Fractions and Repeating Decimals

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

$fractions and repeating decimals$

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.

$\frac{8}{18}$
$\frac{7}{11}$
$\frac{6}{9}$
$\frac{3}{8}$
$\frac{8}{11}$

Answer Key

$fractions9$

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