ICSE Class 8 Maths Numbers Chapter 03 Fractions

3 - Fractions

Fractions

Fractions represent parts of a whole. For example, if an apple is divided into four equal parts, each part is called one fourth, and is denoted by \(\frac{1}{4}\). The line separating 1 and 4 indicates division. \(\frac{1}{4}\) is a fraction. Similarly, \(\frac{1}{2}, \frac{2}{3}, \frac{7}{7}, \frac{9}{11}\) and \(\frac{15}{19}\) are also fractions. In any fraction \(\frac{a}{b}\), a is called the numerator and b is called the denominator. For example, in the fraction \(\frac{5}{9}\), the numerator = 5 and the denominator = 9.

Classification Of Fractions

Common (or simple or vulgar) Fraction

The numerator is any integer and the denominator is a nonzero integer other than 10, 100, 1000, etc., in a common fraction. \(\frac{1}{8}, \frac{3}{7}\) and \(\frac{10}{77}\) are some common fractions.

Decimal Fraction

The denominator is some power of ten such as 10, 100, 1000, etc., in a decimal fraction. \(\frac{3}{10}, \frac{17}{100}\) and \(\frac{1}{1000}\) are some decimal fractions.

Complex Fraction

A fraction in which the numerator or denominator or both contain fractions is called a complex fraction. \(\frac{\frac{2}{7}}{\frac{9}{8}}\) and \(\frac{\frac{11}{9}}{\frac{25}{37}}\) are some complex fractions.

Proper Fraction

The numerator is less than the denominator in a proper fraction. \(\frac{1}{4}, \frac{11}{38}\) and \(\frac{125}{1373}\) are some proper fractions.

Improper Fraction

The numerator is equal to or greater than the denominator in an improper fraction. \(\frac{3}{3}, \frac{125}{77}\) and \(\frac{1435}{389}\) are some improper fractions.

Mixed Fraction (or number)

An integer together with a proper fraction is called a mixed fraction. \(3\frac{5}{7}, 8\frac{9}{25}\) and \(14\frac{17}{31}\) are some mixed fractions. In \(3\frac{5}{7}\), 3 is called the integral part and \(\frac{5}{7}\) is called the fractional part.

Mixed fractions can be written as improper fractions.

Examples (i) \(3\frac{5}{7} = \frac{3 \times 7 + 5}{7} = \frac{26}{7}\). (ii) \(8\frac{9}{25} = \frac{8 \times 25 + 9}{25} = \frac{209}{25}\).

(iii) \(14\frac{17}{31} = \frac{14 \times 31 + 17}{31} = \frac{451}{31}\).

Improper fractions can be written as mixed fractions.

Examples (i) \(\frac{23}{4} = 5\frac{3}{4}\) because 23 - 4 gives quotient 5 and remainder 3.

(ii) \(\frac{25}{12} = 2\frac{1}{12}\) because 25 - 12 gives quotient 2 and remainder 1.

Equivalent (or equal) Fractions

If the numerator and the denominator of a fraction are multiplied or divided by the same nonzero number, we get an equivalent fraction. The value of the fraction remains unchanged.

Examples (i) \(\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6}, \frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}, \frac{2}{3} = \frac{2 \times 7}{3 \times 7} = \frac{14}{21}\) etc.

\(\therefore \frac{2}{3} = \frac{4}{6} = \frac{10}{15} = \frac{14}{21}...\)

(ii) \(\frac{18}{24} = \frac{18 \div 2}{24 \div 2} = \frac{9}{12}, \frac{18}{24} = \frac{18 \div 3}{24 \div 3} = \frac{6}{8}, \frac{18}{24} = \frac{18 \div 6}{24 \div 6} = \frac{3}{4}\).

Thus, \(\frac{18}{24} = \frac{9}{12} = \frac{6}{8} = \frac{3}{4}\) are equivalent fractions.

Two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\) are equivalent if \(ad = bc\).

The fractions \(\frac{7}{9}\) and \(\frac{20}{27}\) are not equivalent fractions because \(7 \times 27 \neq 9 \times 20\).

Simplest Form Of A Fraction

A fraction is said to be in the simplest form if its numerator and denominator have no factor in common except 1.

Examples \(\frac{1}{2}, \frac{32}{79}, \frac{99}{211}, \frac{157}{371}\) are some fractions in the simplest form.

\(\frac{3}{12}, \frac{30}{135}, \frac{125}{625}\) are some fractions which are not in the simplest form.

A fraction can be reduced to its simplest form just by cancelling out the common factors in the numerator and the denominator.

Examples (i) \(\frac{3}{12} = \frac{3 \times 1}{3 \times 4} = \frac{1}{4}\). (ii) \(\frac{30}{135} = \frac{2 \times 3 \times 5}{3 \times 5 \times 9} = \frac{2}{9}\).

Like And Unlike Fractions

Fractions with the same denominator are called like fractions.

\(\frac{3}{7}, \frac{1}{7}, \frac{5}{7}\) are like fractions.

Fractions with different denominators are called unlike fractions.

\(\frac{2}{11}, \frac{5}{7}, \frac{5}{17}\) are unlike fractions.

Conversion Of Unlike Fractions Into Like Fractions

Steps 1. Find the LCM of the denominators of the fractions.

2. Divide the LCM by the respective denominators.

3. Multiply the numerator and the denominator of each fraction by the corresponding quotient obtained in Step 2.

Example Convert \(\frac{11}{16}, \frac{13}{20}\) and \(\frac{19}{25}\) into like fractions.

Solution First, we find the LCM of the denominators.

\(\therefore\) LCM = 2 × 2 × 5 × 4 × 5 = 400.

Now, 400 ÷ 16 = 25, 400 ÷ 20 = 20, 400 ÷ 25 = 16.

\(\therefore \frac{11}{16} = \frac{11 \times 25}{16 \times 25} = \frac{275}{400}, \frac{13}{20} = \frac{13 \times 20}{20 \times 20} = \frac{260}{400}\) and \(\frac{19}{25} = \frac{19 \times 16}{25 \times 16} = \frac{304}{400}\)

Hence, \(\frac{275}{400}, \frac{260}{400}\) and \(\frac{304}{400}\) are the required like fractions.

Teacher's Note

When sharing pizza slices or dividing a chocolate bar among friends, you are using fractions in everyday situations. Understanding fractions helps you make fair divisions and understand portions in cooking and recipes.

Comparison Of Fractions

Among like fractions, the fraction with the greatest numerator is the greatest.

To compare unlike fractions, we first convert them into like fractions and then compare their numerators.

Example Arrange the fractions \(\frac{17}{25}, \frac{5}{12}, \frac{13}{18}\) and \(\frac{11}{15}\) in ascending order.

Solution To change the fractions into like fractions, we find the LCM of the denominators.

LCM of the denominators 25, 12, 18 and 15

= 2 × 3 × 5 × 5 × 2 × 3 = 900.

Now, 900 ÷ 25 = 36, 900 ÷ 12 = 75, 900 ÷ 18 = 50 and 900 ÷ 15 = 60.

\(\therefore \frac{17}{25} = \frac{17 \times 36}{25 \times 36} = \frac{612}{900}, \frac{5}{12} = \frac{5 \times 75}{12 \times 75} = \frac{375}{900}, \frac{13}{18} = \frac{13 \times 50}{18 \times 50} = \frac{650}{900}\) and \(\frac{11}{15} = \frac{11 \times 60}{15 \times 60} = \frac{660}{900}\)

\(\therefore 375 < 612 < 650 < 660\), \(\therefore \frac{375}{900} < \frac{612}{900} < \frac{650}{900} < \frac{660}{900}\). So, \(\frac{5}{12} < \frac{17}{25} < \frac{13}{18} < \frac{11}{15}\).

Hence, the fractions in ascending order are \(\frac{5}{12}, \frac{17}{25}, \frac{13}{18}, \frac{11}{15}\)

Teacher's Note

When comparing prices per unit or deciding which store offers the best deal, you are comparing fractions in the form of ratios and proportions used in real shopping situations.

This is a preview of the first 3 pages. To get the complete book, click below.