You might have witnessed showcasing a particular numeric value in multiple ways. For instance, representing 20 as 5 x 4, 15 + 5, and many other ways. In the same way, you can represent fractions having similar values. These fractions are known as equivalent fractions. So, let’s learn what equivalent fractions are.

**What is an equivalent fraction? **

Equivalent fractions are those fractional values whose numerator and denominator are different, but they represent the same value. For example, 5/4 and 30/24. They both have different numerators and denominators, but they represent the same value, i.e. 5/4 or 1.25.

Equivalent fractions are those fractional values that represent similar values when reduced to the simplest form of fractions. As shown above, 30/24 can be further simplified as 15/12, which can further be simplified as 5/4.

**Understanding equivalent fractions by examples**

Let us consider a circle cut into two parts, as shown below. If you further cut the circle into more parts, the shaded portion still shows the same fractional value. If seen as a whole, the shaded parts represent the same equivalent fraction as that represented by the first one.

Hence, ½ is equivalent to 2/4, 3/6, 4/8, etc. Therefore, ½, 2/4, 3/6, and 4/8 will be equivalent fractions.

**How to find the equivalent fractions?**

You might have come across ways where you were multiplying the numerator and denominator with the same number to make them equal to add or subtract. The same method needs to be applied here but in a different way.

Equivalent fractions can be determined using multiplying or dividing the numerator and denominator by the same number. Let us learn about these two methods in detail.

**Multiplying and dividing the numerator and denominator**

You must multiply both the numerator and denominator with the same number to make the fractions equivalent. For example, to make equivalent fractions of ⅚, you must multiply numerator and denominator with the same number. The number can be any, ranging from 2 to infinity.

However, the value must be positive. Otherwise, the fractional value will become negative. If you take one number positive and the other negative, the resulting value will be negative. If you take both the value negative and multiply with the numerator and denominator, the value will be positive as the negative terms will cancel each other.

Now, coming to making equivalent fractions of 5/6. Multiply the numerator and denominator with numbers greater than 2. Hence, the equivalent fractions can be represented as:

⅚ = 5 x 3/6 x 3= 15/18

⅚ = 5 x 4/6 x 4= 20/24

⅚ = 5 x 5/6 x 5= 25/30

⅚ = 5 x 6/6 x 6= 30/36, and so on.

You can find several equivalent fractions of any number using the multiplication method.

Next is using the dividing method. If the fractional value is big and you need to find the equivalent fractions of the same, you need to use the dividing method. For example, 56/84 is a big fraction. So, you have to use the dividing method as illustrated below.

The first step is to find the common factor between these numerators and denominators. The smallest number, to begin with, is 2, and 56 and 84 are divisible by 2.

Therefore, the number now becomes 28/42. If you further divide the number by 2, the equivalent fraction becomes 14/21. Finally, the numerator and denominator are divisible by 7. Hence, the smallest equivalent fraction of 56/84 is ⅔.

To ace this method, you must know the factors of various numbers to find the simplest equivalent fractions quickly.

**How to determine whether two fractions are equivalent or not?**

To determine whether two fractions are equivalent, you can use two methods – make their numerator and denominator equal or use the cross multiplication method.

- You have to equate the numerator and denominator to the same value in the first method. For example, let us check whether 2/6 and 3/9 are equivalent. Multiplying the first fraction by 3 and the other by 2. Once you multiply the numerators and denominators by 3 and 2, respectively, you will get 6/18 in both terms.

2/6 = 2 x 3/6 x 3= 6/18

3/9 = 3 x 2/9 x 2= 6/18

Hence, you will get 6/18 in both terms, which denotes that these two fractions are equivalent.

Similarly, you could have used the dividing method. You can see the numerator and denominator of 2/6 are divisible by 2, and that of 3/9 is divisible by 3. Hence, the simplified fraction will be ⅓ in both the terms, which denotes they are equivalent.

**Cross multiplication method**

Let us try to find the equivalent fraction using the cross multiplication method. In the above example, 2/6 and 3/9 were given. In this method, you need to cross multiply the numerator and denominator. If you get the same value, then these fractions are equivalent. For example, 2 will be multiplied by 9 and 6 by 3. Hence, in both cases, you get 18, which means these fractions are equivalent.

Similarly, taking another example of ⅘ and 16/20. Cross multiply the numerator and denominator of these fractions. You will get 4 x 20 and 5 x 16. Both the operations will yield 80, which denotes these fractions are equivalent.

**Points to Ponder**

- If the values of two fractions are the same, or even their decimal values are equal, these fractions are known as equivalent fractions.
- You can use two methods to get equivalent fractions – multiply the numerator and denominator by the same number, or divide the numerator and denominator by the factors of the same number.
- Equivalent fractions represent the same amount of distance or points on a number line.
- All equivalent fractions reduce to the same fraction in their simplest form.
- You can only multiply or divide, not add or subtract, to get equivalent fractions.
- The cross multiplication method is the simplest method to find equivalent fractions.
- Make the denominator the same to find equivalent fractions. Use this method if you need to find equivalent fractions of more than one fractional value.
- The equivalent fraction of ⅔ are 2/3, 4/6, 6/9, 8/12, 10/15, 12/18, 14/21, 16/24, 18/27, 20/30, 22/33, 24/36, 26/39, 28/42, 30/45, 32/48, 34/51, 36/54, 38/57, 40/60, and so on.
- The equivalent fractions of ¾ are 3/4, 6/8, 9/12, 12/16, 15/20, 18/24, 21/28, 24/32, 27/36, 30/40, 33/44, 36/48, 39/52, 42/56, 45/60, 48/64, 51/68, 54/72, 57/76, 60/80, and so on.
- If you are looking for what fraction is equivalent to ⅓, then here are the equivalent fractions of ⅓ – 1/3, 2/6, 3/9, 4/12, 5/15, 6/18, 7/21, 8/24, 9/27, 10/30, 11/33, 12/36, 13/39, 14/42, 15/45, 16/48, 17/51, 18/54, 19/57, 20/60, and so on.

**Example 1: Find the equivalent fractions for the following: 2/7, 4/11, 3/13, 7/9, 5/7. **

**Solution:** To find the equivalent fraction of the following terms, you must use the multiplication method and multiply their numerators and denominators by numbers starting from 2.

The equivalent fractions for the above fractions are:

2/7 – 2/7, 4/14, 6/21, 8/28, 10/35, 12/42, 14/49, 16/56, 18/63, 20/70, 22/77, 24/84, 26/91, 28/98, 30/105, 32/112, 34/119, 36/126, 38/133, 40/140

4/11 – 4/11, 8/22, 12/33, 16/44, 20/55, 24/66, 28/77, 32/88, 36/99, 40/110, 44/121, 48/132, 52/143, 56/154, 60/165, 64/176, 68/187, 72/198, 76/209, 80/220

3/13 – 3/13, 6/26, 9/39, 12/52, 15/65, 18/78, 21/91, 24/104, 27/117, 30/130, 33/143, 36/156, 39/169, 42/182, 45/195, 48/208, 51/221, 54/234, 57/247, 60/260

7/9 – 7/9, 14/18, 21/27, 28/36, 35/45, 42/54, 49/63, 56/72, 63/81, 70/90, 77/99, 84/108, 91/117, 98/126, 105/135, 112/144, 119/153, 126/162, 133/171, 140/180

5/7 – 5/7, 10/14, 15/21, 20/28, 25/35, 30/42, 35/49, 40/56, 45/63, 50/70, 55/77, 60/84, 65/91, 70/98, 75/105, 80/112, 85/119, 90/126, 95/133, 100/140