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Multiplication of Fractions – How to Multiply Fractions, Steps, and Examples

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Multiplication of Fractions- Multiplication of Fractions is a concept that helps us multiply two fractions or to find the products of two fractions.

Fractions are numerical expressions represented by the division of two numbers where both the numbers are generally coprime. To multiply any two given fractions we multiply the numerator of both the fractions and multiply the denominator of both fractions separately which will be discussed in detail further in the article.

In this article, we will explore how to do multiplication of fractions by fractions, and multiplication of fractions with the same and different denominators, along with examples. Other than that we will also learn about the rules of multiplication of fractions with whole numbers, multiplication of mixed fractions, multiplication of improper fractions, etc.

Multiplication of Fractions

Multiplication of fractions is a mathematical operation that involves finding the product of two or more fractions. To multiply fractions, you simply multiply the numerators (the top numbers of the fractions) together to get a new numerator, and you multiply the denominators (the bottom numbers of the fractions) together to get a new denominator. The result is a fraction that represents the product of the original fractions.

How to Multiply Fractions

Multiplication of fractions does not hold any resemblance with the addition or subtraction of fractions. Multiplication of fractions does not depend on like or unlike fractions and we simply find the multiplication of the fractions by multiplying the numerators and the denominators of the fractions separately.

Both multiplication of like fractions and unlike fractions are attained by the same methods. To find the product of the two fractions we should follow the steps added below,

Step 1: Multiply the numerator of both fractions separately.

Step 2: Multiply the denominator of both fractions separately.

Step 3: Write the result of the product of the fraction in the form

(Product of Numerators)/ (Product of Denominators)

Step 4: Simplify the resultant fractions to get the fractions in the simplest form.

Example of How to Multiply Fractions

Let’s consider an example of how to multiply fractions to understand the process better.

Example: Find the multiplication of 7/6 × 4/5.

Solution:

7/6 × 4/5

= (7×4)/(6×5)

= 28/30 [Simplifying it further]

= 14/15

Rules of Multiplication of Fractions by Fractions

Multiplication of fractions involves specific rules that allow fractions to be multiplied.There are 4 rules for multiplication of fractions:

Rule 1: Convert the mixed fractions into improper fractions.

Rule 2: Multiply the numerators separately.

Rule 3: Multiply the denominators separately and write the number in the form of Numerator/Denominator

Rule 4: Simplify the fraction to get the simplest form of the fraction

How to Multiply Fractions with Same Denominators

Fractions with the same denominator, are also called like fractions. Multiplication of Fractions with the same denominators is not very different from the multiplication of fractions already discussed in the article. In this case, we also multiply the denominator with the denominator and the numerator with the numerator to get the product of the fraction.

Suppose we have two fractions say, a/b and c/b, here we see that the denominator of both fractions is the same. Now we can easily find their product as,

a/b × c/b = (a×c)/(b×b) = ac/b2

The image added below shows the same,

Multiplication-of-Fraction-with-same-denominator

An example of the multiplication of the like fraction is given below.

Example: Find the product of 2/3 and 7/3.

Answer:

2/3 × 7/3

= (2×7) / (3×3) [Simplifying it further]

= 14/9

How to Multiply Fractions with Different Denominators

Fractions with different denominators are also called, unlike fractions. Multiplication of unlike fractions is similar to multiplication of like fractions. In this case, also we simply multiply the numerator with the numerator and the denominator with the denominator to get the required product of the fraction.

Suppose we have two fractions say, a/b and c/d, here we see that the denominator of both fractions is different. To Multiply Fractions with Different Denominators we can use the following formula:

a/b × c/d = (a × b) / (c × d) = ab/cd

The image added below shows the same,

Multiplication-of-fraction-with-Different-Denominator

An example of the multiplication of the unlike fraction is,

Find the product of 2/3 and 3/4

2/3 × 3/4

= (2×3) / (3×4)

= 6 / 12 [Simplifying it further]

= 1/2

Multiplication of Improper Fractions

Improper fractions are the fractions in which the numerator is smaller than the denominator and multiplication of these fractions is similar to multiplication of normal fraction. This is explained suppose if we have two improper fractions a/b {where a < b} and c/d {where c < d} then,

a/b × c/d = (a×c)/(b×d)

For example, find the product of 2/3 and 7/8

2/3 × 7/8

= (2×7) / (3×8)

= 14/24 [Simplifying it further]

= 7/12

Multiplication of Fractions with Whole Numbers

Multiplication of fractions with whole numbers can be easily achieved by converting the whole number into a proper fraction by taking the denominator of the whole number as one and then multiplying the fraction normally as in the denominator part the multiplication with one does not change the denominator part thus we can also say that, if we multiply the fraction with whole number then we multiply the whole number with the numerator of the reaction and then convert it into the simplest form to get the product.

This is achieved as,

a × b/c = a/1 × b/c = (ab)/c

Multiplication of Fractions with Whole Numbers can be explained using these examples:

Example: Multiply 3 × 11/5

Solution:

3 × 11/5

= 3/1 × 11/5

= (3 × 11)/5

= 33/5

How to Multiply Mixed Fractions

We multiply a fraction with mixed fractions by simply changing the mixed fractions into improper fractions and then multiplying them accordingly. A mixed fraction is written as, a(b/c)

a(b/c) = (ac + b)/c

Now changing the fractions into improper fractions we can easily multiply the mixed fractions. This is shown as,

a(b/c) × d(e/f) = (ac + b)/c × (df + e)/c

Now the mixed fractions are multiplied as normal fractions. This can be explained using the example as,

Example: Multiply 3(4/5) × 11(2/3)

Solution:

3(4/5) × 11(2/3)

= (3×5 + 4)/5 × (11×3 + 2)/3

= 19/5 × 35/3

= (19 × 35) + (5 × 3)

= 665/15 [Simplifying it further]

= 133/3

Related Resources

Dividing Fractions by Fractions

Dividing a fraction is the reverse operation of multiplication of fraction. suppose we have to divide two fractions and then multiply the first fraction with the reciprocal of the second fraction to get the required result of division. The following example explains this,

Example: 2/7 ÷ 3/5

Solution:

2/7 ÷ 3/5

= 2/7 × 5/3

= (2 × 5)/(7 × 3)

= 10/21

Division and Multiplication of Fractions

The key differences between Division of Fractions and Multiplication of Fractions are listed in the following table:

Aspect

Multiplication of Fractions

Division of Fractions

Operation

Multiplying two or more fractions togetherDividing one fraction by another fraction

Mathematical Expression

a/b ​× c/da/b ​÷ c/d​

General Formula

(a × c)/(b ​× d)​(a × d)/(b ​× c)​​

Example

2/3 ​× 4/5 = 8/152/3 ​÷ 4/5 = 5/6

Commutative Property

Multiplication is commutative: a × b = b × aDivision is not commutative: a ÷ b ≠ b ÷ a

Simplification of Fractions

Simplification of fractions is reducing the fractions to the simplest form. Suppose we find the product of two fractions then converting the fraction into the simplest form means simplification of the fraction. This is explained by the example,

Example: Simplify, 3/4 × 5/6

Solution:

(3 × 5)/(4 × 6)

= 15/24

= 5/8 {In simplest form}

Related Resources:

Solved Examples of Multiplication of Fractions

You have been provided with some Examples of Multiplication of Fractions down below:

Example 1: Find the product of the unlike fractions 2/5 and 3/4.

Solution:

2/5 × 3/4

= (2×3)/(5×4)

= 6/20 [Simplifying it further]

= 3/10

Example 2: Find the product of the like fractions 11/5 and 3/5.

Solution:

11/5 × 3/5

= (11 × 3)/(5 × 5)

= 33/25

Example 3: Find the fraction to be multiplied by 5/6 to get a product equal to 3/2.

Solution:

Let the fraction to be multiplied by 5/6 be y.

y × 5/6 = 3/2

⇒ y = 3/2 ÷ 5/6

⇒ y = 3/2 × 6/5

⇒ y = (3 × 6)/( 2 × 5)

⇒ y = 18/10 = 9/5

Thus, 9/5 should be multiplied by 5/6 to get a product of 3/2.

Example 4: Divide the fractions 21/9 and 7/9.

Solution:

21/9 ÷ 7/9

= 21/9 × 9/7

= (21 × 9)/(9 × 7)

= 189/63   [Simplifying it further]

= 3/1 = 3

Practice Problems on Multiplication of Fractions

Here are a few practice problems on multiplication of fractions for you to solve:

1. Multiply the following:

  • 3/5 × 2/7
  • 2/3 × 3/4
  • 3/4 × 10/9
  • 5/12 × 8/25

2. Simplify the following:

  • 3(4/5)× 3(4/5)
  • 3(4/5)× 3(4/5)
  • 4/5 × 15
  • 3(12/13)× 26

3. Simplify the followings:

  • 1/2 × 1/4 ÷ 3/8
  • 3/7 × 4/10 ÷ 5/6
  • 1/9 ÷ 4/5 × 2/3
  • 9/10 ÷ 4/5 ÷ 2/3

Multiplication of Fractions- FAQs

1. What are Fractions?

Fractions are the number that are represented in the form of a numerator divided by a denominator. Such as 3/4, 7/9, 4/3, etc.

2. What is Multiplication of Fractions?

Multiplication of fractions is a mathematical operation used to find the product of two or more fractions. When you multiply fractions, you multiply the numerators (top numbers) together to get the new numerator, and you multiply the denominators (bottom numbers) together to get the new denominator.

2. How to Multiply Fractions?

We can simply multiply two fractions by simply multiplying the numerators and denominators of the fractions individually with each other.

3. How to Multiply Mixed Fractions?

Mixed fractions are multiplied by first changing mixed fractions into improper fractions and then multiplying them normally.

4. How do Multiply Fractions with Whole Fractions?

To multiply a fraction with the whole number we simply multiply the numerator of the fractions with the whole number and then after simplifying we get the required answer.

5. How to Multiply Fractions with Decimals?

A fraction is multiplied with a decimal by first changing the decimal to the fraction and then multiplying the two fractions normally.

6. How to Multiply Fractions Step by Step?

To multiply fractions we can use the following steps:

  • Step 1: Write down the fractions.
  • Step 2: Multiply the numerators to get the new numerator.
  • Step 3: Multiply the denominators to get the new denominator.
  • Step 4: Simplify the fraction if possible.

The result of the simplification is the required product of fractions.

7. Can you Multiply More than Two Fractions?

Yes, we can multiply any number of fractions by multiplying their numerators and denominators with each other.

8. Is there a Shortcut for Multiplication of Fractions?

Sometimes it’s easier to multiply the fraction if we simplify the fraction before multiplying them together i.e., cancel common factors between the numerators and denominators before multiplying.

9. What is the Rule for Signs in Multiplication of Fractions?

In multiplication of fractions, signs follow the usual rules of multiplication. If both fractions are positive or both are negative, the product is positive. If one fraction is positive and the other is negative, the product is negative.



Last Updated : 24 Nov, 2023
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