9. Vulgar fraction, Decimal fractions and numbers .

 

9. Vulgar fraction, Decimal fractions and numbers .

Learn and Remember:

 1. Vulgar fractions : Expressions of rational numbers as ratios rather than decimal fractions are known as vulgar fractions. Such fractions are

(i) proper fractions 

(ii) improper fractions (or mixed fractions).

1. Since, the numerator and the denominator of a fraction are integers and the denominator is a non-zero integer, a fraction is a rational number. Opposites of all fractions are also rational numbers.

2. Comparison of vulgar fractions :

(a) In fractions with the equal denominator, fraction having the greatest numerator is the greatest.

 e.g. the greatest fraction out of    ² ⁄ ₇ ,³ ⁄ ₇,⁵ ⁄ ₇ and  ⁶ ⁄ ₇  is  ⁶ ⁄ ₇

(b) In fractions with the equal numerator, the fraction having the greatest denominator is the smallest.

 e.g. the smallest fraction out of ¹ ⁄ ₂,¹ ⁄ ₃, ¹ ⁄ ₄  and = ¹ ⁄ ₇  is ¹ ⁄ ₇ 

3. In fractions with numerators smaller than the denominators by 1 (or 2, 3, ...), the fraction with the greatest denominator is the greatest.

e.g. (i) the greatest fraction out of , ⁷ ⁄ ₈,  ⁵ ⁄ ₆,  ¹¹ ⁄ _{12  and    ³ ⁄ 4}  is  ¹¹ ⁄ ₁₂

 (ii) the greatest fraction out of  ¹ ⁄ ₄,² ⁄ ₅,⁷ ⁄ ₁₀ and ¹⁴ ⁄ _{17  is   ¹⁴ ⁄ 17}

 [Note : For comparing negative fractions, consider the opposites of 2 and 3.]

4. Observe that the expression

¹⁄_1x2  +¹⁄_2x3  +¹ ⁄_3x4  +¹⁄_4x5 +........+ ¹⁄_27x28   gets simplified to ²⁷⁄₂₈

♦ Decimal fractions :

1. A fraction whose denominator is 10 or any other ten times multiple of 10 is called a decimal fraction.

 e.g. ⁷ ⁄ ₁₀ , ³³³ ⁄ ₁₀₀ ,⁶⁷ ⁄ ₁₀₀₀    - are decimal fractions.

2. When a decimal fraction is multiplied by 10, 100, 1000, ..., the decimal point is shifted to

the right by 1, 2, 3, ..., places respectively.

e.g. 3.574  x  100 = 357.4;    357.4  x  1000 = 357400.0

3. When a decimal fraction is divided by 10, 100, 1000, ..., the decimal point is shifted to the

left by 1, 2, 3, ..., places respectively.

e.g. 563.2 ÷ 10 =56.32; 5.632 ÷ 100 = 0.05632.

4. In order to multiply two decimal fractions, multiply two numbers as usual without decimal points and then put the decimal point after the total number of decimal places (in the multiplicand and the multiplier) starting from units place to the left.

e.g. multiply 26.56 x 3.2.

2656 x 32 = 84992

∴ 26.56 x 3.2 = 84.992

4. For division of decimal fractions, first convert the divisor to an integer (by multiplying it suitably by 10, 100, 1000, ..., etc.). Multiply the dividend also by the same number and then carry out the division.

e.g. (i) 20.6 ÷ 1.03

      ^{∴   20.6} ⁄ _1.03

∴ =   ^{20.6 x 100} ⁄ _{1.03 x 100}

∴ = ²⁰⁶⁰ ⁄ ₁₀₃

∴ = 20

(2)  ^0.206 ⁄ _1.03

∴ =  ^{0.206 x 100} ⁄ _{1.03 x 100}

∴ =  ^20.6 ⁄ ₁₀₃

∴=  ²⁰⁶ ⁄ ₁₀ x ¹ ⁄ ₁₀₃

∴ = ¹ ⁄ ₁₀=0.2

6. While adding or subtracting decimal fractions, numbers are written in such a way that the decimal points fall one below the other. Then, addition or subtraction are performed by usual method and the decimal point in the answer is put just below the other decimal points.

7. Any number of zeros after the last digit of the decimal fraction do not change its value.

8. If the numerator of a vulgar fraction is divided by its denominator, the vulgar fraction is converted into a decimal fraction.