ICSE Class 8 Maths Chapter 04 Fractions

Chapter 4

Fractions

(Including Decimals, Rounding off and Significant Figures)

4.1 Review

Fraction

(i) A fraction is a part of the whole and is written in the form \(\frac{a}{b}\).

Such as: \(\frac{2}{5}\), \(\frac{3}{8}\), \(\frac{4}{11}\), etc.

(ii) The fraction \(\frac{2}{5}\) means: 2 parts out of 5 equal parts of the whole (given) quantity.

(iii) In any fraction \(\frac{a}{b}\), a and b are called its terms. Also, in fraction \(\frac{a}{b}\), its upper term a is called its numerator and its lower term b is called its denominator. Thus, Fraction = \(\frac{\text{Numerator}}{\text{Denominator}}\).

1. The value of a fraction is equal to one (unity), if its numerator is equal to its denominator.

2. The value of a fraction is equal to zero, if its numerator is zero and denominator is not zero.

Thus, \(\frac{0}{7}\) = 0, \(\frac{0}{-5}\) = ....., \(\frac{0}{28}\) = ..... and so on. But \(\frac{0}{0}\) ≠ 0

3. The value of a fraction is not defined, if its denominator is zero.

Thus, \(\frac{5}{0}\) is not defined, \(\frac{-8}{0}\) is ..................., \(\frac{325}{0}\) is .............., \(\frac{0}{0}\) is .............. and so on.

4. If both the terms of a fraction be multiplied or divided by the same non-zero number, the value of the fraction remains unaltered (unchanged).

5. A fraction should always be expressed in its lowest terms. To reduce a given fraction to its lowest terms, divide each of its terms by their H.C.F. A fraction is said to be in its lowest terms (or, in its simplest form), if its numerator and the denominator have no common factor.

4.2 Kinds Of Fractions

Simple fraction

A fraction, whose both the terms are integers, is called a simple fraction.

e.g. \(\frac{3}{8}\), \(\frac{5}{17}\), \(\frac{-7}{-53}\), etc.

Complex fraction

A fraction, whose one or both the terms are fractional numbers, is called a complex fraction.

e.g. \(\frac{\frac{2}{3}}{7}\), \(\frac{5}{\frac{6}{11}}\), \(\frac{2\frac{1}{3}}{7\frac{5}{8}}\), etc.

Decimal fractions

Fractions, with denominators 10, 100, 1000, etc., are called decimal fractions.

e.g. \(\frac{3}{10}\), \(\frac{57}{100}\), \(\frac{9}{1000}\), \(\frac{323}{10^8}\), etc.

Vulgar fractions

Fractions, whose denominators are not 10, 100, 1000, etc., are called vulgar fractions.

Proper fraction

A fraction, whose numerator is positive and also less than its denominator, is called a proper fraction.

e.g. \(\frac{5}{12}\), \(\frac{19}{100}\), \(\frac{131}{200}\), etc.

Improper fraction

A fraction, whose numerator is greater than its denominator, is called improper fraction.

e.g. \(\frac{7}{5}\), \(\frac{100}{17}\), \(\frac{213}{200}\), etc.

Mixed fraction

A fraction, which is expressed as a combination of an integer and a proper fraction, is called a mixed fraction.

e.g. \(2\frac{3}{5}\), which is the combination of an integer (2) and a proper fraction \(\left(\frac{3}{5}\right)\).

1. An improper fraction can always be expressed as a mixed fraction.

e.g. \(\frac{49}{11}\) = \(\frac{44 + 5}{11}\) = \(\frac{44}{11}\) + \(\frac{5}{11}\) = \(4 + \frac{5}{11}\) = \(4\frac{5}{11}\)

2. A mixed fraction can also be converted to an improper fraction by multiplying the integer with the denominator and adding numerator to the product.

e.g. \(5\frac{7}{8}\) = \(5 + \frac{7}{8}\) = \(\frac{5 \times 8 + 7}{8}\) = \(\frac{47}{8}\)

4.3 Decimal Fraction

Decimal

It is a fraction whose denominator is 10 or any integral power of 10.

e.g. \(\frac{7}{10}\), \(\frac{5}{10^3}\), \(\frac{29}{100}\), \(\frac{357}{10^5}\), etc.

2. A dot, called a decimal point is properly placed to remove the denominator of a decimal fraction.

e.g. (i) \(\frac{37}{10}\) = 3.7; read as: three-point-seven.

(ii) \(\frac{37}{100}\) = 0.37; read as: point-three-seven or zero-point-three-seven.

(iii) \(\frac{37}{1000}\) = 0.037; read as: zero-point-zero-three-seven and so on.

Decimal places

The number of digits which follow the decimal point is called the number of decimal places.

e.g. (i) 3.47 has 2 decimal places,

(ii) 0.0849 has 4 decimal places and so on.

4. The value of a decimal number remains unchanged by annexing cipher (or ciphers) at its extreme right or by removing them.

e.g. (i) 0.3 = 0.30 = 0.300 = 0.3000 and so on.

(ii) 3.7000 = 3.70 = 3.7 and so on.

5. In decimal number 3.47, 3 is its integral part and 0.47 is its decimal part.

6. An integer may be expressed as a decimal number by writing zero (or zeroes) in the decimal part:

e.g 15 = 15.0 = 15.000 and so on.

7. In a decimal number, the first place to the right of decimal is called tenth's place; the second place to the right of decimal is called hundredth's place and so on.

e.g. in number 5.628; 6 is at tenth's place, 2 is at hundredth's place and 8 is at thousandth's place.

Test Yourself

1. Evaluate: \(\frac{0}{8}\) = ....., \(\frac{8}{0}\) = ....................., \(\frac{0}{-32}\) = ....., \(\frac{8}{8}\) = ....., \(\frac{-8}{8}\) = ......

2. \(\frac{5}{13}\) is a .................. fraction, \(\frac{13}{5}\) is an .................. fraction, \(5\frac{3}{13}\) is a .................. fraction, \(\frac{29}{1000}\) is a ..................... fraction.

3. 0.562 × 100 = ........., \(\frac{0.562}{100}\) = ................., \(\frac{97}{1000}\) = .................

4.4 Converting A Decimal Fraction Into A Vulgar Fraction

Steps:

1. Remove the decimal point and write the resulting number as numerator. At the same time, in the denominator, write as many zeroes to the right of one (1) as the decimal places in the given number.

2. Reduce the vulgar fraction, obtained in Step 1, to its lowest terms

e.g. 0.9 = \(\frac{9}{10}\), 3.48 = \(\frac{348}{100}\) = \(\frac{87}{25}\), 0.088 = \(\frac{88}{1000}\) = \(\frac{11}{125}\) and so on.

4.5 Four Fundamental Operations

1. Addition and Subtraction:

Example 1:

Evaluate: \(6\frac{2}{5}\) - \(4\frac{4}{15}\) + \(3\frac{5}{9}\) - \(2\frac{3}{10}\)

Solution:

\(\frac{32}{5}\) - \(\frac{64}{15}\) + \(\frac{32}{9}\) - \(\frac{23}{10}\) = \(\frac{32 \times 18 - 64 \times 6 + 32 \times 10 - 23 \times 9}{90}\) [L.C.M. of 5, 15, 9 and 10 is 90]

= \(\frac{576 - 384 + 320 - 207}{90}\)

= \(\frac{896 - 591}{90}\) = \(\frac{305}{90}\) = \(\frac{61}{18}\) = \(3\frac{7}{18}\) (Ans.)

After simplification, if required:

(i) the fraction should be reduced to its lowest terms.

(ii) an improper fraction should always be expressed as mixed fraction.

Teacher's Note

When you share snacks with friends or divide a pizza into equal parts, you are using fractions in everyday life to ensure fair distribution.

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