# Equivalent Fractions

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## Introduction to Equivalent Fractions

Often, there is more to meets the eye when it comes to math, and that especially holds true for fractions. Have a look at these two fractions:

$${3 \over 4}$$ and $${15 \over 20}$$

They look so different! Both the numerators and denominator are different, however, mathematically, they represent the same portion of the whole, and that makes them equal!

## The Big Idea: What are Equivalent Fractions?

Would you say $${3 \over 4}$$ is greater than or less than $${9 \over 12}$$? 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 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.

## Why equivalent fractions? Why are equivalent fractions important?

Remember how fractions were introduced? As parts of a whole. The fundamental principle of parts of any whole is that if you want to compare them, it’s imperative that they are parts of the same whole. This is the governing principle of equivalent fractions. Ensuring that fractions are comparable. Consider the above example, $${3 \over 4}$$ and$${9 \over 12}$$, both the fractions are equal because they represent the same portion of the whole.

Consider this interesting concept: If you add two equivalent fractions, the result is always two times the original fraction. Similarly, if you subtract a fraction from its equivalent fraction, then the result is always zero. Find that difficult to believe?

Consider $${2 \over 5}$$ and $${4 \over 10}$$. Both are equivalent fractions. Adding them would give $${8 \over 10}$$ by the LCM method described above, but $${8 \over 10}$$ is actually $${4 \over 5}$$, which is two times $${2 \over 5}$$!! This will be the same for any pair of equivalent fractions.

And $${2 \over 5}$$$${4 \over 10}$$ by the LCM method would give the result 0. This would be the same result for any pair of equivalent fractions!

## 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 \begin{align}3+1=4\end{align} and \begin{align}8+1=9\end{align}
To address this 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.

## 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}.