Pure Recurring Decimal Represented With A Bar

There must have been a lot of instances that when dividing two numbers, the quotient comes out to be in a repetitive, infinite manner. That’s precisely what pure recurring decimals are. When the digits after the decimal get repeated infinitely, they’re known as repeating, non-term, or pure recurring decimals.

Example: 1.5555555…. is a pure recurring decimal, which can also be written as 1.5.

Bar Representation Of Recurring Decimals

Let’s take up a simple example.

83= 2.666666… 

Here, the answer comes out to be 2.6666… An easy and convenient way of writingthe repeated digits is to use a bar representation over the recurring digits. In this case, 2.666666… would be represented as 2.6

Converting Pure Recurring Decimals Into Fractions

There are some steps to convert recurring decimals to fractions, which are:

1. Let there be a number x. The number x would be a recurring decimal.
2. Let there be n, which represents the number of recurring digits.
3. Next, you need to multiply the recurring decimal by 10n.
4. Now subtract (1) from (3) to get a whole number. With this step, we’d be removing the repeating part.
5. Solve for x. The answer should come as a fraction in its simplest form.

For example, let x=1.2222… or 1.2

Here, there is only one recurring digit, 2.

So, n = 1

10x = 12.222…

 x = 1.222…

10x – x = 11

9x = 11

X = 119

Let’s take up another example with two recurring digits.

Let x=2.34

 or 2.343434…

Here, there are two recurring digits, 3 and 4.

So, n=2.

102=100

100x= 234.343434…

     x= 2.343434…

100x-x=232

99x=232

X = 23299

Mixed Recurring Decimals

These numbers have both recurring and non-recurring digits after the decimal point, hence called mixed recurring decimals. Mixed recurring decimals consist of those numbers having at least one repetitive digit after the decimal, along with the recurring ones. 

Example: 5.68444. This number has 4 as the recurring digit and 6 and 8 as non-recurring digits.

Converting Mixed Recurring Decimals Into Fractions

The steps for the conversion are as follows:

1. Remove the bar from the recurring digits of the number and simply write it in a decimal form. Let that number be x.
2. Count the number of non-repetitive digits occurring after the decimal.
3. Let the number of non-recurring digits be n. 
4. Multiply both sides by 10n.
5. Now, count the number of recurring digits after the decimal.
6. Let the number of recurring digits be n. Multiply both sides by 10n.
7. Subtract (1) from (2).
8. Next, divide both sides.
9. Solve for x. The answer should come as a fraction in its simplest form.

For example, let x=0.16.

No. of non-recurring digits, n=1.

Multiplying both sides by 10, we get,

10x=1.66…   (I)

No. of recurring digits, n=1.

Multiplying both sides by 10, we get,

100x=16.66…    (II)

Subtracting (I) from (II), we get,

100x-10x= 16.66.. – 1.66..

90x = 15

X=1590

The simplest form, x=16

Conclusion:

This article intended to explain the pure recurring decimal in an elaborated way. The examples given in this article will help the students to understand the concepts in a detailed way.