Write three equivalent fractions of the following fractions: (i) \frac{2}{3} (ii) \frac{4}{5} (iii) \frac{3}{7} (iv) \frac{3}{8}
The Correct Answer and Explanation is:
Answer
(i) For \frac{2}{3}, three equivalent fractions are:
\frac{4}{6}, \frac{6}{9}, \frac{8}{12}
(ii) For \frac{4}{5}, three equivalent fractions are:
\frac{8}{10}, \frac{12}{15}, \frac{16}{20}
(iii) For \frac{3}{7}, three equivalent fractions are:
\frac{6}{14}, \frac{9}{21}, \frac{12}{28}
(iv) For \frac{3}{8}, three equivalent fractions are:
\frac{6}{16}, \frac{9}{24}, \frac{12}{32}
Explanation
Equivalent fractions are different fractions that represent the same value or the same part of a whole. Although their numerators and denominators are different, they hold an equal proportional relationship. For example, half of a cake can be represented as \frac{1}{2}, but it is also the same amount as \frac{2}{4} or \frac{4}{8} of the cake.
The fundamental rule for creating equivalent fractions is to multiply (or divide) both the numerator and the denominator of a fraction by the same non-zero number. This action does not change the actual value of the fraction. The reason this works is that multiplying a fraction by a form of 1, such as \frac{2}{2} or \frac{3}{3}, results in an equivalent fraction. Since any number divided by itself is 1, this operation preserves the fraction’s original value.
To find an equivalent fraction for \frac{2}{3}, one can choose any integer and multiply both the top and bottom numbers by it.
- Multiplying by 2:
2×23×2=463×22×2=64 - Multiplying by 3:
2×33×3=693×32×3=96 - Multiplying by 4:
2×43×4=8123×42×4=128
Therefore, \frac{4}{6}, \frac{6}{9}, and \frac{8}{12} are all equivalent to \frac{2}{3}. This same procedure is applied to find equivalent fractions for any given fraction. An infinite number of equivalent fractions can be generated for any single fraction by simply choosing different integers to multiply with. This principle is a cornerstone of understanding and manipulating fractional values in mathematics.
