VULGAR FRACTIONS

A vulgar fraction is nothing but a common fraction or simply called as fraction which can be written in the usual way where one integer above the another integer separated by a line.

Examples of vulgar fractions :

1/2, 3/4, 100/300

When dealing with fractions, we often use some special mathematical language. Instead of using the words 'top number' and 'bottom number' we use the words numerator and denominator. So, in 3/4, 3 is the numerator and 4 is the denominator.

Mathematically, vulgar fraction is a part of the whole or combination of several equal parts of the whole.

The number in denominator indicates how many parts the whole has been divided into. The number in numerator indicates how many parts are taken.

For example, in the fraction 3/4, the denominator 4 indicates that whole has been divided into 4 parts. And the numerator 3 indicates that 3 out of the four parts of the whole are taken.

Proper Fraction :

In a vulgar fraction, the numerator is always smaller than the denominator. 

Examples :

2/5,  3/11

Improper Fraction :

In a vulgar fraction, the numerator is equal to or larger than the denominator. 

Examples : 

7/5,  3/2, 1/1

Operations with Vulgar Fractions

The value of any vulgar fraction is unchanged, if the numerator and denominator are multiplied or divided by one and the same number.

Numerically,

A vulgar fraction can be simplified to its lowest term or simplest form, if the numerator and denominator have the same divisors.

If the denominators of two vulgar fractions are the same, the fractions can be added by adding their numerators; to subtract them, subtract the numerator of the subtrahend from the numerator of the minuend.

To multiply a vulgar fraction by another vulgar fraction, multiply the numerators together for the numerator of the product and multiply the denominator together for the denominator of the product.

To divide a number by a vulgar fraction, multiply the number by the reciprocal of the vulgar fraction.

Comparing Vulgar Fractions

If two vulgar fractions have the same numerator, the larger fraction is the one with the smaller denominator.

1/2 > 5/11

If two vulgar fractions have the same denominator, the larger fraction is the one with the larger denominator.

7/13 > 5/13

Converting Recurring Decimals to Vulgar Fractions

How to convert recurring decimals to vulgar fractions :

1. Make sure that there are only recurring digits after the decimal point.

2. If there is only one digit in the recurring pattern, subtract 1 from 10, the result is 9. 

3. Take the recurring digit in the numerator and 9 in the denominator. 

4. Simplify the fraction, as needed. 

Note :

If there are two digits in the recurring pattern, subtract 1 from 100, for three digits, subtract 1 from 1000 and so on. Continue the rest of the process as explained above. 

Covert the recurring decimals to vulgar fractions :

Example 1 :

0.33333..........

Solution : 

There are only recurring digits after the decimal point in  

0.33333..........

There is only one digit in the recurring pattern, that is 3. 

Subtract 1 from 10, the result is 9.

Write a vulgar fraction with numerator 3 and denominator 9.

= 3/9

= 1/3

Example 2 :

0.77777..........

Solution : 

There are only recurring digits after the decimal point in  

0.77777..........

There is only one digit in the recurring pattern, that is 7. 

Subtract 1 from 10, the result is 9.

Write a vulgar fraction with numerator 7 and denominator 9.

= 7/9

Example 3 :

0.36363636..........

Solution : 

There are only recurring digits after the decimal point in  

0.36363636..........

There are two digits in the recurring pattern, that is 36. 

Subtract 1 from 100, the result is 99.

Write a vulgar fraction with numerator 36 and denominator 99.

= 36/99

= 4/11

Example 4 :

0.507507507507..........

Solution : 

There are only recurring digits after the decimal point in  

0.507507507507..........

There are three digits in the recurring pattern, that is 507. 

Subtract 1 from 1000, the result is 999.

Write a vulgar fraction with numerator 507 and denominator 999.

= 507/999

= 169/333

Example 5 :

0.06666..........

Solution : 

There is one digit between the decimal point and the recurring digits, that is 0.

0.06666.......... = (0.6666..........) ÷ 10 ----(1)

There are only recurring digits after the decimal point in  

0.6666..........

There is only one digit in the recurring pattern, that is 6. 

Subtract 1 from 10, the result is 9.

Write a vulgar fraction with numerator 6 and denominator 9 for 0.6666.......... in (1).

0.06666.......... = (6/9) ÷ 10

= (2/3) ÷ (10/1)

= (2/3) ⋅ (1/10)

= (2 ⋅ 1)/(3 ⋅ 10)

= 2/30

= 1/15

Example 6 :

0.00151515..........

Solution : 

There are two digits between the decimal point and the recurring digits, that is 00.

0.00151515.......... = (0.151515..........) ÷ 100 ----(1)

There are only recurring digits after the decimal point in  

0.151515..........

There are two digits in the recurring pattern, that is 15. 

Subtract 1 from 100, the result is 99.

Write a vulgar fraction with numerator 15 and denominator 99 for 0.151515.......... in (1). 

0.00151515.......... = (15/99) ÷ 100

= (5/33) ÷ (100/1)

= (5/33) ⋅ (1/100)

= (5 ⋅ 1)/(33 ⋅ 100)

= 5/3300

= 1/660

Example 7 :

0.37777..........

Solution : 

There is one digit between the decimal point and the recurring digits, that is 3.

0.37777.......... = 0.3 + 0.07777..........

In 0.07777.........., there is one digit between the decimal point and the recurring digits, that is 0.

0.37777.......... = 0.3 + (0.7777..........) ÷ 10 ----(1)

There are only recurring digits after the decimal point in  

0.7777..........

There is only one digit in the recurring pattern, that is 7. 

Subtract 1 from 10, the result is 9.

Write a vulgar fraction with numerator 7 and denominator 9 for 0.7777.......... in (1). 

0.39999.......... = 0.3 + (7/9) ÷ 10

= 3/10 + 7/90

= 27/90 + 7/90

= (27 + 7)/90

= 34/90

= 17/45

Example 8 :

1.21212121..........

Solution : 

1.21212121.......... = 1 + 0.21212121.......... ----(1)

There are only recurring digits after the decimal point in  

0.21212121..........

There are two digits in the recurring pattern, that is 21. 

Subtract 1 from 100, the result is 99.

Write a vulgar fraction with numerator 21 and denominator 99 for 0.21212121.......... in (1). 

1.21212121.......... = 1 + 21/99

= 1 + 7/33

= 33/33 + 7/33

= (33 + 7)/33

= 40/33

Example 9 :

2.342342342..........

Solution : 

2.342342342.......... = 2 + 0.342342342.......... ----(1)

There are only recurring digits after the decimal point in  

0.342342342..........

There are three digits in the recurring pattern, that is 342. 

Subtract 1 from 1000, the result is 999.

Write a vulgar fraction with numerator 342 and denominator 999 for 0.342342342.......... in (1). 

2.342342342.......... = 2 + 342/999

= 2 + 38/111

= 222/111 + 38/111

= (222 + 38)/111

= 260/111

Example 10 :

2.05555..........

Solution : 

2.05555.......... = 2 + 0.05555.......... ----(1)

In 0.05555.........., there is one digit between the decimal point and the recurring digits, that is 0.

2.05555.......... = 2 + (0.5555..........) ÷ 10 ----(1)

There are only recurring digits after the decimal point in  

0.5555..........

There is only one digit in the recurring pattern, that is 5. 

Subtract 1 from 10, the result is 9.

Write a vulgar fraction with numerator 5 and denominator 9 for 0.5555.......... in (1). 

2.05555.......... = 2 + (5/9) ÷ 10

= 2 + 5/90

= 2 + 1/18

= 36/18 + 1/18

= (36 + 1)/18

= 37/18

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