ICSE Class 6 Maths Chapter 05 Fractions

Chapter 5

Fractions

5.1 Basic Concept

If a certain quantity of rice is divided into four equal parts, each part so obtained is said to be one-fourth \(\frac{1}{4}\) of the whole quantity of the rice.

Similarly, if an apple is divided into five equal parts, each part is one-fifth \(\frac{1}{5}\) of the whole apple. Now, if two parts of these 5 equal parts are eaten, three parts are left, and we say three-fifths \(\frac{3}{5}\) of the apple is left.

The numbers \(\frac{1}{4}\), \(\frac{1}{5}\) and \(\frac{3}{5}\) discussed above, each representing a part of the whole quantity, are called fractions.

A fraction is a quantity that expresses a part of the whole.

To Make The Concept Of Fractions More Clear

Draw a circle with any suitable radius.

Divide the circle into three equal parts (sectors).

If two parts of the three equal parts be shaded, we say \(\frac{2}{3}\) (two-thirds) of the circle is shaded and \(\frac{1}{3}\) (one-third) of the circle is not.

In the fraction \(\frac{a}{b}\), a is the numerator of the fraction and b is its denominator.

Fraction = \(\frac{\text{Numerator}}{\text{Denominator}}\)

Thus, in fraction \(\frac{7}{11}\), numerator = 7 and denominator = 11.

The numerator and the denominator are also known as the terms of a fraction.

Every fraction must be expressed in its lowest terms. In other words, the terms of a fraction must not have any common factor except 1(one).

Fractions \(\frac{3}{5}\), \(\frac{15}{11}\) and \(\frac{7}{10}\), are in their lowest terms, because the terms of each of these fractions have only 1 (one) as common factor.

5 out of 7 means a given quantity is divided into seven equal parts and five of these equal parts are taken.

Thus, 5 out of 7 = \(\frac{5}{7}\).

Teacher's Note

Fractions are used every day when we cook, measure ingredients, or split a pizza with friends. Understanding how parts make a whole is essential for practical math.

5.2 Types Of Fractions

1. Proper Fraction

A fraction whose numerator is less than its denominator is called a proper fraction, e.g. \(\frac{4}{5}\), \(\frac{3}{7}\), \(\frac{101}{235}\), \(\frac{4}{7}\), \(\frac{9}{14}\), etc.

2. Improper Fraction

A fraction whose numerator is greater than or equal to its denominator is called an improper fraction.

e.g. (i) \(\frac{7}{5}\), \(\frac{25}{12}\), \(\frac{181}{62}\), etc.

(ii) \(\frac{3}{3}\), \(\frac{4}{4}\), \(\frac{5}{5}\), etc.

If the numerator and the denominator of a fraction are equal, the value of the fraction is unity (1).

e.g. \(\frac{4}{4} = 1\), \(\frac{-3}{-3} = 1\), etc.

3. Mixed Fraction

A mixed fraction consists of two parts: (i) an integer and (ii) a proper fraction.

e.g. \(4\frac{2}{3}\) is a mixed fraction, consisting of an integer (4) and a proper fraction \(\left(\frac{2}{3}\right)\).

\(3\frac{2}{5} = 3 + \frac{2}{5}\), \(8\frac{5}{6} = 8 + \frac{5}{6}\), \(-2\frac{1}{8} = -\left(2 + \frac{1}{8}\right)\) and so on.

Conversely,

\(2 + \frac{3}{8} = 2\frac{3}{8}\), \(7 + \frac{4}{9} = 7\frac{4}{9}\), \(-8 - \frac{5}{6} = -\left(8 + \frac{5}{6}\right) = -8\frac{5}{6}\) and so on.

4. Like And Unlike Fractions

Two or more fractions with the same denominator but different numerators are called like fractions.

e.g. \(\frac{3}{5}\), \(\frac{1}{5}\), \(\frac{2}{5}\), \(\frac{4}{5}\), \(\frac{7}{5}\), etc. are like fractions.

Two or more fractions with different denominators are called unlike fractions.

e.g. \(\frac{5}{9}\), \(\frac{7}{8}\), \(\frac{3}{4}\), \(\frac{1}{3}\), etc.

5. Equivalent Fractions

If two or more fractions have the same value, they are called equivalent or equal fractions.

e.g.the fractions \(\frac{1}{3}\), \(\frac{3}{9}\), \(\frac{6}{18}\) and \(\frac{9}{27}\) are equivalent fractions as \(\frac{1}{3} = \frac{3}{9} = \frac{6}{18} = \frac{9}{27}\).

The value of a fraction does not change if its numerator and denominator are both multiplied or divided by the same non-zero number.

e.g. \(\frac{4}{7}\) and \(\frac{4 \times 2}{7 \times 2}\) i.e. \(\frac{4}{7}\) and \(\frac{8}{14}\) are equivalent fractions.

Also, \(\frac{15}{20}\) and \(\frac{15 \div 5}{20 \div 5}\) i.e. \(\frac{15}{20}\) and \(\frac{3}{4}\) are equivalent fractions.

Teacher's Note

When sharing snacks or cutting fabric, equivalent fractions help us understand that different representations can mean the same amount, like one-half being the same as two-fourths.

5.3 Converting A Mixed Fraction Into An Improper Fraction

Multiply the integral part by the denominator and add the numerator to the product. The result so obtained is the numerator of the required improper fraction. The denominator of the required fraction will be the same as the denominator of the given mixed fraction.

Thus, for the mixed fraction \(3\frac{7}{15}\),

the required improper fraction = \(\frac{\text{Integral part} \times \text{Denominator} + \text{Numerator}}{\text{Denominator}}\)

\(= \frac{(3 \times 15) + 7}{15} = \frac{45 + 7}{15} = \frac{52}{15}\)

Similarly, \(5\frac{3}{4} = \frac{5 \times 4 + 3}{4} = \frac{20 + 3}{4} = \frac{23}{4}\),

\(7\frac{5}{6} = \frac{7 \times 6 + 5}{6} = \frac{42 + 5}{6} = \frac{47}{6}\) and so on.

5.4 Converting An Improper Fraction Into A Mixed Fraction

Divide the numerator by the denominator. The quotient of this division is the integral part and the remainder obtained is the numerator of the required mixed fraction.

Of course, the denominator will remain the same.

Thus, \(\frac{23}{4}\) = Quotient \(\frac{\text{Remainder}}{\text{Denominator}} = 5\frac{3}{4}\)

On dividing 23 by 4, quotient = 5 and remainder = 3.

Similarly, \(\frac{37}{8}\) = Quotient \(\frac{\text{Remainder}}{\text{Denominator}} = 4\frac{5}{8}\)

\(\frac{41}{9} = 4\frac{5}{9}\), \(\frac{73}{12} = 6\frac{1}{12}\) and so on.

5.5 Converting Unlike Fractions Into Like Fractions

Steps

Find the L.C.M. of the denominators of all the given fractions.

Multiply the numerator and the denominator of each fraction by a same suitable number so that the denominator of each fraction becomes equal to the L.C.M. obtained in step 1.

Example 1

Convert \(\frac{3}{7}\), \(\frac{4}{5}\) and \(\frac{1}{3}\) into like fractions.

Solution

L.C.M. of denominators 7, 5 and 3 = 105

Now, \(\frac{3}{7} = \frac{3 \times 15}{7 \times 15} = \frac{45}{105}\), \(\frac{4}{5} = \frac{4 \times 21}{5 \times 21} = \frac{84}{105}\) and \(\frac{1}{3} = \frac{1 \times 35}{3 \times 35} = \frac{35}{105}\)

Therefore, \(\frac{3}{7}\), \(\frac{4}{5}\) and \(\frac{1}{3} = \frac{45}{105}\), \(\frac{84}{105}\) and \(\frac{35}{105}\) respectively

Teacher's Note

Converting fractions to the same denominator is like finding a common language between different groups so everyone can be compared fairly and equally.

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