Multiplying Fractions

Multiplying fractions starts with the multiplication of the given numerators, followed by the multiplication of the denominators. Then, the resultant fraction is simplified further and reduced to its lowest terms, if needed. Learn all about multiplying fractions in this article.

1.	How to Multiply Fractions?
2.	Rules of Multiplying Fractions
3.	Multiplying Fractions with Same Denominator
4.	Multiplying Fractions with Different Denominators
5.	Multiplying Fractions with Whole Numbers
6.	Multiplying Fractions with Mixed Numbers
7.	Multiplication of Improper Fractions
8.	FAQs on Multiplying Fractions

The multiplication of fractions is not like the addition or subtraction of fractions, where the denominator should be the same. Here, any two fractions with different denominators can easily be multiplied. The only thing to be kept in mind is that the fractions should not be in the mixed form, they should either be proper fractions or improper fractions. Let us learn how to multiply fractions through the following steps:

• Step 1: Multiply the numerators.
• Step 2: Multiply the denominators.
• Step 3: Reduce the resultant fraction to its lowest terms.

For example, let us multiply the following fractions: 1/3 × 3/5. We start by multiplying the numerators: 1 × 3 = 3, then, the denominators: 3 × 5 = 15. This can be written as: (1 × 3)/(3 × 5) = 3/15. Now, reduce this value to its lowest form. 3 is the greatest common factor of 3 and 15, so, divide both 3 and 15 by 3 to simplify the fraction. Therefore, 1/3 × 3/5 = 1/5.

Multiplication of Fractions Using Visual Models

Now, let us see the visual representation for the multiplication of fractions. Visualizing the multiplication of fractions using fractional squares is a very interesting method to understand the concept. Let's multiply these two fractions: 1/4 × 1/3 using the visual model. Draw a rectangle and divide its length into 4 equal parts. Each column will represent 1/4 of the whole rectangle. Now, divide its width into 3 equal parts such that each portion will represent 1/3. Now, we just have to look for the portion that is common to both 1/4 and 1/3, which is 1/12^th of the whole rectangle (marked in light orange color in the figure below).

In the figure given above, we can clearly see that 1/12 of the rectangle is common to both 1/4 and 1/3. It is an overlapping portion. Thus, 1/4 × 1/3 = 1/12. Now that you have insight into multiplying fractions, let us explore this topic further.

While multiplying fractions, the following rules have to be kept in mind:

• Rule 1: The first rule is to convert mixed fractions to improper fractions if any. Then, multiply the numerators of the given fractions.
• Rule 2: Multiply the denominators separately.
• Rule 3: Simplify the value obtained to its lowest term.

These three rules can be applied to any two fractions to find their product. Now, let us learn the individual cases of multiplying fractions with different types of fractions.

Multiplying fractions with the same denominator does not change the rule of multiplication of fractions. Fractions that have the same denominator are termed like fractions. Although addition and subtraction of like fractions are different from the addition and subtraction of unlike fractions, in the case of multiplication and division the method remains the same. We multiply the numerators, then the denominators, and then the fraction is reduced to its lowest terms.

Example: Multiply 1/3 × 5/3

Solution: We can multiply these fractions using the following steps.

• Step 1: Multiply the numerators, 1 × 5 = 5.
• Step 2: Multiply the denominators, 3 × 3 = 9.
• Step 3: The product that we get is 5/9. This cannot be reduced any further, therefore 5/9 is the answer.

Multiplying fractions with unlike denominators is exactly the same as the multiplication of like fractions. Let us understand this with an example.

Example: Multiply 4/12 × 16/24.

We can multiply these fractions using the following steps:

• Step 1: Multiply the numerators, 4 × 16 = 64.
• Step 2: Multiply the denominators, 12 × 24 = 288.
• Step 3: The product that we get is 64/288. This can be reduced to 2/9. Therefore, 2/9 is the answer.

Alternative Method

The same fractions can be multiplied using another method in which we simplify the fractions among themselves and then multiply the numerators, then the denominators to get the final product.

Example: Multiply 4/12 × 16/24.

Let us multiply the given fractions using the following steps:

• Step 1: We will simplify the given fractions among themselves. In other words, these fractions can be reduced to 1/3 × 2/3.
• Step 2: Let us multiply the numerators, 1 × 2 = 2.
• Step 3: Now, let us multiply the denominators, 3 × 3 = 9.
• Step 4: Therefore, the product that we get is 2/9.

Multiplying fractions by whole numbers is an easy concept. As we know that multiplication is the repeated addition of the same number, this fact can be applied to fractions as well.

Multiplying Fractions with Whole Numbers Visual Model

Let us consider this example: 4 × 2/3. This means 2/3 is added 4 times. Let us represent this example using a visual model. Four times two-thirds is represented as:

Steps of Multiplying Fractions with Whole Numbers

In order to multiply fractions with whole numbers, we use the simple rule of multiplying the numerators, then multiplying the denominators, and then reducing them to the lowest terms. However, in the case of whole numbers, we write them in the fractional form by placing '1' in the denominator. Let us understand this with an example.

Example: Multiply: 5 × 3/4.

Let us use the following steps to multiply the given fraction with a whole number.

• Step 1: Here, 5 is a whole number that can be written as 5/1, and then it can be multiplied as we multiply regular fractions.
• Step 2: This means, we need to multiply 5/1 × 3/4.
• Step 3: Multiply the numerators, 5 × 3 = 15.
• Step 4: Multiply the denominators, 1 × 4 = 4.
• Step 5: The resultant product is 15/4 which cannot be reduced further.
• Step 6: Since 15/4 is an improper fraction, we will change it to a mixed fraction, 15/4 = \(3\frac{3}{4}\).

Mixed numbers or mixed fractions are fractions that consist of a whole number and a proper fraction, like \(2\frac{3}{4}\), where 2 is the whole number and 3/4 is the proper fraction. For multiplying mixed fractions, we need to change the mixed fractions into an improper fraction before multiplying. For example, if the number is \(2\frac{2}{3}\), we need to change this to 8/3. Let us understand this with the help of an example.

Example: Multiply \(2\frac{2}{3}\) and \(3\frac{1}{4}\).

The following steps can be used to multiply fractions with mixed numbers.

• Step 1: Change the given mixed fractions to improper fractions, i.e. (8/3) × (13/4).
• Step 2: Multiply the numerators of the improper fractions, and then multiply the denominators. This will give 104/12.
• Step 3: Now, reduce the resultant fraction to its lowest terms, which will make it 26/3.
• Step 4: Further, convert the answer back to a mixed fraction which will be \(8\frac{2}{3}\).

Now let us understand the multiplication of improper fractions. We already know that an improper fraction is one where the numerator is bigger than the denominator. When multiplying two improper fractions, we frequently end up with an improper fraction. For example, to multiply 3/2 × 7/5 which are two improper fractions, we need to take the following steps:

• Step 1: Multiply the numerators and denominators. (3 × 7)/(2 × 5) = 21/10.
• Step 2: The fraction 21/10 cannot be reduced further to its lowest terms.
• Step 3: Hence, the answer is 21/10 which can be written as \(2\frac{1}{10}\).

Tips and Tricks of Multiplying Fractions:

Here are a few important tips and tricks which are helpful in the multiplication of fractions.

• Generally, students simplify a fraction after multiplication. However, to make calculations easier, check if the two fractions to be multiplied are already in their lowest forms. If not, first simplify them and then multiply. For example, 4/12 × 5/13 will be difficult to multiply directly.
• Now, if we simplify the fraction first, we get:1/3 × 5/13 = 5/39.
• Simplification can also be done across two fractions. If there is a common factor between the numerator of one of the fractions and the denominator of the other fraction, you can simplify them and proceed. For example, 5/28 × 7/9 can be simplified to 5/4 × 1/9 before multiplying.

► Related Topics

• Multiplying Fractions Calculator
• Reciprocal of Fractions
• Multiplying Decimals
• Multiplication and Division of Integers

 

What is Multiplying Fractions?

Multiplying fractions means finding the product of two or more fractions. The method that is used for the multiplication of fractions is different from the addition and subtraction of fractions. To multiply any two fractions, we follow the steps given below. Let us multiply 7/8 × 2/6 to understand the steps.

• Multiply the numerators. So, 7 × 2 = 14.
• Multiply the denominators. This means, 8 × 6 = 48.
• The resultant fraction is 14/48. Simplify the resultant fraction to its lowest terms. Simplifying the fraction 14/48 gives us 7/24. Therefore, the answer is 7/24.

What are the Rules for Multiplying Fractions?

There are three simple rules for multiplying fractions. First, multiply the numerators, and then the denominators of both the fractions to obtain the resultant fraction. Then, we need to simplify the obtained fraction to get the final answer. This can be understood by a simple example → 2/6 × 4/7 = (2 × 4)/(6 × 7) = 8/42 = 4/21.

How to Multiply Fractions with Mixed Numbers?

The following steps can be used for the multiplication of mixed fractions. Let us multiply 1/4 × \(3\frac{1}{2}\).

• Change the mixed fraction to an improper fraction. Here, \(3\frac{1}{2}\) will become 7/2. So, now we need to multiply 1/4 × 7/2.
• Multiply the numerators and then the denominators. This means, (1 × 7)/(4 × 2) = 7/8.
• Ensure that the answer is in the lowest terms. Since 7/8 cannot be reduced any further, 7/8 is the answer.

How to Multiply Fractions with Whole Numbers?

To understand the multiplication of a fraction with a whole number, we can take a simple numerical example, 2/7 × 3. Start by rewriting the whole number (3 in this example) as a fraction, 3/1. Now, we can apply the steps that we use to multiply fractions. This means, 2/7 × 3/1 = (2 × 3)/(7 × 1) = 6/7.

How to Multiply Fractions with Same Denominators?

Multiplying fractions with same denominators is the same as multiplying other regular fractions. Let us understand this with an example. Let us multiply 4/5 × 3/5. We multiply the numerators, that is, 4 × 3 = 12. Then, we multiply the denominators, that is, 5 × 5 = 25. This will give us the product as 12/25. Since this cannot be reduced any further, 12/25 will be the answer.

How to Multiply Fractions with Different Denominators?

Multiplying fractions with different denominators does not change the rule for the multiplication of fractions. Let us understand this with an example. Multiply 2/6 × 3/4. We can multiply these fractions using the following steps:

• Multiply the numerators, 2 × 3 = 6.
• Multiply the denominators, 6 × 4 = 24.
• The product that we get is 6/24. This can be reduced to 1/4, therefore 1/4 is the answer.

How to Multiply a Fraction by a Fraction?

The multiplication between two fractions is the simplest form of arithmetic operations between two fractions. The numerators of both the fractions are to be multiplied first, followed by the multiplication of the denominators. Then, the resultant fraction is simplified to its lowest terms, if needed.

How is Multiplying Fractions Different from Adding Fractions?

The addition of fractions is different from the multiplication of fractions. In multiplication, first, the numerators of the two fractions are multiplied, then the denominators are multiplied to obtain the resultant fraction. However, in the process of addition of fractions, we first need to make the denominators of both the fractions equal and then we add the numerators to obtain the resultant fraction. In addition or subtraction of fractions, we do not add or subtract the denominators separately.

How to Multiply Fractions with Decimals?

In order to multiply fractions with decimals, we convert the decimal number to a fraction, and then we use the same rules of multiplication of fractions. For example, let us multiply 5/7 × 0.6.

• Here, we will convert 0.6 to its fraction form which will make it 6/10.
• Now, we will multiply 5/7 × 6/10 in a regular way.
• We will multiply the numerators, 5 × 6 = 30.
• We will multiply the denominators, 7 × 10 = 70.
• Thus, the resultant fraction will be 30/70.
• On simplifying the resultant fraction, we will get the product as 3/7.

How to Teach Multiplication of Fractions?

The multiplication of fractions can be taught in the same way as the multiplication of whole numbers. The important aspect, before the multiplication of fractions, is to convert the mixed fraction into an improper fraction. After this step, we multiply the numerators of both the fractions and then the denominators of both the fractions to obtain the resultant fraction. The following ways can be used to teach multiplying fractions:

• Use visual models as much as possible to introduce the concept. Make learners understand the use and process of multiplication of fractions.
• Use multiplying fractions worksheets after teaching a concept.