Equivalent Fractions

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What are equivalent fractions?

Would you say \(\begin{align} \frac {3}{4} \end{align}\) is greater than or less than \(\begin{align} \frac {9}{12} \end{align}\)? The correct answer is that they are both equal. That’s because they represent the same part of the whole. That is, at its core the whole concept of equivalent fractions. Visually:

If you multiply both the numerator and denominator of a fraction by the same number, the resulting fraction is exactly equal in value. Such a fraction is referred to as an Equivalent Fraction of the first fraction.

Let’s consider the mathematical equivalence of these equivalent fractions. In \(\begin{align} \frac {3}{4} \end{align}\) if both the numerator and denominator are multiplied by the number \(3,\) the fraction would be \(\begin{align} \frac {3 \times 3}{4 \times 3}=\frac {9}{12}.  \end{align}\)

In other words, \(ab\) is a fraction, where \(a\) and \(b\) are natural numbers. If \(m\) is a natural number, then \(a \times m b \times m\) is equivalent to the fraction \(ab.\) 

In other words, \(\begin{align} \frac {a}{b}  \end{align}\) is a fraction, where \(a\) and \(b\) are natural numbers. If \(m\) is a natural number, then \(\begin{align} \frac {(a \times m )}{(b \times m )} \end{align}\) is equivalent to the fraction \(\begin{align} \frac {a}{b}.  \end{align}\)

This procedure works because '\(m\)' is a non-zero whole number, and if both the numerator and denominator are multiplied or divided by the same whole number, the fractions will be equivalent. 

Reducing a fraction to its lowest form

Representing a fraction using as few parts as possible is referred to as the lowest form. A fraction is said to be in its lowest form if its numerator and denominator have no common divisor besides \(1.\) 

Pick a small number. For example, \(2,\;3,\;4,\;5,\;6 \) or \(7.\) Ensure that both the numerator and denominator are divisible by the same number.

Divide the numerator and denominator of a fraction by that number, let’s try and understand this using an example, 
\(\begin{align} \frac {4}{16} = \frac {(4 \div 4 )} {(16 \div 4 )}= \frac {1}{4} \end{align}\)

Tips and Tricks

  • When reducing a fraction to its lowest form (simplest form), students try to find the highest common factor. Sometimes, it’s easier to start with smaller common factors like \(2,\;3\) and \(5.\) If both the numerator and denominator are even, just divide them both by \(2.\) Similarly, if they are divisible by \(3,\) then start by dividing by \(3.\)
  • Some exercises require students to identify two equivalent fractions from a set of \(4\) or \(5\) fractions. In such cases instead of directly trying to find the equivalent pair, it helps to reduce all fractions to their lowest form.
    E.g. \(\begin{align} \frac {6}{14} \end{align}\) and \(\begin{align} \frac {9}{21} \end{align}\) are equivalent fractions but it is easier to spot that if we reduce both to their lowest form \(\begin{align} -\frac {3}{7}. \end{align}\)

Common mistakes or misconceptions

  • Misconception: Often children may add the same number to numerator and denominator to arrive at an equivalent fraction. E.g. \(\begin{align} \frac {3}{8} = \frac {4}{9} \end{align}\) because \(3 + 1 = 4\) and \(8 + 1 = 9\)
    To address this, a visual model should be used where such additive fractions are shown. When children see directly that they don’t look equal, this misconception gets addressed.
  • Misconception: Thinking that multiplying the numerator and the denominator by the same whole number increases the value of the fraction; similarly, dividing by the same whole number reduces the value.
    To address this misconception, strips and number lines can be used. Here, different equivalent fractions should be represented using these models. Children need to understand that all representations are of the same fraction.

Test Your Knowledge

  1. Write the lowest form of the fractions.
    \(\begin{align} \frac {6}{21}, \; \frac {8}{20}, \; \frac {4}{10} \end{align}\)
  2. Write \(3\) equivalent fractions for each fraction
    \(\begin{align} a.\;\; \frac {3}{5} \end{align}\)
    \(\begin{align} b.\;\; \frac {6}{9} \end{align}\)
    \(\begin{align} c.\;\; \frac {12}{16} \end{align}\)
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