Fractions and Repeating Decimals

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Fractions and Repeating Decimals

We have already discussed how to convert a given fraction into the decimal form with terminating decimals, https://www.etutorworld.com/fractions-to-decimals.html. Here, we will try to understand how to convert a given fraction to repeating decimal.

fractions and repeating decimals
  1. Now, we consider each of the above cases separately and understand them with examples.

Terminating and non-repeating decimal

Example 1:    \frac{1}{2}=\frac{1\times5}{2\times5}=\frac{5}{10}=0.5 or by long division method

fractions and repeating decimals

So, we observe that in this case the remainder is zero and hence the long division terminates and given fraction has a terminating & non-repeating decimal representation.

Non-terminating and repeating decimal.

fractions and repeating decimals

 Example 2: Consider \frac{1}{3}

fractions and repeating decimals

Example 3: Consider \frac{5}{6}

 

So, we observe in this case the remainder is 2 from second step onwards and hence the long division does not terminate and the given fraction has a non-terminating & repeating decimal representation.
fractions and repeating decimals

 

So decimal representation of = 0.83333… and this repetition is denoted by putting a bar over the repeated number i.e. = 0.83333… = fractions11

From the above examples, we see that the decimal representation of a fraction is non-terminating and repeating if its long division does not terminate & we have the same remainder, which cannot be exactly divided by the divisor.

CHECK POINT

Convert the following fractions to their decimal form and identify whether it is a repeating decimal  or not.

  1. \frac{8}{18}
  2. \frac{7}{11}
  3. \frac{6}{9}
  4. \frac{3}{8}
  5. \frac{8}{11}
Answer Key

fractions9