1144 LearnersLast updated on June 6th, 2025
Distributive Property
As the name says, the distributive property is about distributing the values in an operation. The distributive property, also known as distributive law, is applicable for multiplication, addition, division, and subtraction. In this topic, we are going to learn about the distributive property and how it is used in various operations.
What is the Distributive Property?
Distributive property is the way the number is distributed throughout the operations. It is also known as the distributive law of multiplication over addition and subtraction. It states that A (B + C) = AB + AC, that means the product of multiplying a number with the sum of two more numbers is the same as the sum of multiplying the addends separately.
For example, 5 × (2 +3) = (5 × 2) + (5 × 3) = 25
5 × (7 - 3) = (5 × 7) - (5 × 3) = 20
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How to Use the Distributive Property?
The distributive property formula can be expressed as a (b × c) = ab + ac. Now let’s learn about how to use distributive property.
Step 1: Identifying the outside term, the outside term here refers to the term which will be distributed.
Step 2: Multiply the term with the terms inside, by keeping the operation as it is, for example, P(Q + R) = PQ + PR.
Step 3: Doing the operation, either the addition or subtraction.
For example, 6 (5 + 2)
Step 1: Here the outside term is 6
Step 2: Multiply 6 with 5 and 2, and find the sum of the products. (6 × 5) + (6 × 2)
Step 3: Finding the sum, (6 × 5) + (6 × 2) = 30 + 12 = 42
Distributive Property of Multiplication
The distributive property of multiplication is used when we multiply a number by the sum or difference of two numbers. This property breaks down the multiplication operation into separate addition or subtraction operations. For instance, the distributive property of multiplication for any three numbers is p, q, and r
p × (q + r) = pq + pr
p × (q - r) = pq - pr
Distributive property of multiplication over addition
The formula we use is a × (b + c) = ab + ac
For example; 20 (5 + 8)
Here a = 20
b = 5
c = 8
As, a × (b + c) = ab + ac
20 (5 + 8) = (20 × 5) + (20 × 8)
= 100 + 160 = 260
Distributive property of multiplication over subtraction
The formula we use is a × (b - c) = ab - ac
For example; 10 (9 - 5)
Here a = 10
b = 9
c = 5
As, a × (b - c) = ab - ac
10 (9 - 5) = (10 × 9) - (10 × 5)
= 90 - 50 = 40
Distributive Property of Division
When dividing a number with the sum or difference of two or more numbers we follow the same pattern as in the distributive property of multiplication. The distributive property of division can be expressed as:
(b + c) ÷ a = (b ÷ a) + (c ÷ a)
(b - c) ÷ a = (b ÷ a) - (c ÷ a)
For example: (40 + 9) ÷ 7
Using the distributive property of division (b + c) ÷ a = (b ÷ a) + (c ÷ a)
Here, a = 7
b = 40
c = 9
So, (40 + 9) ÷ 7 = (40 ÷ 7) + (9 ÷ 7)
= 7
Real-world applications of the Distributive Property
Real-world applications of the Distributive Property
Distributive property is one of the fundamental properties in math that is used to distribute one operation over the other. Here are a few real-world applications of the distributive property:.
- While shopping, we use the distributive property to quickly calculate the total cost of multiple items.
- In interior designing and painting, multiply the cost per square foot by calculating the total area of various walls.
- For calculating the time management and scheduling, we use the distributive property
Common Mistakes and How to Avoid Them in the Distributive Property
When learning about the distributive property, students tend to make a few mistakes. Here are some mistakes that students make in distributive property and ways to avoid them.
Mistake 1
Forgetting to distribute to all terms
When applying the distributive property sometimes students forget to distribute all the terms, for example 3 (x + 5) is written as 3x + 5 instead of 3x + 15 can lead to errors. So when using distributive property, try to verify the answer and check if the inside terms are multiplied with the outside term.
Mistake 2
Incorrect application of distributive property to subtraction
When applying the distributive property to subtraction students get confused with which symbol to add, either addition or subtraction. That is for, 6(x - 3) = 6x + 18 instead of 6x - 18, so it is important to understand that we multiply the outer term with the inner terms and find the difference between the two products.
Mistake 3
Not distributing a negative sign
When the outside term has the negative sign, sometimes students apply it only for the first term and avoid the second one, which is wrong. For example -4 (x + 5) = -4x + 20 instead of -4x - 20. So whenever the outside term has a negative sign we multiply the inner term with the sign.
Mistake 4
Ignoring multiple coefficients and variables in terms
When the outside term has both the value and variable, students usually multiply it only with the numerical part, and leaving the variable part is wrong. For example, 2x (5x + 2) = 10x + 4 which is wrong. So we should multiply the numerical value and the variable; that 2x (5x + 2) = 10x2 + 4x.
Mistake 5
Confusing distribution with commutative property
Students sometimes get confused with the distributive and commutative property as they think the distributive property allows reordering. But it is wrong in distributive property, we multiply the outside term with all the internal terms individually. That is, 2 (x + 5) = 2x + 10
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Solved Examples of the Distributive Property
Problem 1
Sarah wants to distribute 5 apples to each of her 3 friends. How many apples does she give in total?
The number of apples she gives in total is 15
Explanation
The number of apples Sarah wants to distribute is 5
The number of people they gave is 3
So the total number of apples she gives is 5 × 3 = 15
Problem 2
A shop sells packs of 4 pencils. If Jason buys 6 packs and gives 2 packs to his sister, how many pencils does he have left?
The number of pencils they have left with is 16
Explanation
The number of pencils in a packet = 4
Number of pack Jason buy = 6 packets
Number of packs he bought for sister = 2 packets
The number of pencils he left with = 4 × (6 - 2)
= (4 × 6) - (4 ×2)
= 24 - 8 = 16
So, Jason is left with 16 pencils.
Problem 3
A concert hall sells 12 tickets per row. If 9 rows are sold in the morning and 5 more rows are sold in the evening, how many tickets were sold in total?
The number of tickets sold is 168
Explanation
The number of tickets sold in row = 12
The number of rows sold in the morning = 9
The number of tickets sold in the evening = 5
The number of tickets sold = 12 (9 + 5)
= (12 × 9) + (12 × 5)
= 108 + 60 = 168
Problem 4
Each box contains 6 chocolates. If Liam buys 4 boxes for himself and 3 more for his friends, how many chocolates does he have?
The number of chocolates he bought is 42
Explanation
The number of chocolates in a box = 6
The number of chocolates Liam bought for him = 4 boxes
The number of chocolates Liam bought for his friends = 3 boxes
So, the number of chocolates he bought = 6 (4 +3)
= (6 × 4) + (6 × 3)
= 24 + 18 = 42
Problem 5
Each carton holds 7 bottles of water. If a company ships 11 cartons in one order and 6 cartons in another, how many bottles are shipped?
The number of bottles shipped is 119
Explanation
The number of water bottles in a cartons = 7
The number of cartons shipped = 11 + 6
So, the number of bottles shipped = 7 × (11 +6)
= 77 + 42 = 119
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FAQs on the Distributive Property
1.What is the distributive property of 3 × 6?
2.What is the distributive property of 39 × 5?
3.What is the distributive property of 15 and 45?
4.What is the greatest common factor of 27 and 36?
5.What is the distributive property of 13 × 6?
6.How can children in Australia use numbers in everyday life to understand Distributive Property?
7.What are some fun ways kids in Australia can practice Distributive Property with numbers?
8.What role do numbers and Distributive Property play in helping children in Australia develop problem-solving skills?
9.How can families in Australia create number-rich environments to improve Distributive Property skills?
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Hiralee Lalitkumar Makwana
About the Author
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
Fun Fact
: She loves to read number jokes and games.
