \: \: \: \: \: }$

$x = \frac{1.11 \times 100}{9 \times 100} = \frac{111}{900}$

$\therefore \: 0.12 \overline{3} = \frac{111}{900}$

To know:-

We are multiplying both sides by 10 because the one number is repeating here.