Multiplying Fractions

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.

How to Multiply 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.

Rules of Multiplying Fractions

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 Same Denominator

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 Different Denominators

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 with Whole Numbers

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 = 334334.

Multiplying Fractions with Mixed Numbers

Mixed numbers or mixed fractions are fractions that consist of a whole number and a proper fraction, like 234234, 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 223223, we need to change this to 8/3. Let us understand this with the help of an example.

Example: Multiply 223223 and 314314.

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 823823.

Multiplication of Improper Fractions

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 21102110.

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.