In Class 7 Mathematics Chapter 2.3, students learn about the distributive property of multiplication over addition and subtraction. This property helps us simplify arithmetic expressions and perform calculations more easily.
Instead of multiplying a number by a complete sum or difference, we can multiply it separately by each number inside the brackets and then add or subtract the results.
For example:
5 × (4 + 3) = 5 × 4 + 5 × 3
= 20 + 15
= 35
Both methods give the same answer. Let us understand this property with simple examples, diagrams and exercise solutions.
The distributive property explains how multiplication works with addition and subtraction inside brackets.
When a number is multiplied by a sum, we can multiply that number by each term separately and then add the products.
a × (b + c) = a × b + a × c
For example:
4 × (6 + 3)
= 4 × 6 + 4 × 3
= 24 + 12
= 36
Check the original expression:
4 × (6 + 3) = 4 × 9 = 36
Both methods give the same result.
When a number is multiplied by a difference, multiply it by both numbers separately and subtract the second product from the first.
a × (b − c) = a × b − a × c
For example:
5 × (10 − 3)
= 5 × 10 − 5 × 3
= 50 − 15
= 35
Check:
5 × (10 − 3) = 5 × 7 = 35
Both methods give the same answer.
Let us understand why the distributive property works.
Consider:
3 × (4 + 2)
This means three groups of (4 + 2).
We can write it as:
(4 + 2) + (4 + 2) + (4 + 2)
Now arrange the numbers:
= 4 + 2 + 4 + 2 + 4 + 2
Group the 4s and 2s:
= 4 + 4 + 4 + 2 + 2 + 2
= 3 × 4 + 3 × 2
= 12 + 6
= 18
Therefore:
3 × (4 + 2) = 3 × 4 + 3 × 2 = 18
The distributive property is especially useful when multiplying larger numbers.
Solution:
We can write 13 as 10 + 3.
13 × 98 = (10 + 3) × 98
Using the distributive property:
= 10 × 98 + 3 × 98
= 980 + 294
= 1,274
Answer: 1,274
Solution:
Both terms contain multiplication by 10.
We can take 10 as a common factor.
14 × 10 − 6 × 10
= (14 − 6) × 10
= 8 × 10
= 80
Answer: 80
We can break a number into convenient parts to simplify multiplication.
Solution:
97 is close to 100.
Therefore, write:
97 = 100 − 3
Now:
97 × 25 = (100 − 3) × 25
Using the distributive property:
= 100 × 25 − 3 × 25
= 2,500 − 75
= 2,425
Answer: 2,425
Solution:
We know:
53 × 18 = 954
Now write 63 as 53 + 10.
63 × 18 = (53 + 10) × 18
Using the distributive property:
= 53 × 18 + 10 × 18
= 954 + 180
= 1,134
Answer: 1,134
Solution:
Write 95 as 100 − 5.
95 × 8 = (100 − 5) × 8
= 100 × 8 − 5 × 8
= 800 − 40
= 760
Answer: 760
Solution:
Write 104 as 100 + 4.
104 × 15 = (100 + 4) × 15
= 100 × 15 + 4 × 15
= 1,500 + 60
= 1,560
Answer: 1,560
Solution:
Write 49 as 50 − 1.
49 × 50 = (50 − 1) × 50
= 50 × 50 − 1 × 50
= 2,500 − 50
= 2,450
Answer: 2,450
| Property | Formula | Example |
|---|---|---|
| Multiplication over addition | a × (b + c) = a × b + a × c | 3 × (4 + 5) = 3 × 4 + 3 × 5 |
| Multiplication over subtraction | a × (b − c) = a × b − a × c | 4 × (9 − 2) = 4 × 9 − 4 × 2 |
| Taking a common factor | a × b + a × c = a × (b + c) | 5 × 7 + 5 × 3 = 5 × (7 + 3) |
| Taking a common factor from subtraction | a × b − a × c = a × (b − c) | 8 × 9 − 8 × 4 = 8 × (9 − 4) |
Solution:
Using the distributive property:
3 × (6 + 7) = 3 × 6 + 3 × 7
= 18 + 21 = 39
Answer: 3 × 6 + 3 × 7
Solution:
Multiply both terms inside the brackets by 4.
(8 + 3) × 4 = 8 × 4 + 3 × 4
= 32 + 12 = 44
Answer: 8 × 4 + 3 × 4
Solution:
Multiply 3 by both numbers inside the brackets.
3 × (5 + 8) = 3 × 5 + 3 × 8
Answer: 8
Solution:
Both terms are multiplied by 4.
(9 + 2) × 4 = 9 × 4 + 2 × 4
Answer: 4
Solution:
One valid completion is:
3 × (5 + 4) = 3 × 5 + 3 × 4
Answer: 5, 5 and 3 × 4
Solution:
The first missing number is 13. The second term must be 6 × 4.
(13 + 6) × 4 = 13 × 4 + 6 × 4
Answer: 13 and 6 × 4
Solution:
Compare both terms on the right side.
3 × (5 + 2) = 3 × 5 + 3 × 2
Answer: 5 and 2
Solution:
Both terms on the right are multiplied by 4.
(2 + 3) × 4 = 2 × 4 + 3 × 4
Answer: 2, 3 and 4
Solution:
Using the distributive property over subtraction:
5 × (9 − 2) = 5 × 9 − 5 × 2
Answer: 2
Solution:
(5 − 2) × 7 = 5 × 7 − 2 × 7
Answer: 7
Solution:
Because the expression contains subtraction, the sign between the products is minus.
5 × (8 − 3) = 5 × 8 − 5 × 3
Answer: Minus (−) and 3
Solution:
(8 − 3) × 7 = 8 × 7 − 3 × 7
Answer: Minus (−)
Solution:
One valid completion is:
5 × (12 − 2) = 12 × 5 − 5 × 2
Answer: 2, 12 and 2
Solution:
Compare the two terms on the right side.
(15 − 6) × 7 = 15 × 7 − 6 × 7
Answer: 6 and 15
Solution:
5 × (9 − 4) = 5 × 9 − 5 × 4
Answer: 9 and 4
Solution:
(17 − 9) × 7 = 17 × 7 − 9 × 7
Answer: 17, 9 and 7
Solution:
8 − 3 = 5, which is positive.
3 − 8 = −5, which is negative.
Multiplying a positive number by 29 gives a positive result. Multiplying a negative number by 29 gives a negative result.
Therefore:
(8 − 3) × 29 > (3 − 8) × 29
Solution:
The left side has 15 added to 9 × 18.
The right side multiplies both 15 and 9 by 18.
The right side is greater because it includes an additional 15 × 17 compared with the left side.
Answer: <
Solution:
Using the distributive property:
23 × (17 − 9) = 23 × 17 − 23 × 9
The expression on the right uses addition instead of subtraction.
Therefore, the left side is smaller.
Answer: <
Solution:
By the distributive property:
(34 − 28) × 42 = 34 × 42 − 28 × 42
Answer: =
The textbook gives one example:
2 × (1 + 6) = 14
Find four other possibilities.
Solution:
Here are four valid examples:
(a) 1 × (9 + 5) = 1 × 14 = 14
(b) 2 × (3 + 4) = 2 × 7 = 14
(c) 7 × (1 + 1) = 7 × 2 = 14
(d) 14 × (1 + 0) = 14 × 1 = 14
Answer: All four expressions give 14.
(a) First picture
The picture contains five yellow squares with the number 4 and four blue circles with the number 8.
Method 1: Group equal numbers.
= 5 × 4 + 4 × 8
= 20 + 32
= 52
Method 2: Add the rows.
= (4 + 8 + 4) + (8 + 4 + 8) + (4 + 8 + 4)
= 16 + 20 + 16
= 52
Answer: 52
(b) Second picture
The picture contains eight circles showing 5 and eight circles showing 6.
Method 1: Group equal numbers.
= 8 × 5 + 8 × 6
= 40 + 48
= 88
Method 2: Group the four rows.
= 4 × (5 + 6 + 6 + 5)
= 4 × 22
= 88
Answer: 88
Rahim supplies 9 kg of mangoes every day and Shyam supplies 11 kg every day to a market. The market operates seven days a week. Find the total mango supply in one week.
Solution:
Daily supply = 9 + 11 = 20 kg
Weekly supply = 7 × (9 + 11)
Using the distributive property:
= 7 × 9 + 7 × 11
= 63 + 77
= 140 kg
Answer: 140 kg
Binu earns ₹20,000 every month. She spends ₹5,000 on rent, ₹5,000 on food and ₹2,000 on other expenses. Find her savings at the end of one year.
Solution:
Monthly income = ₹20,000
Total monthly expenses:
= 5,000 + 5,000 + 2,000
= ₹12,000
Monthly savings:
= 20,000 − 12,000
= ₹8,000
Annual savings:
= 12 × 8,000
= ₹96,000
Answer: ₹96,000
A snail climbs 3 cm during the day and slips down 2 cm during the night. The post is 10 cm high. How many days will the snail take to reach the top?
Solution:
During one complete day and night, the snail gains:
3 − 2 = 1 cm
However, on the final day, the snail reaches the top before slipping back.
After seven nights, the snail has reached 7 cm.
On the eighth day, it climbs 3 cm:
7 + 3 = 10 cm
Answer: 8 days
Try these original questions to practise the distributive property.
Question 1:
Question 2:
Question 3:
The distributive property states that multiplying a number by a sum or difference is the same as multiplying it by each term separately and then adding or subtracting the products.
The formula is a × (b + c) = a × b + a × c.
The formula is a × (b − c) = a × b − a × c.
It allows us to split a number into convenient parts. For example, 99 × 12 can be calculated as (100 − 1) × 12.
A common factor is a number or quantity that multiplies each term. For example, 5 is the common factor in 5 × 7 + 5 × 3.
Yes. It can help calculate shopping totals, quantities, savings, repeated groups and large multiplication problems.
The distributive property is an important concept in Class 7 Mathematics. It helps students simplify expressions, calculate large products more efficiently and understand the relationship between multiplication, addition and subtraction.
Practise the solved examples and Figure It Out questions to improve your understanding. Once you are comfortable with these rules, try the additional questions without looking at the answers.
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