In the first figure, the shaded portion is 1 part out of 3 equal parts.

So, the fraction indicated by the shaded portion is **\(\frac{1}{3}\)**

In the second figure, the shaded portion is 2 parts out of 6 equal parts.

So, the fraction indicated by the shaded portion is **\(\frac{2}{6}\)**

In the third figure, the shaded portion is 3 parts out of 9 equal parts.

So, the fraction indicated by the shaded portion is **\(\frac{3}{9}\)**.

However, the shaded portions in the three figures indicate equal portions of the whole i.e., one third of the whole. This means, the fractions **\(\frac{1}{3}\), \(\frac{2}{6}\), and \(\frac{3}{9}\)** have equal value.

So, \(\frac{1}{3}\) = \(\frac{2}{6}\) = \(\frac{3}{9}\) Such fractions, which have equal value, are called **equivalent** fractions.

So, \(\frac{1}{3}\), \(\frac{2}{6}\), and \(\frac{3}{9}\) are equivalent fractions.

We can obtain equivalent fractions by multiplying or dividing the numerator and the denominator of a fraction by the same counting number (counting numbers are 1, 2, 3, 4, … )

1) Consider the fraction \(\frac{1}{3}\) Let us obtain equivalent fractions of the fraction \(\frac{1}{3}\) by multiplying the numerator and the denominator of the fraction by the same counting numbers.

\(\frac{1}{3}\), \(\frac{1 X 2}{3 X 2} = \frac{2}{6}\), \(\frac{1 X 3}{3 X 3} = \frac{3}{9}\), \(\frac{1 X 4}{3 X 4} = \frac{4}{12}\), \(\frac{1 X 5}{3 X 5} = \frac{5}{15}\).

The fractions so obtained are \(\frac{1}{3}\), \(\frac{2}{6}\), \(\frac{3}{9}\), \(\frac{4}{12}\), \(\frac{5}{15}\). All these are **equivalent fractions.**

2) Consider the fraction \(\frac{8}{16}\) Let us obtain equivalent fractions of the fraction \(\frac{8}{16}\) by dividing the numerator and the denominator of the fraction \(\frac{8}{16}\) by the same counting numbers.

\(\frac{8}{16}\), \(\frac{8 \div 2}{16 \div 2} = \frac{4}{8}\), \(\frac{8 \div 4}{16 \div 4} = \frac{2}{4}\), \(\frac{8 \div 8}{16 \div 8} = \frac{1}{2}\)

The fractions so obtained are \(\frac{8}{16}\), \(\frac{4}{8}\), \(\frac{2}{4}\), \(\frac{1}{2}\). All these are **equivalent fractions**.

A fraction has unlimited number of equivalent fractions.

How do we know whether a given pair of fractions are equivalent or not?

Study the following examples:

1) Consider the pair of fractions \(\frac{2}{3}\) and \(\frac{12}{18}\).

In order to know whether \(\frac{2}{3}\) and \(\frac{12}{18}\) are equivalent, find the cross products.

So, \(\frac{2}{3}\) and \(\frac{12}{18}\) are** equivalent fractions**.

Let us verify the above result.

\(\frac{2}{3} = \frac{2 X 6}{3 X 6} = \frac{12}{18}\). So, \(\frac{2}{3}\) and \(\frac{12}{18}\) are equivalent fractions.

2) Consider the pair of fractions \(\frac{4}{7}\) and \(\frac{5}{8}\).

In order to know whether \(\frac{4}{7}\) and \(\frac{5}{8}\) are equivalent, find the cross products.

So, we conclude that the fractions \(\frac{4}{7}\) and \(\frac{5}{8}\) are **not equivalent**.

Two fractions are equivalent if their cross products are equal.