Express $0.6\overline{8}$ as a rational number?

Answer

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Hint: Write the given repeating decimal as \[0.6\overline{8}=0.6888......\]. Assume this expression as $x=0.6888....$. Now, multiply both the sides with 10 and assume the obtained relation as equation (1). Further, multiply $x=0.6888....$ with 100 and consider it as equation (2). Subtract equation (1) from equation (2) and solve.

Complete step by step answer:

Here we have been provided with the decimal number $0.6\overline{8}$ and we are asked to write it in the rational form.

Now, clearly we can see that the bar is present over 8 only in the given decimal number that means 8 is repeating up to infinite places after 6, therefore we cannot directly remove the decimal. So, we need some other and better method. Let us assume the given decimal number as x. So, we have,

$\Rightarrow x=0.6\overline{8}$

Removing the bar sign we have,

\[\Rightarrow x=0.6888......\]

Multiplying both the sides with 10 we get,

\[\Rightarrow 10x=6.888......\] - (1)

Now, multiplying $x=0.6888....$ with 100 we get,

\[\Rightarrow 100x=68.888......\] - (2)

Subtracting equation (1) from equation (2) we get,

$ \Rightarrow 90x=62.000..... $

$ \Rightarrow 90x=62 $

Dividing both sides with 90 we get,

\[\Rightarrow x=\dfrac{62}{90}\]

Hence, \[\dfrac{31}{45}\] represents the rational form of the decimal number $0.6\overline{8}$.

Note: Note that the given number in the question is a rational number (non – terminating repeating) and that is why we were able to convert it in the rational form. If the number is non – terminating and non – repeating then it is called an irrational number and we cannot write an irrational number into the fractional form. Do not directly remove the decimal point because we cannot write ${{10}^{\infty }}$ in the denominator of the rational number.

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