Multiplying Fractions

Multiplying fractions: When a fraction is multiplied by another fraction the resultant is a fraction or a whole number. We know, a fraction has two parts: numerator and denominator. Thus, when we multiply any two fractions, then numerators and denominators are multiplied, respectively. Example of multiplying fractions is ⅔ x ¼ = (2 x 1)/(3 x 4) = 2/12 = ⅙. Multiplying fractions is not like the addition or subtraction of fractions, where the denominators of both the fractions should be the same. Two fractions can be multiplied easily, even if the denominators are different. While multiplying fractions, it should be kept in mind that the fractions can be in a proper fraction or improper fraction, but they cannot be a mixed fraction.   

Fractions and Types

If a fraction is written in the form of a/b, then a and b are the parts of the fraction, where a is called a numerator and b is called the denominator.

For example,

Suppose ⅖ is a fraction, then 2 is the numerator and 5 is the denominator.

There are three main types of fractions which are proper fractions, improper fractions, and mixed fractions. Below is a brief explanation on each of the types.

Proper fractions

When the numerator of a fraction is less than the denominator.

Example: ½, ⅗, 7/9

Improper fractions

When the numerator is greater than the denominator

Example: 3/2, 5/4, 8/3

Mixed Fraction

The combination of a whole number and a fraction. It is also called a mixed number.

Example: 13/4 =  3 ¼ 

21/5 =  4 ⅕ 

How to Multiply Fractions?

Multiplying fractions is defined as the product of a fraction with a fraction or with an integer or with the variables. The procedure to multiply the fractions are:

• Multiply the numerator with numerator
• Multiply the denominator with the denominator
• Simplify the fractions, if required

For example,

3/2 and ⅓ are the two fractions

The multiplication of two fractions is given by:

(3/2)× (⅓) = [3×1]/[2×3]

(3/2)× (⅓) = 3/6

Now, simplify the fraction, we get ½

Therefore, the multiplication of two fractions 3/2 and ⅓ is ½. 

Simply, we can write the formula for multiplication of fraction as;

If “a/b” and “p/q are the multiplicand and multiplier, then the product of (a/b) and (p/q) is given by “ap/bq”

Multiplication of Fractions
$\begin{array}{l}\frac{�}{�}\times \frac{�}{�}\end{array}$	$\begin{array}{l}\frac{�\times �}{�\times �}\end{array}$	$\begin{array}{l}\frac{��}{��}\end{array}$

Thus, 

The product of Fraction = Product of Numerator/Product of Denominator

Dividing Fractions

When we divide the fraction by another fraction, we convert the latter into reciprocal and then multiply with the former fraction. Learn dividing fractions in detail at BYJU’S.

Example: ⅔ ÷ ¾

Solution: Convert ¾ into its reciprocal, to get 4/3.

Now multiply ⅔ by 4/3

⇒ ⅔ x 4/3 

⇒ (2×4)/(3×3)

⇒ 8/9.

Simplification of Fractions

In multiplying fractions, we generally multiply the top numbers (numerators) with each other, and the bottom numbers (denominators) with each other. To make the fractional multiplication simpler, we can reduce the fraction by cancelling off the common factors. It means that you can cancel out the common factors from one side of the fraction, which is duplicated on the other side of the fractional part.

For example, (4/9) and (3/16) are the two fractions.

(4/9) can be written as (2×2)/(3×3)

(3/16) can be written as (1×3)/(2×2×2×2)

Therefore, 

$\begin{array}{l}\frac{4}{9}\times \frac{3}{16}=\frac{2\times 2}{3\times 3}\times \frac{1\times 3}{2\times 2\times 2\times 2}\end{array}$

Now, cancel out the common factors, we get

$\begin{array}{l}\frac{4}{9}\times \frac{3}{16}=\frac{1}{3}\times \frac{1}{4}\end{array}$

Now, we can multiply numerator with numerator and denominator with denominator.

(4/9) × (3/16)= 1/12

In case, if the fraction has no common factors, then we should directly multiply the numerators and denominators to get the product of the fractions.

Multiplication of Fractions with Fractions

Multiplying Proper Fractions

Multiplication of proper fractions is simple, as we can directly multiply the numerator of one fraction with the other fraction and the denominator of one fraction with the other fraction. If required, we can simplify the resultant fractions into their lowest term.

For example, the multiplication of 5/9 and 2/3.

(5/9)×(2/3)= (5×2)/(9×3) = 10/27.

Multiplying Improper Fractions

We know that in an improper fraction, the numerator is greater than the denominator. While multiplying two improper fractions, it will also result in the improper fraction. For example, multiplying two improper fractions, such as 9/2 and 6/5, results in:

(9/2)×(6/5) = (9/1)×(3/5)= 27/5.

If required, we can convert the improper fraction into a mixed fraction.

Example 1: Solve ⅔×½

Solution: ⅔×½ = 2×1/3×2 = 2/6 = ⅓

Therefore, from the above example, we can observe, by multiplying two fractions we get a fraction number. This is a proper fraction.

Example 2: Multiply ⅘×⅞

Solution: ⅘×⅞ = 4×7 / 5×8 = 28/40

We can further simplify it as;

28/40 = 7/10

If we have to multiply three fractions, then the above formula remains the same.

Example 3: Multiply ¼×⅖×⅛

Solution: Multiplying the given fraction ¼×⅖×⅛, we get

Product = 1×2×1 / 4×5×8

= 2 / 160

= 1 / 80

Multiplying Fractions with Whole numbers

If a whole number or real number is multiplied with a fraction, then it is equal to the real number times the fraction is added.

Example 4: Multiply 5×½

Solution: 5×½ means 5 times of ½

This means, if we add ½ five times, we get the answer.

Therefore,

½ + ½ + ½ + ½ +½ = (1+1+1+1+1)/2 = 5/2 = 2.5

Example 5: Multiply 8/7×10

Solution: Given, 8/7×10

We can write it as 8×10/7

Therefore, 80/7 is the answer. In decimal, it is 11.42.

• Note: If we multiply a mixed fraction to a whole number or real number, then we get a fraction itself.

Example 6: Multiply

$\begin{array}{l}3\frac{1}{5}\end{array}$

×12

Solution: Simplifying the value

$\begin{array}{l}3\frac{1}{5}\end{array}$

we get,

16/5×12 = 16×12 / 5 = 192 / 5 = 38.4

Multiplying Fractions with Variables

Now, consider the fraction is multiplied with a variable, then the results or outcome will be as per the below example.

Example 7: Multiply 5x/2y × 2x/3z

Solution: Given, 5x/2y × 2x/3z

Therefore, we can solve the above-given expression as;

$\begin{array}{l}\frac{5�\times 2�}{2�\times 3�}=\frac{10{�}^2}{6��}\end{array}$

Properties of Fractional Multiplication

The following are the properties of multiplication of fractions:

• If the two given fractional numbers are multiplied in either order, the product of the fraction remains the same. 

           For example, (⅔) × (4/6) = 8/18 = 4/9

           Similarly, (4/6)×(⅔) = 8/18 = 4/9

• If the given fractional number is multiplied by (1/1), the product remains the same fractional number.

           For example, (⅘)× (1/1) = (⅘)

• If a given fractional number is multiplied by 0, the product remains zero.

           For example, (½)× 0 = 0

Multiplying Mixed Fractions

Multiplication of simple fractions is easy, we just need to multiply numerators and denominators respectively. But to multiply mixed numbers or fractions we need to add one more step. 

• First, convert the given mixed fraction into improper fractions
• Now multiply the fractions
• Simplify the answer
• Again convert into mixed numbers

Multiplying Fractions Tricks:

• Usually, we will simplify the fraction after the fractions are multiplied. However, to make the calculations easier, we can simplify the fractions into their lowest terms before multiplication if possible.  For example, (6/2)×(8/10) can be simplified as (3/1)×(4/5). Now, we can multiply the fractions (3/1)×(4/5)= 12/5.
• Simplification can be performed across two fractions also. For example, (10/7)×(21/5) can be simplified into (2/1)×(3/1) and we can multiply fractions, which will result in 6/1.

Solved Examples

1. Multiply 21/3 and 3.

We can write, 

21/3 = 7/3

Now multiply 7/3 and 3

7/3 x 3 = 7

2. Multiply 11/2 and 21/5.

We can write, 

11/2 = 3/2

21/5 = 11/5

Now multiply both the fractions.

3/2 x 11/5

(3 x 11)/(2 x 5)

33/10

Now convert 33/10 into a mixed fraction

33/10 = 33/10

Problems and Solutions

Q.1: Multiply ⅖ and 6/7.

Solution: ⅖ x 6/7 

⇒ (2×6)/(5×7)

⇒ 12/35

Q.2: Multiply ⅓ and 1/10.

Solution: ⅓ x 1/10

⇒ 1/(3 x 10)

⇒ 1/30

Q.3: Find the product of ⅝ and 4/10.

Solution: ⅝ x 4/10

⇒ (5 x 4)/(8 x 10)

⇒ 20/80

⇒ 1/4

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