If you’re learning about fractions, you’ve probably heard the term “equivalent fractions” before. But what exactly are they?

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Simply put, equivalent fractions are fractions that represent the same value but are written in different ways.
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For example, \frac{1}{2} and \frac{2}{4} are equivalent fractions because they both represent half of a whole. Similarly, \frac{3}{6} and \frac{5}{10} are equivalent fractions because they also both represent half of a whole.

As you can see, there can be many equivalent fractions for the same value.

Understanding equivalent fractions is important because they allow us to compare and manipulate fractions more easily. By finding equivalent fractions, we can add and subtract fractions with different denominators, simplify fractions, and compare fractions to see which is larger or smaller.

## What Are Equivalent Fractions?

Equivalent fractions are fractions that have different numerators and denominators but represent the same value. In other words, they are different ways of representing the same part of a whole.

### Definition

A fraction is made up of two parts: the numerator and the denominator. The numerator represents the part of the whole that is being considered, while the denominator represents the total number of equal parts that the whole is divided into. Equivalent fractions have different numerators and denominators, but they still represent the same value.

\frac{numerator}{denominator}

For example, \frac{1}{3} and \frac{2}{6} are equivalent fractions because they represent the same value, which is one-third of a whole.

### Examples

There are many examples of equivalent fractions. Here are a few:

\frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{4}{8} = \frac{5}{10} \ ...

\frac{2}{3} = \frac{4}{6} = \frac{6}{9} = \frac{8}{12} = \frac{10}{15} \ ...

\frac{3}{4} = \frac{6}{8} = \frac{9}{12} = \frac{12}{16} = \frac{15}{20} \ ...

Equivalent fractions can be useful in many situations, such as simplifying fractions, comparing fractions, and adding or subtracting fractions with different denominators. Understanding equivalent fractions is an important part of working with fractions in mathematics.

## How To Find Equivalent Fractions

To find equivalent fractions, you need to understand the concept of multiplying and dividing fractions.

### Multiply By Common Factor

To find equivalent fractions, you can multiply or divide both the numerator and denominator by the same number known as a
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factor
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.

For example, if you have the fraction \frac{3}{4} , you can find an equivalent fraction by multiplying both the numerator and denominator by a factor of 2, which gives you \frac{6}{8} .

\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}

Knowing how to multiply fractions to find their higher equivalents helps when you are adding or subtracting fractions as you will need to find a common denominator before performing the addition or subtraction operation.

For example, if you are adding the fractions \frac{1}{4} + \frac{1}{6} , you can find an equivalent fraction by finding the lowest common multiple (LCM) of 4 and 6 , which is 12 .

Then, you can multiply both the numerator and denominator of \frac{1}{4} by the number that gets the denominator to 12 , this would be the number 3 :

\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}

You would apply the same process for the second fraction \frac{1}{6} .

What number would you need to multiply the denominator 6 by to change it to 12 ? The number 2 . Therefore, \frac{1}{6} would be changed into the equivalent fraction of:

\frac{1}{6} = \frac{1 \times 2}{6 \times 12} = \frac{2}{12}

Now that the denominators of both fractions are the same their numerators can be added together like so:

\frac{3}{12} + \frac{2}{12} = \frac{3 + 2}{12} = \frac{5}{12}

### Divide By Common Factor

Similarly, you can divide both the numerator and denominator by a common factor to find an equivalent fraction.

For instance, if you have the fraction \frac{6}{12} , you can divide both the numerator and denominator by 6 , which gives you \frac{1}{2} .

\frac{6}{12} = \frac{6 \div 6}{12 \div 6} = \frac{1}{2}

This process of dividing both the numerator and the denominator is known as
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simplify a fraction
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. If you found the greatest (or highest) common factor (GCF or HCF) between both the numerator and denominator and divide both numbers by it, you can find the equivalent fraction in its most reduced or simplest form.

For example, if you have the fraction \frac{8}{24} , you can simplify it by finding the common factors of 8 and 24 , which are {1, 2, 3, 4, 8} . The greatest factor in that list is the number 8 , therefore, dividing both parts of the fraction by this number will help you to obtain the fraction in simplest form:

\frac{8}{24} = \frac{8 \div 8}{24 \div 8} = \frac{1}{3}

This will give you \frac{1}{3} .

By understanding these concepts, you can easily find equivalent fractions and simplify them to their lowest or simplest terms.

## Are These Fractions Equivalent?

One way of checking if two or more fractions are the same equivalent type is to convert all fractions in the list to the same
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denominator
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.

For example, are these fractions all equivalent (the same)?

\frac{3}{4}, \frac{12}{16}, \frac{33}{44}

One way to know is to try and either simplify (divide) the fractions into their simplest form, or multiply the fractions into their highest form.

To represent each fraction in simplest form you would need to divide both the numerator and denominator by its greatest (or highest) common factor. Therefore, inspecting each fraction in the example would produce the following GCF of each fraction:

\frac{3}{4} = \frac{3 \div 1}{4 \div 1} = \frac{3}{4}

\frac{12}{16} = \frac{12 \div 4}{16 \div 4} = \frac{3}{4}

\frac{33}{44} = \frac{33 \div 11}{44 \div 11} = \frac{3}{4}

As you can see each of the fractions above when represented in simplest form produces the same fraction, therefore, you could conclude that these fractions are all equivalent (the same).

## Checking Equivalent Fractions By Decimal

Another way you can check if two or more fractions are equivalent is by changing the fractions into decimals. This process can be longer than the above, but is another way you can check your answers.

To convert a fraction to decimal form, you simply divide the numerator (the top number) by the denominator (the bottom number). For example, to convert \frac{3}{4} to decimal form, you would divide 3 by 4 , which gives you 0.75 .

Performing the same process on the other two fractions \frac{12}{16} and \frac{33}{44} would produce the following results:

\frac{12}{16} = 12 \div 16 = 0.75

\frac{33}{44} = 33 \div 44 = 0.75

As can be seen from the above calculations all fractions return the same decimal number, and you can safely conclude that they are equivalent fractions.

## Equivalent Fractions Summary

Equivalent fractions are fractions that represent the same value, but they are written differently. Two fractions are equivalent if they have the same value, but different numerators and denominators. In other words, if you can simplify or reduce one fraction to get the other, they are equivalent.

For example, \frac{1}{2} and \frac{2}{4} are equivalent fractions because they represent the same value.

Equivalent fractions can be found by multiplying or dividing both the numerator and denominator by the same number.

For instance, to find an equivalent fraction of \frac{2}{3} , you can multiply both the numerator and denominator by 2 to get \frac{4}{6} or for a fraction like \frac{6}{9} divide both the numerator and denominator by 3 to get \frac{2}{3} .

Equivalent fractions are useful for comparing fractions, adding and subtracting fractions, and solving problems involving fractions. It is important to understand equivalent fractions to work with fractions effectively.