How to find equivalent fractions

Why are they the same? Because when you multiply or divide both the top and bottom by the same number, the fraction keeps it’s value.

The rule to remember is:

“Change the bottom using multiply or divide,
And the same to the top must be applied”

Here is why those fractions are really the same:

1 /2 2 /4 4 /8
How to find equivalent fractions = How to find equivalent fractions = How to find equivalent fractions

See Fractions on the Number Line .

. it shows many equivalent fractions.

Also see the Chart of Fractions with many examples of equivalent fractions.

Dividing

Here are some more equivalent fractions, this time by dividing:

÷ 3 ÷ 6
18 = 6 = 1
36 12 2
÷ 3 ÷ 6

Choose the number you divide by carefully, so that the results (both top and bottom) stay whole numbers.

If we keep dividing until we can’t go any further, then we have simplified the fraction (made it as simple as possible).

Find equivalent fractions. Enter a fraction, mixed number or integer to get fractions that are equivalent to your input. Example entries:

  • Fraction – like 2/3 or 15/16
  • Mixed number – like 1 1/2 or 4 5/6
  • Integer – like 5 or 28

What are Equivalent Fractions?

Equivalent fractions are fractions with different numbers representing the same part of a whole. They have different numerators and denominators, but their fractional values are the same.

For example, think about the fraction 1/2. It means half of something. You can also say that 6/12 is half, and that 50/100 is half. They represent the same part of the whole. These equivalent fractions contain different numbers but they mean the same thing: 1/2 = 6/12 = 50/100

How to Find Equivalent Fractions

Multiply both the numerator and denominator of a fraction by the same whole number. As long as you multiply both top and bottom of the fraction by the same number, you won’t change the value of the fraction, and you’ll create an equivalent fraction.

Example Equivalent Fractions

Find fractions equivalent to 3/4 by multiplying the numerator and denominator by the same whole number:

Therefore these are all equivalent fractions:

Note that if you reduce all of these fractions to lowest terms, they equal 3/4.

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Two or more fractions are said to be equivalent if they result in the same value on simplification. Let us consider two fractions \(\frac < u > < v >\), \(\frac < w > < x >\) they are said to be equivalent if they result in same value \(\frac < y > < z >\) on simplification.

Usually, Equivalent Fractions denote the same portion on a whole. Find details like equivalent fractions definition, how to find equivalent fractions using different methods like multiplication, division with the same numbers. Check out few solved examples on finding the equivalent fractions explained step by step in the later sections.

Equivalent Fractions – Definition

Equivalent Fractions are the fractions having the same value irrespective of the numerators and denominators they have. All Equivalent Fractions on reducing will result in the same value after simplification. For Example: \(\frac <3 > < 4 >\), \(\frac < 6> < 8 >\), \(\frac < 9 > < 12 >\), \(\frac < 12 > < 16 >\), etc. are all equivalent fractions.

How to Find Equivalent Fractions?

There are two different ways to find the Equivalent Fractions and they are explained in detail below. They are as follows

  • Multiply both the numerator and denominator of the fraction with the same number.
  • Divide both the numerator and denominator of the fraction with the same number.

Multiplying Numerator and Denominator of the Fraction with the Same Number

To get the Equivalent Fraction of any number simply multiply both the numerator and denominator with the same number.
Example:
Find Equivalent Fractions of the Fraction \(\frac < 1 > < 2 >\)?
Solution:
Equivalent fraction of \(\frac < 1 > < 2 >\) can be obtained by multiplying with non-zero numbers.
\(\frac < 1 > < 2 >\)*2 = \(\frac < 2 > < 4 >\)
\(\frac < 1 > < 2 >\)*3 = \(\frac < 3 > < 6 >\)
\(\frac < 1 > < 2 >\)*4 = \(\frac < 4 > < 8 >\), etc.

Therefore, \(\frac < 2 > < 4 >\), \(\frac < 3 > < 6 >\), \(\frac < 4 > < 8 >\) are all equivalent fractions of \(\frac < 1 > < 2 >\).

Dividing Numerator and Denominator of the Fraction with the Same Number

To get an Equivalent Fraction of a Number simply divide both the numerator and denominator with the same value other than zero. To divide the fraction with the same number we need to evaluate the common factors that both the denominator and numerator share and then divide with them.

Example:
Find the equivalent fractions of the fraction \(\frac < 16 > < 24 >\)?
Solution:
To obtain the equivalent fractions of \(\frac < 16 > < 24 >\) we will first find the common factors that both 16, 24 share.
Common Factors of 16, 24 are 1,2,4,8
Dividing with common factors we have
\(\frac < 16 > < 24 >\) = \(\frac < 16÷2 > < 24÷2 >\)
= \(\frac < 8 > < 12 >\)
\(\frac < 16 > < 24 >\) = \(\frac < 16÷4 > < 24÷4 >\)
= \(\frac < 4 > < 6 >\)
\(\frac < 16 > < 24 >\) = \(\frac < 16÷8 > < 24÷8 >\)
= \(\frac < 2 > < 3 >\)
Therefore, equivalent fractions of \(\frac < 16 > < 24 >\) are \(\frac < 8 > < 12 >\), \(\frac < 4 > < 6 >\), \(\frac < 2 > < 3 >\), etc. and so on.

Equivalent Fractions Examples

Example 1. Check if the following fractions are equivalent or not?

Solution:

The first figure is shaded half so representing in the fraction we have \(\frac < 1 > < 2 >\)
In the same way, the second and third figures are given by \(\frac < 4 > < 8 >\) and \(\frac < 5 > < 10 >\)
As all the figures are shaded half they are said to be equivalent fractions.
We can write \(\frac < 4 > < 8 >\) = \(\frac < 1 > < 2 >\) * 2
\(\frac < 5 > < 10 >\) = \(\frac < 1 > < 2 >\) * 5
Thus, equivalent fractions are obtained by multiplying numerators and denominators with non-zero numbers.

Example 2.
Find the Equivalent Fractions of the fraction \(\frac < 8 > < 24 >\)?
Solution:
We can find the Equivalent Fractions of a fraction by either multiplying or dividing the fraction with the same number.
Let us find the equivalent fractions of the fraction \(\frac < 8 > < 24 >\) by applying division method.
\(\frac < 8 > < 24 >\) = \(\frac < 8÷2 > < 24÷2 >\)
= \(\frac < 4 > < 12 >\)
\(\frac < 8 > < 24 >\) = \(\frac < 8÷4 > < 24÷4 >\)
= \(\frac < 2 > < 6 >\)
\(\frac < 8 > < 24 >\) = \(\frac < 8÷8 > < 24÷8 >\)
= \(\frac < 1 > < 3 >\)

Therefore, Equivalent Fractions of fraction \(\frac < 8 > < 24 >\) are \(\frac < 4 > < 12 >\), \(\frac < 2 > < 6 >\), \(\frac < 1 > < 3 >\)

Understanding equivalent fractions is crucial for being able to complete other math computations such as adding and subtracting fractions.

  • What is an equivalent fraction?
  • How to find equivalent fractions?
  • Cross multiplication to double-check equivalent fractions

What Is An Equivalent Fraction?

Equivalent fractions are fractions with different numerators and denominators that are still equal in value. They can be found by multiplying or dividing the numerator and denominator by whole numbers. An example would be 1/2 is equivalent to 2/4, which represents equal parts of an overall whole.

Let’s look at a few examples to explain this. How to find equivalent fractions

Here you will see in the first row that we have 3 boxes with 2 of them being colored. The fraction of boxes that are colored would be written as 2/3.

Your denominator always shows the total pieces of the whole.

The numerator always shows the total number that you are looking for. In this example, we are looking for the number of boxes that are colored.

You can think of the numerator and denominator this way.

Numerator starts with the letter N. Think about the direction north – which on a compass is pointing up top.

Denominator starts with the letter D. Think about the word down – as it will always be the bottom number.

The second row shows a total of 6 boxes with 4 of them being colored. You would write the fraction of colored boxes as 4/6.

Finally, the last row has a total of 12 boxes with 8 of them being colored. The fraction of colored boxes is stated as 8/12.

First, we look at all 3 of these bars together. Do you agree that they are the same length and that the amount of squares that are colored are the same?

Each row has the same portion or piece of the overall whole as being colored, yet the fractions are written differently. They are equal in value, but are written differently.

These are equivalent fractions!

Let’s look a little closer at the 3 fractions we have written above.

If we look at the fraction 2/3 and compare it to 4/6 you will notice that the numerator and denominator are exactly doubled.

Another way to look at it is to say that if you multiply the numerator in 2/3 times 2 and the denominator in 2/3 times 2 you end up with 4 over 6.

This is an example of the identity property of multiplication.

What is the identity property of multiplication and division?

The identity property of multiplication states that any number multiplied by 1 will always equal itself. The identity property of division also states that any number divided by 1 will always equal itself.

In other words, 2 x 1 will always equal 2.

For division, 4 divided by 1 will always equal 4.

This concept is important for creating equivalent fractions. When you have a numerator and a denominator with the same number, it is the same thing as being equivalent to 1.

How Do You Find Equivalent Fractions?

Equivalent fractions are found by multiplying or dividing fractions by whole numbers. The key is to multiply or divide the numerator and denominator by the same number. The identity property of multiplication and division states that any number multiplied or divided by 1 will always equal the original number.

Looking at the fraction of 2/3 we can see that by multiplying both the numerator and denominator by 2 we end up with 4/6. If we continue that thought, we can multiply 4/6 by 2 and get 8/12. This proves that all 3 fractions above are equivalent.

How to find equivalent fractions

Cross Multiplication with Equivalent Fractions

Now that you have had some practice with finding equivalent fractions, let’s go over a simple way that you can double check your answers to ensure they are correct.

Cross multiplication can be used as a way to validate that the fractions are equivalent. Cross multiplication means multiplying the numerator of one fraction times the denominator of the other and vice versa.

Let’s look at a quick example:

We know from above that 2/3 is equivalent to 4/6. Let’s use cross multiplication to prove that out.

How to find equivalent fractions

Multiplying 3 x 4 = 12

Multiplying 2 x 6 = 12

These fractions are indeed equivalent!

Be sure to check out my YouTube video on equivalent fractions above to walk through more examples.

For additional practice on equivalent fractions, head over to my additional resources page for extra practice problems and answer sheets or find the direct links below. Also, check out my Knowledge Check video on Equivalent Fractions for more practice and answers.

This Equivalent Fractions Calculator will show you, step-by-step, equivalent fractions to any fraction you input.

See below the step-by-step solution on how to find equivalent fractions.

How to find equivalent fractions?

Two fractions are equivalent when they are both equal when written in lowest terms. The fraction 2 6 is equal to 1 3 when reduced to lowest terms. To find equivalent fractions, you just need to multiply the numerator and denominator of that reduced fraction ( 1 3 ) by the same integer number, ie, multiply by 2, 3, 4, 5, 6 .

  • 2 6 is equivalent to 1 3 because 1 x 2 = 2 and 3 x 2 = 6
  • 3 9 is equivalent to 1 3 because 1 x 3 = 3 and 3 x 3 = 9
  • 4 12 is equivalent to 1 3 because 1 x 4 = 4 and 3 x 4 = 12

At a glance, equivalent fractions look different, but if you reduce then to the lowest terms you will get the same value showing that they are equivalent. If a given fraction is not reduced to lowest terms, you can find other equivalent fractions by dividing both numerator and denominator by the same number.

What is an equivalent fraction? How to know if two fractions are equivalent?

Finding equivalent fractions can be ease if you use this rule:

Equivalent fractions definition: two fractions a b and c d are equivalent only if the product (multiplication) of the numerator (a) of the first fraction and the denominator (d) of the other fraction is equal to the product of the denominator (b) of the first fraction and the numerator (c) of the other fraction.

In other words, if you cross-multiply ( a b and c d ) the equality will remain, i.e, a.d = b.c . So, here are some examples:

  • 2 6 is equivalent to 1 3 because 2 x 3 = 6 x 1 = 6
  • 3 9 is equivalent to 1 3 because 3 x 3 = 9 x 1 = 9
  • 4 12 is equivalent to 1 3 because 4 x 3 = 12 x 1 = 12

Equivalent Fractions Table / Chart

This Equivalent Fractions Table/Chart contains common practical fractions. You can easily convert from fraction to decimal, as well as, from fractions of inches to millimeters.

When fractions have different numbers in them, but have the same value, they are called equivalent fractions.

Let’s take a look at a simple example of equivalent fractions: the fractions ½ and 2/4. These fractions have the same value, but use different numbers. You can see from the picture below that they both have the same value.

How can you find equivalent fractions?

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

How does this work?

We know from multiplication and division that when you multiply or divide a number by 1 you get the same number. We also know that when you have the same numerator and denominator in a fraction, it always equals 1. For example:

So as long as we multiply or divide both the top and the bottom of a fraction by the same number, it’s just the same as multiplying or dividing by 1 and we won’t change the value of the fraction.

Since we multiplied the fraction by 1 or 2/2, the value doesn’t change. The two fractions have the same value and are equivalent.

You can also divide the top and bottom by the same number to create an equivalent fraction as shown above.

Cross Multiply

There is a formula you can use to determine if two fractions are equivalent. It’s called the cross multiply rule. The rule is shown below:

This formula says that if the numerator of one fraction times the denominator of the other fraction equals the denominator of the first fraction times the numerator of the second fraction, then the fractions are equivalent. It’s a bit confusing when written out, but you can see from the formula that it’s fairly simple to work out the math.

If you get confused on what to do, just remember the name of the formula: “cross multiply”. You are multiplying across the two fractions like the pink “X” shown in the example below.

Comparing Fractions

How can you tell if one fraction is bigger than another?

In some cases it’s pretty easy to tell. For example, after working with fractions for a while, you probably know that ½ is bigger than ¼. It’s also easy to tell if the denominators are the same. Then the fraction with the larger numerator is bigger.

However, sometimes it’s difficult to tell which is bigger just by looking at two fractions. In these cases you can use cross multiplication to compare the two fractions. Here is the basic formula:

Operating with fractions will be extremely difficult for your child unless they can grasp the concept of equivalent fractions.

Teaching equivalent fractions

Explain to your child why equivalent fractions are important. They may not yet have tried addition or subtraction with fractions but you can introduce equivalency by comparing fractions.

Use hands-on activities. For example, take a pizza, cut it in half then cut one half into three equal slices. The single half and the half cut into three should appear equal. 1⁄2 = 3⁄6

How to find equivalent fractions

The example below uses a pie, cut into equal pieces, to show equivalent fractions.

How to find equivalent fractions

Look at the pie example above. Notice how the top and bottom (numerator and denominator) of the fraction is increasing by a factor of 2. In other words, they are both being multiplied by 2.

Multiplying or dividing both the numerator and denominator of a fraction will result in an equivalent fraction. Here are some more examples:

How to find equivalent fractions

Do the same to both the numerator and the denominator

Questions often require equivalent fractions to be written when only the numerator or denominator are given. The examples below show how these questions can be answered.

How to find equivalent fractions

Remember: Only use multiplication or division when finding equivalent fractions. Do not use addition or subtraction.

Before moving on to work with fractions, it is important that your children understand equivalency of fractions. Be sure they can accurately determine larger and smaller equivalent fractions. Ask them to describe what they are doing to determine equivalent fractions?

Worksheets

Practice working with equivalent fractions using the worksheets below.

    e.g. 1/3 = 3/9 e.g. 9/3 = 1/3

Online Fraction Games

The two fraction games below will help with practicing equivalent fractions.

A 60 minute lesson in which students will identify equivalent fractions for one half, one third, one quarter and one fifth.

This lesson plan includes the following resources:

Preparation

Equipment
  • Cuisenaire rods

Lesson Plan

To see the rest of this lesson plan, upgrade to the Plus Plan.

NSW Curriculum alignment

  • MA2-7NA

Fractions and Decimals – represents, models and compares commonly used fractions and decimals

Victorian Curriculum alignment

  • VCMNA157

Investigate equivalent fractions used in contexts

Australian Curriculum alignment

  • ACMNA077

Investigate equivalent fractions used in contexts

Find more resources for these topics

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