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Multiplying fractions is the product of two fractions where numerators are multiplied with each other, and so are denominators. Multiplication is the mathematical operation defined by two or more numbers and variables.
- Fraction is the number in the form of p/q where q≠0 and p is the numerator, and q is the denominator.
- Multiplication of fractions is similar to the multiplication of whole numbers.
- It is totally different from the addition and subtraction of fractions, where denominator should be the same.
There are three cases in multiplying fractions, which are multiplying fraction with a whole number, multiplying fraction with a fraction and multiplying fraction with a variable.
- While multiplying the fraction, it should be noted that the fraction can be proper or improper.
- The process becomes easier when denominators are different.
- First, start by multiplying the numerator followed by the multiplication of denominators.
Key Terms: Multiplication, Multiplying Fractions, Addition, Subtraction, Numerator, Denominator, Whole number, Fraction, Variable, Improper fraction, Proper Fraction, Mixed Fraction
Fractions and Types
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Fraction number are numbers which are written in a form of p/q where q≠0. Here, the top number i.e., q is called numerator and q is denominator. Generally, the fraction is used to define a part of the whole thing.
- There are three types of fractions which are:
Proper fraction
The proper fraction is a fraction in which the denominator is bigger than the numerator i.e., p < q. The value of fraction is always less than 1.
Example of Proper FractionExample: 5/6 and 3/13. |
Improper fraction
The improper fraction is a fraction in which the denominator is smaller than the numerator i.e., q < p. The value of fraction is always more than 1.
Example of Improper FractionExample: 12/5 and 99/13. |
Mixed fraction
Mixed fraction is the type of fraction which contains both, a whole number and fraction together. It forms an improper fraction when simplified.
Example of Mixed FractionExample: 5 ½ is example of mixed fraction. |
Read More:
| Chapter Related Concepts | ||
|---|---|---|
| symbols | Arithmetic | closure property |
| Basic Proportionality Theorem | Commutative Property | Number Systems |
How to Multiply Fractions
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Multiplying fraction refers to the process of product of a fraction with a fraction or with an integer or with the variables. The process for multiplying fractions are as follows:
- First multiply both the numerators.
- In next step multiply denominator with denominator.
- Simplify the fraction to the lowest value.
Example of How to Multiply Fractions?Example: 3/16 × 4/5
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Multiplying Fraction
- There are three cases where the fraction is multiplied to whole number, variable and fraction:
Multiplication of a Fraction by Whole number
When a whole number or real number is multiplied by the fraction, the multiplication operator is applied between the number and the numerator of the fraction whereas the denominator remains the same.
- The resultant of multiplication of fraction with whole number is fraction itself.
- Also, it is the real number of times the fraction is added.
Example of Multiplication of a Fraction by Whole numberExample: Fraction (3/5) multiplied by whole number 4. (3/5) × 4 = (3 ×4)/5 (3/5) × 4 = 12/5 |
Multiplication of a Fraction by Fraction
In multiplication of a fraction by fraction, the numerator of the first fraction is multiplied by the numerator of the second fraction. Similarly, the denominator of the two fractions is multiplication with each other.
- Moreover, the resultant can be simplified further to the smallest terms by dividing it if possible.
- Also, the multiplication of two improper fractions gives an improper fraction.
Example of Multiplication of a Fraction by FractionExample 1: Proper fraction (2/7) multiplied by proper fraction (3/5). (2/7) × (3/5) = (2 × 3)/ (7 × 5) (2/7) × (3/5) = 6/21 Example 2: Improper fraction (5/3) multiplied by improper fraction (3/2). (5/3) × (3/2) = (5 × 3)/ (3 × 2) (5/3) × (3/2) = 15/6 (5/3) × (3/2) = 5/2 |
Multiplication of a Fraction by Variable
When a fraction is multiplied with the variable, it results in the term of variables with the fraction as its coefficient. If the fraction contains a variable then the multiplication with the same or different variable is same as the multiplication of variables.
Example of Multiplication of a Fraction by VariableExample 1: Fraction (11/5) multiplied by variable y. (11/5) × y = (11 × y)/5 (11/5) × y = 11y/5 Example 2: Fraction (3x/4) multiplied by variable (y/5). (3x/4) × (y/5) = (3x × y)/ (4 × 5) (3x/4) × (y/5) = 3xy/20 |
Multiplication of Mixed Fractions
In multiplication of mixed fractions, the first step is to convert the mixed fraction into improper fraction then multiply the numerator of the first fraction with other and denominators with each other.
Example of Multiplication of Mixed FractionsExample: The multiplication of 5 ½ and 7 ½. Firstly, converting mixed fractions to improper fractions. 5 ½ is converted to 11/2 and 7 ½ is equal to 15/2 5 ½ × 7 ½ = (11/2) × (15/2) 5 ½ × 7 ½ = (11 × 15)/ (2 × 2) 5 ½ × 7 ½ = (165/4) |
Dividing Fractions
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Division is another mathematical operator used between two variables or numbers. Division of fraction is one of the applications of multiplication of fraction. Two fractions are divided if the first fraction is multiplied with the reciprocal of the second fraction.
- The procedure of dividing fraction is similar for all types of fractions.
Example of Dividing FractionsExample: 5/7 ÷ 2/3 First reciprocal the 2/3 then multiply it with 5/7 5/7 ÷ 2/3 = 5/7 × 3/2 5/7 ÷ 2/3 = (5 × 3)/ (7 × 2) 5/7 ÷ 2/3 = (15/ 14) Simplifying it to its lower term: 5/7 ÷ 2/3 = 15/14 |
Simplification of Fractions
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Simplifying the fractions to its lowest term means that the numerator and denominator cannot be divided further. It can be done by dividing the numerators with the denominator if possible.
- In multiplying fractions, the simplification can be done before or after multiplying both the fractions.
Example of Simplification of FractionsExample: Multiplication of 3/8 and 4/5 3/8 × 4/5 = (3 × 4)/ (8 × 5) 3/8 × 4/5 = (12/ 40) Simplifying it to its lower term: 3/8 × 4/5 = 3/10 Here, the simplification is done after multiplying. Now, the fractions will be simplified before the multiplication 3/8 × 4/5 = (3 × 4)/ (8 × 5) Here, 8 is multiple of 4 so it gives 3/8 × 4/5 = (3 × 1)/ (2 × 5) 3/8 × 4/5 = 3/10 |
Properties of Fractional Multiplication
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Multiplication of fraction has following properties which are as follows:
Multiplication with 0
Any fraction multiplied by 0 gives the only result 0.
Example of Multiplication with 0Example: (11/5) × 0 = 0 |
Multiplication with 1
Whenever a fraction is multiplied by 1, the product is same as the fraction.
Example of Multiplication with 1Example: (3/5) × 1 = (3/5) |
Order of Multiplication
Multiplication of the two fractional numbers in any order gives the same result i.e., multiplying fraction is irrespective of the order.
Example of Order of MultiplicationExample 1: (2/3) × (4/6) = 8/18 = 4/9 Example 2: Similarly, (4/6) × (2/3) = 8/18 = 4/9 |
Things to Remember
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- Multiplying fractions is based on finding a common denominator first.
- It involves multiplication of the numerators followed by denominators and simplifies if necessary.
- The denominator does not need to be the same for multiplying fractions.
- The product of fractions can be a fraction or a real number.
- The value of the product of an improper fraction is an improper fraction.
- Simplification of the product can be done before multiplication.
Read More:
| Class 7 Mathematics Related Concepts | ||
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| Polynomials | Quadratic Equations Formula | Degree of polynomial |
| Descending Order | Area of Sector | Number Lines |
Sample Questions
Ques. Explain “the value of the product of two proper fractions is smaller than each of the two fractions”. (5 Marks)
Ans. Proper fraction is the type of fraction in which the numerator is smaller than the denominator. This means that the value of fraction is always less than 1. When two proper fractions are multiplied gives a proper fraction. One fraction is multiplied with the other whose value is less than 1 gives the value smaller than both the fractions.
For example, two proper fractions 3/5 = 0.6 and ½ = 0.5
The product of 3/5 and ½ is
(3/5) × (1/2) = (3×1)/ (5×2)
(3/5) × (1/2) = 3/10 = 0.3
Hence, 3/10 < 3/5 and 3/10 < ½
Therefore, the value of the product of two proper fraction is smaller than each of the two fractions.
Ques. What happens when two improper fractions are multiplied? (3 Marks)
Ans. Improper fraction is the type of fraction in which the numerator is greater than the denominator. This means that the value of fraction is always more than 1. When two improper fractions are multiplied, the product is an improper fraction. Also, the value of the product of two improper fractions is more than each of the two fractions.
For example, two improper fractions 7/5 = 1.4 and 3/2 = 1.5
The product of 7/5 and 3/2 is
(7/5) × (3/2) = (7×3)/ (5×2)
(7/5) × (3/2) = 21/10 = 2.1
Hence, 7/5 < 21/10 and 3/2 < 21/10
Therefore, when two improper fractions are multiplied, they give the value greater than each of the fraction.
Ques. Find the value of: (3 Marks)
a) 2 × 3/7
b) 9/7 × 6
c) 3/11 × 6
Ans. Multiplying fraction with a real number is simply multiplying numerator with number and leaving the denominator same.
- 2 × 3/7
2 × (3/7) = (2 × 3)/7
2 × (3/7) = 6/7
- 9/7 × 6
9/7 × 6 = (9 × 6)/7
9/7 × 6 = 54/7
- 13/11 × 6
13/11 × 6 = (13 × 6)/11
13/11 × 6 = 78/11
Ques. Multiply the fraction with fraction: (2 Marks)
a) 3/8 × 4/5
b) 2/3 × 1/5
Ans. Multiplication of the fraction with the fraction is the done by multiplying the numerator of the one fraction with other and denominator of one with denominator of other.
- 3/8 × 4/5
3/8 × 4/5 = (3 × 4)/ (8 × 5)
3/8 × 4/5 = (12/ 40)
Simplifying it to its lower term:
3/8 × 4/5 = 3/10
- 2/3 × 1/5
2/3 × 1/5 = (2 × 1)/ (3 × 5)
2/3 × 1/5 = 2/ 15
Ques. Simplify the following: (2 Marks)
a) 4x2/5 × 3x/2
b) 2yz/7 × y3z/2
Ans. Multiplication of fractions with variable is same as that of fraction with fraction.
- 4x2/5 × 3x/2
(4x2/5) × (3x/2) = (4x2 × 3x)/ (5 × 2)
(4x2/5) × (3x/2) = (4× 3) x2+1/ (5 × 2)
(4x2/5) × (3x/2) = 12x3/ 10
- 2yz/7 × y3z/2
(2yz/7) × (y3z/2) = (2yz × y3z)/ (7 × 2)
(2yz/7) × (y3z/2) = (2 × 1) y3+1 z1+1/ (7 × 2)
(2yz/7) × (y3z/2) = 2y4z2/ 14
Ques. What are characteristics of fractional multiplication? (3 Marks)
Ans. Multiplication of fraction has following properties:
- Multiplication with 0: Any fraction multiplied by 0 gives the only result 0.
For example, (11/5) × 0 = 0
- Multiplication with 1: Whenever a fraction is multiplied by 1, the product is same as the fraction.
For example, (3/5) × 1 = (3/5)
- Order of multiplication: Multiplication of the two fractional number in any order gives the same result i.e., multiplying fraction is irrespective of the order.
For example, (2/3) × (4/6) = 8/18 = 4/9
Similarly, (4/6) × (2/3) = 8/18 = 4/9
Ques. What are fractions? (2 Marks)
Ans. A number which is written in a form of p/q where q≠0 is known as fraction number. Here, the top number i.e., q is called numerator and q is denominator. Generally, the fraction is used to define a part of the whole thing. There are three types of fractions which are named as proper fraction, improper fraction and mixed fraction. Examples of fraction are 4/5, 2/5 and 8/3.
Ques. Do denominator need to be same for multiplication? (3 Marks)
Ans. Fraction is consisted of two parts: numerator and denominator. For addition and subtraction of the fraction, the denominator of the fractions should be same but this is not the case with multiplication as well as division. Multiplying fraction with the fraction does not require same denominator because any two fractions can be multiplied by multiplying the numerators together and so does the denominator.
Ques. How to divide the fractions? (3 Marks)
Ans. Division is another mathematical operator used between two variables or numbers. Division of fraction is one of the applications of multiplication of fraction. To divide two fractions, an individual need to reciprocal the second fraction and then multiply it with the first fraction.
For example, 3/8 ÷ 5/4
First reciprocal the 5/4 then multiply it with 3/8
3/8 ÷ 5/4 = 3/8 × 4/5
3/8 ÷ 4/5 = (3 × 4)/ (8 × 5)
3/8 ÷ 4/5 = (12/ 40)
Simplifying it to its lower term:
3/8 ÷ 4/5 = 3/10
Ques. There are total of 48 students in a class and 2/3 of them are girls. Find out how many boys are there? (3 Marks)
Ans. Total number of students = 48
Number of boys + Number of girls = 48
Also, number of girls = 2/3 of total students
So, Number of girls = 2/3 × 48
Number of girls = (2 × 48)/3
Number of girls = 96/3
Number of girls = 32
Therefore, Number of boys = Total students – number of girls
Number of boys = 48 – 32
Number of boys = 16
So, the class of 48 students has 32 girls and 16 boys.
Ques. Vidya and Pratap went for a picnic. Their mother gave them a water bottle that contained 5 litres of water. Vidya consumed 2/5 of the water. Pratap consumed the remaining water. How much water did Vidya drink? What fraction of the total quantity of water did Pratap drink? (5 Marks)
Ans. Litre of water bottle contains = 50
Vidya consumed water = 2/5 of the water in bottle
Vidya consumed water = 2/5 of 50 litre
Vidya consumed water = 2/5 × 50 litre
Vidya consumed water = (2 × 50)/ 5 litre
Vidya consumed water = 100/5 litre
Vidya consumed water = 20 litre
Fraction of water Pratap drink = 1 – Fraction of water Vidya Drink
Fraction of water Pratap drink = 1- 2/5
Fraction of water Pratap drink = (5-2)/5
Fraction of water Pratap drink = 3/5
Ques. Find the product of fractions: 1/4 × 5/12? (2 Marks)
Ans. For multiplying fractions with different denominators, as given in 1/4 × 5/12, we start by multiplying the numerators: 1 × 5 = 5. After this, we multiply the denominators: 4 × 12 = 48. This can be written as: (1 × 5)/(4 × 8) = 5/48
Ques. Find the product of fractions: 7/4 × 7/2 × 7/9? (3 Marks)
Ans. For multiplying three fractions, first we will multiply all three numerators. Next we will multiply all three denominators. Then, simplify the final result.
⇒ 7/4 × 7/2 × 7/9
⇒ (7 × 7 × 7)/(4 × 2 × 9)
⇒ 343/72
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