Multiplying Fractions

A part of a whole is referred to as a fraction, and it has two components: Numerator and denominator. The numerator is the top number of the fraction and shows the number of parts we have. The denominator is the bottom number and indicates the total number of equal parts. A fraction is represented in the form:

Numerator / Denominator

For instance, 2/6 is a fraction, where 2 and 6 is the numerator and denominator respectively.

Unlike addition and subtraction, a common denominator is not required when multiplying fractions. Even if the fractions have different denominators, multiplication can still be performed. The main requirement is that the fractions should be either proper or improper, not in mixed form. To multiply fractions, follow these steps: 

Step 1: Multiply the numerators together.

Step 2: Multiply the denominators together.

Step 3: Simplify the fractions in the lowest form if necessary. 

There are certain rules that we need to follow while multiplying fractions. Take a look at them down below:

Rule 1: If we have a mixed fraction, convert it into an improper fraction. Then multiply the numerators. 

Rule 2: Multiply the denominators together.

Rule 3: Simplify the obtained fraction to its lowest form. 

Rule 4: Another method to multiply fractions is to simplify the fractions before multiplying. It can be performed by canceling out common factors between the numerators and denominators. After that, multiply the numerators together, and then multiply the denominators together.  

Fractions that have the same denominator are also known as like fractions. The rules for addition and subtraction of like fractions differ from those of unlike fractions. However, in the case of multiplying like and unlike fractions, the rule remains the same. To multiply fractions, multiply the numerators together, and then the denominators.  After that, the obtained fraction is simplified to its lowest terms.

For example, multiply 3/2 × 1/2

Here, we will multiply the numerators together. 3 × 1 = 3 

Then, the denominators, 2 × 2 = 4

The product we get is 3/4 

The answer cannot be simplified further, so the result is 3/4 

The method for multiplying fractions with different denominators is the same as for multiplication of fractions with the same denominators. Multiply the numerators together, then the denominators, and simplify the fraction to its lowest term.

For instance, multiply 5/4 × 1/3

First, we can multiply the numerators together, 5 × 1 = 5

Then, the denominators, 4 × 3 = 12

The product we get is 5/12. 

This cannot be simplified further, therefore, 5/12 is the answer. 

There is another method that we can use to multiply fractions in which first, we simplify the fractions themselves then multiply the numerators, and then the denominators. We can understand this method by an example.

For example, multiply 4/12 × 16/24 

Here we will simplify the fractions among themselves.

We can reduce 4/12 to 1/3 
1/3 = (4 ÷ 4/12 ÷ 4)

We can reduce 16/24 to 2/3 
2/3  = (16 ÷ 8/24 ÷ 8)

Now let us multiply the numerators together, 1 × 2 = 2
Then, the denominators, 3 × 3 = 9
Hence, the product is 2/9

Multiplying fractions with whole numbers is a simple process. We must convert the whole number into a fraction by placing 1 as its denominator. Then multiply the numerators, and denominators separately. After obtaining the product, simplify it if necessary. This is represented as, 

a × b/c = a/1 × b/c = (ab)/c

For instance, multiply 4 × 1/3 

Step 1: Convert the whole number (4) into a fraction. Therefore, it becomes 4/1

Step 2: Multiply the numerators together. 
 4 × 1 = 4

Step 3: Multiply the denominators together. 
1 × 3 = 3 

Step 4: The product is 4/3

Step 5: Since 4/3 is an improper fraction, we change it into mixed fraction. In this case, the numerator is greater. 

So, dividing 4 by 3, we get 1 as the quotient and 1 as the remainder. 

So, write the fraction as 1 1/3

Multiplication of mixed numbers involves converting the mixed fractions into improper fractions and multiplying the numerators and denominators separately. Mixed fractions contain a whole number and a proper fraction. Before multiplication, we need to change it to improper fractions. Take a look at this example.

Multiply 3 3/4 and 4 1/5

To convert mixed fractions into improper fractions, we can use the formula:

Improper fraction = (Whole number × Denominator) + Numerator

For 3 3/4: 

Improper fraction = (3 × 4) + 3/4 = 12 + 3/4 = 1/54

For 4 1/5:

Improper fraction = (4 × 5) + 1/5 = 20 + 1/5 = 2/15

So, we get 15/4 × 21/5

Now we can multiply the numerators:

15 × 21 = 315

Denominators:  4 × 5 = 20 

The obtained fraction is: 

315/20

To simplify the fraction, we need to find the greatest common factor (GCF) of 315 and 20. 

To find this, first, we need to identify the prime factorization of both numbers.  

315 = 3² × 5 × 7
20 = 2² × 5 

5 is the only common factor of 315 and 20. 

Hence, 5 is the GCF. 

Now we can divide 315 and 20 by its GCF.
315 ÷ 5 /20 ÷ 5  = 63/4

63/4 is an improper fraction, we must convert it into a mixed number. 

63 ÷ 4 gives 15 as quotient and 3 as remainder. 

So, 
63/4 = 15 3/4

The product is 15 3/4

We refer to a fraction as improper when its numerator is greater than the denominator. Multiplying two improper fractions often gives an improper fraction as a result.

For example, multiply 5/2 and 9/7 

Now, we can multiply the numerators together, 5 × 9 = 45

The denominators: 2 × 7 = 14 

The fraction 45/14 cannot be simplified further. 

Hence, the product is 4514 which can be written as 3 3/14

As always, there are certain tips and tricks that can be used to our advantage while solving problems involving the multiplication of fractions. Some of them are mentioned below:

• Simplify fractions before multiplying them. Instead of multiplying first and then simplifying, check if fractions can be simplified beforehand. For example, when we multiply 154 × 215 before simplifying, it will become complex to solve. 

• If it is possible, we can simplify two fractions. If the numerator of one fraction and the denominator of another share a common factor, we can simplify before multiplying. For example, we can simplify 528 and 79 to 54 and 19