Question

How do you write \[0.\overline 5 \] (\[5\] repeating) as a fraction?

Answer 1

Hint: In the given question, we have been given a decimal number. This decimal number is non-terminating and repeating. We have to solve this number into a fraction that correctly represents the repeating number when solved traditionally, i.e., by dividing the numerator by the denominator. To solve this question, we are going to multiply the number by \[10\], subtract them, get an integer value, and then divide by the difference of the ten times the assumed variable minus the assumed variable, i.e., nine times the assumed variable.

Complete step by step answer:
The given decimal number is \[0.\overline 5 = 0.5555....\]
Let the given number be \[x\],
\[x = 0.5555...\] (i)
Multiply both the sides by \[10\],
\[10x = 5.5555...\] (ii)
Subtracting (ii) and (i), we get,
\[10x - x = 5.5555... - 0.5555...\]
or, \[9x = 5\]
Hence, \[x = \dfrac{5}{9}\]

Thus, the given decimal number is equal to \[\dfrac{5}{9}\].

Additional Information:
In the given question, we divided by \[{10^1} - 1\], because the number of repeating digits was \[1\]. If the number of repeating digits were \[2\], then we would have divided by \[{10^2} - 1 = 99\], or, to generalize, if the number of repeating digits were \[n\], then we would have divided by
\[{10^n} - 1\].

Note:
In this given question, we were given a non-terminating and repeating decimal number. We had to solve this question by converting the decimal number into the fraction which when solved traditionally, i.e., by dividing the numerator by the denominator, gives back the same decimal number with the same repeating pattern. All we needed to do was assume the decimal number to be equal to a variable, multiply them both by ten, subtract the original number and the number obtained by multiplying by ten and then dividing by the difference of the variable counterparts, and that is going to give us the required answer.