Question

The value of 0.854 (bar on 54)+0.23 (23 bar)-0.68 (bar 8) is:​

Answer 1

value of $0.8\bar{54}+0.\bar{23}-0.6\bar{8}$ is $0.3\bar{97}$

we have to find the value of $0.8\bar{54}+0.\bar{23}-0.6\bar{8}$

first we have to express all the given terms in p/q form where q ≠ 0 after that we can easily solve it.

let's try to do it...

$0.8\bar{54}$ = (854-8)/990

= 846/990 = 423/495

$0.\bar{23}$ = (23 - 0)/99 = 23/99

$0.6\bar{8}$ = (68 - 6)/90 = 62/90

= 31/45

now, $0.8\bar{54}+0.\bar{23}-0.6\bar{8}$ = 423/495 - 23/99 + 31/45

= (423 + 115 - 341)/495

= (538 - 341)/495

= 197/495

= 0.39797979....

= $0.3\bar{97}$

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Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

First we explain the process how to calculate

$\displaystyle \: 0.8 \overline{54}$

PROCESS

${ Let} \: \: x = \displaystyle \: 0.8 \overline{54}$

$\implies \: x = 0.854545454......... \: \: \: \: - - - (1)$

Multiplying both sides by 10 we get

$10x = 8.54545454......... \: \: - - - (2)$

Again Multiplying both sides of Equation (1) by 1000

$1000x = 854.54545454....... \: \: - - - (3)$

Now Equation (3) - Equation (2) gives

$990x = 846$

$\implies \displaystyle \: x = \frac{846}{990}$

So

$\displaystyle \: 0.8 \overline{54} = \frac{846}{990}$

Similarly

$\displaystyle \: 0. \overline{23} = \frac{23}{99}$

$\displaystyle \: 0.6 \overline{8} = \frac{62}{90}$

Hence

$\displaystyle \: 0.8 \overline{54} + 0. \overline{23} - 0.6\overline{8}$

$= \displaystyle \sf{ \frac{846}{990} \: + \frac{23}{99} - \frac{62}{90} \: }$

$= \displaystyle \sf{ \frac{(846 + 230 - 62)}{990} \: }$

$= \displaystyle \sf{ \frac{394}{990} \:\: }$

$= \displaystyle \sf{ 0.397979.....\: }$

$= \sf{ \:0.3 \overline{97} \: }$