Question

Find the value of $ 0.32\overline {58} $

Answer 1

Hint: The bar on the digits 58 denotes the repetition of 58 in the given number, so the given number is irrational. To find the value of an irrational number we have to convert it into a fraction, this can be done by multiplying the given number with a power of 10 such that the power is equal to the number of digits after the decimal and before the repeating digits, again multiplying the given number with a power of 10 such that the power is equal to the sum of the number of non-repeating digits and the number of digits that are repeated.

Complete step-by-step answer:
The given number is $ 0.32\overline {58} $ , it can be rewritten as $ 0.32585858..... $ , let $ 0.32\overline {58} = R $
In the given question, the number of non-repeating digits is 2 (32) and the number of digits that are repeated is 2 (58). So we first multiply $ 0.32\overline {58} $ with $ {10^2} $ and then with $ {10^4} $ , then we subtract both the results with each other, as follows –
 $
  {10^2} \times R = {10^2} \times 0.325858.... \\
   \Rightarrow 100R = 32.5858....\,\,\,\,\,\,\,\,\,...(1) \\
  {10^4} \times 4 = {10^4} \times 0.325858... \\
  10000R = 3258.5858.....\,\,\,\,\,\,...(2) \\
  (2) - (1) \\
  10000R - 100R = 3258.5858..... - 32.5858.... \\
   \Rightarrow 9900R = 3226 \\
   \Rightarrow R = \dfrac{{3226}}{{9900}} \\
   \Rightarrow R = \dfrac{{1613}}{{4950}} \;
  $
Hence, the value of $ 0.32\overline {58} = \dfrac{{1613}}{{4950}} $
So, the correct answer is “ $ \dfrac{{1613}}{{4950}} $ ”.

Note: Real numbers are of two types – Rational numbers and Irrational numbers. The numbers that can be expressed as a fraction such that the denominator is not zero are called rational numbers and thus their decimal expansion is terminating and non-repeating, whereas irrational numbers are those which have repeating and non-terminating decimal expansion, the number $ 0.32\overline {58} $ is irrational. So, the irrational numbers can be converted into a fraction using the method shown above.