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