## Arithmetic Chapter 9 Recurring Decimal Number

**Conversion of Recurring Decimal Into Vulgar Fraction:**

There are two types of recurring decimals; pure recurring decimals and mixed recurring decimals. First, we shall discuss the conversion of pure recurring into vulgar fractions.

**Conversion of pure recurring decimal into a vulgar fraction :**

**Question 1: Convert 0.2 into vulgar fractions.**

** Solution**:

**Given:**

** 0.2 **

0.2 x 10 = (2222 ) x 10 = 2.2222 ………… (1) (∵ Multiplying 0.2 by 10)

0.2x 1 = (0.2222 ) x 1 = 0.2222 ……… (2) (∵ Multiplying 0.2 by 1)

Subtracting (2) from (1), we get,

0.2 (10 – 1) = (2.2222 ) – (2222……) = 2

or, 0.2 x 9 = 2

or 0.2 = 2/9

0.2 into vulgar fractions is 2/9

**Question 2: Convert 0**

**.**

**35 into a vulgar fraction.**** Solution **:

**Given:**

** 0****.****35**

0.35 = 0.353535

Multiplying both sides by 100 and 1 respectively, we get,

0.35 x 100 = (0.353535 ) x 100 = 35.353535………(1)

0.35 x 1 = (0.353535 ) x 1 = 0.353535……(2)

Subtracting (2) from (1), we get,

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0.35 (100 – 1) = 35

or, 0.35 x 99 = 35

or, 0.35 = 35/99

∴ 0.35 = 35/99

0.35 into a vulgar fraction is 35/99

From the above examples, we get the following rule of conversion of pure recurring decimal into a vulgar fraction

**For Example** 0.54632 = 54632/99999;

0.025 = 205/999;

0.51 = 51/99

= 17/13.

**2. Conversion of mixed recurring decimal into a vulgar fraction: **

**For this observe the following examples :**

**Question 1: Convert 0.1275 into a vulgar fraction.**

**Solution: **

**Given: 0.1275**

0.1275 = 0.12757575 ……..(1)

Multiplying both sides by 10000, we get,

0.1275 x 10000 = (0.12757575 ) x 10000 = 1275.757575 …………(2)

Multiplying both sides of (1) by 100, we get,

0.1275 x 100 = (0.12757575 ) x 100 = 12.757575.

Subtracting (3) from (2), we get,

(10000 – 100) x 0.1275 = 1275 – 12

or, 9900 x 0.1275 = 1275 – 12

or, 0.1275 = (1275-12) / 9900

= 1263/9900

= 421/3300

0.1275 into a vulgar fraction is 421/3300

**Question 2. Convert 0.26321 into a vulgar fraction.**

**Solution:**

**Given: 0.26321**

0.26321 = 0.26321321321…………………..(1)

Multiplying both sides of (1) by 100000, we get,

0-2632i x 100000 = (0-26321321321 ) x 100000 = 26321.321321……………………….(2)

Again multiplying both sides of (1) by 100, we get,

0.26321 x 100 = (0.26321321321……….) x 100 = 26-321321321…………………..(3)

Subtracting (3) from (2) we get,

0.26321 (100000 – 100) = 26321 – 26

or, 0.26321 x 99900 = 26321 – 26

or, 0.26321 = (26321 – 26) / 99900

= 26295/99900

= 8765 / 33300

0.26321 into a vulgar fraction is 8765 / 33300

**Question 3:** **Convert 3.128 into a vulgar fraction.**

** Solution **:

**Given: 3.128**

3.128 = 3.1282828……………………..(1)

Multiplying both sides of (1) by 1000, we get,

3.128 x 1000 = (3.1282828 ) x 1000 = 3128.282828……………………….(2)

Again multiplying both sides of (1) by 10, we get,

3.128 x 10 = (3.1282828 ) x 10 = 31.282828……………………….(3)

Subtracting (3) from (2), we get,

(1000 – 10) x 3.128 = 3128 – 31

or, 990 x 3.128 = 3128 – 31

or, 3.128 = (3128-31) / 990

= 3097 / 990

= 3 127 / 990.

∴ 3.128 = 3.127 / 990.

3.128 into a vulgar fraction is 3.127 / 990

**Some Examples Are given Below:**

**1. 0.02028 **

0.02028 = (2028 – 2) / 99900

= 2026 / 99900

= 1013 / 49950.

**2. 10.293 **

10.293 = (10293 – 102) / 990

= 10191 / 990

= 3397 / 330

= 10 97 / 330.

**3. 5.2476 **

5.2476 = (52476 – 52) / 9990

= 52424 / 9990

= 5 2474 / 9990

= 5 1237 / 4995

**This can also be done in the following way:**

If the recurring part contains only 9 (one or more), then omitting the recurring part, 1 is added to the number just before the recurring part.

**For Example:**

0.9 = 9/9

= 1;

0.19 =(19 – 1) / 90

= 18 / 90

= 1/5

= 0.2;

2.349 = (2349 – 234) / 900

= 2115 / 900

= 423 / 180

= 141 / 60

= 2.35;

1.099 = (1099 – 10) / 990

= 1089 / 990

= 121 / 110

= 11 / 10

= 1.1