Distributive Property
2026-02-28 13:46 Diff
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Last updated on December 10, 2025

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?

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The 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 \(A (B + C) = AB + AC\), which means the product of a number and the sum of two other numbers is the same as the sum of multiplying the addends separately.

 
For example, \(5 \times (2 +3) =  (5 \times 2) + (5 \times 3) = 25\)
 

\( 5 \times (7 - 3) = (5 \times 7) - (5 \times 3) = 20\)

How to Use the Distributive Property?

The distributive property formula can be expressed as  \(a × (b + c) = (a × b) + (a × c)\). 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, \(6 (5 + 2)\) 


Here the outside term is 6


Multiply 6 with 5 and 2, and find the sum of the products. \((6 \times 5) + (6 \times 2)\) 


Finding the sum, \((6 \times 5) + (6 \times 2) = {30 + 12} = {42}\)

Distributive Property Formula

The distributive property explains how multiplication can be applied to each term inside a bracket. According to the distributive property, an expression of the form \(A × (B + C) \)can be expressed as: 

\(A × (B + C) = A × B + A × C \)

Here, the number A is distributed and multiplied by both B and C. 

The distributive property over addition formula: \(A × (B + C) = A × B + A × C\)

The distributive property over subtraction formula: \(A × (B + C) = A × B + A × C\)

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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.

\( p \times (q + r) = pq + pr\)


\(p \times (q - r) = pq - pr\)

Distributive Property of Multiplication Over Addition

The formula we use is \(a \times (b + c) = ab + ac\)


For example; \(20 (5 + 8)\) 


Here a = 20


b = 5


c = 8 


As, \(a \times (b + c) = ab + ac\)
 

\( 20 (5 + 8) = (20 \times 5) + (20 \times 8) \)
 

\(= 100 + 160 = 260\)

Distributive Property of Multiplication Over Subtraction

The formula we use is\( a \times (b - c) = ab - ac\)


For example; \(10 (9 - 5) \)


Here a = 10


b = 9

c = 5 

As, \(a \times (b - c) = ab - ac \)

\( 10 (9 - 5) = (10 \times 9) - (10 \times 5)  \)
 

\(= 90 - 50 = 40\)

Verification of Distributive Property

To understand how the distributive property works, let us verify it separately for addition and subtraction. This helps us know how multiplication spreads over the terms inside the brackets.

Distributive Property of Addition: The distributive property of multiplication over addition is given by: \(A × (B + C) = AB + AC\). 

Example: Verify the property using \(2(1 + 4)\)


\(2(1 + 4) = (2 × 1) + (2 × 4) \\ \ \\ = 2 + 8 = 10 \)

Applying the BODMAS rule to verify:

\(2(1 + 4) = 2(5) = 10 \)

Hence, the distributive property is verified.

Distributive Property of Subtraction: The distributive property of multiplication over subtraction is written  as: \(A × (B - C) = A × B - A × C\)


Let’s verify it with an example, \(2(4 - 1)\)

\(2(4 - 1) = (2 × 4) - (2 × 1) \\ \ \\ = 8 - 2 = 6 \)
 

Using BODMAS to verify: 

\(2(4 - 1) = 2(3) = 6 \)

Since the result in the distributive property and BODMAS is the same, the distributive property of subtraction is verified.

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:

\((b + c) \div a = (b \div a) + (c \div a)\)

\( (b - c) \div a = (b \div a) - (c \div a)\)
 

For example: \((40 + 9) \div 7\)

Using the distributive property of division, 

\((b + c) \div a = (b \div a) + (c \div a)\)

Here, a = 7


b = 40


c = 9


So,  \((40 \div 7) + (9 \div 7)\)

\(= (40 + 9) \div 7\)

\(= 49 \div 7\)


= 7

Tips and Tricks to Master Distributive Property

The distributive property is used to simplify complex multiplication and algebraic expressions. By learning the distributive property, students can solve problems faster and also improve their mental math more easily. Here are a few tips and tricks to master the distributive property. 

  • Memorize the formula for the distributive property of addition and multiplication. For any three numbers, a, b, and c, the distributive property is: 
    \(a × (b + c) = ab + ac \)
    \(a × (b - c) = ab - ac\)
     
  • When distributing negative numbers or subtracting terms, always remember that a negative times a negative is equal to a positive. For example, \(-2(x - 4) = (-2 \times x) + (-2 \times -4) = -2x + 8.  \)
     
  • When multiplying bigger numbers, use the distributive property to break the bigger number. For example, \(5 \times 27 = 5 \times (20 + 7) = 100 + 35 = 135. \)
     
  • To master the distributive property, use it in real-life applications like shopping, calculating total cost, or finding area to make learning fun. 
     
  • Always remember that FOIL is the specific application of the distributive property. FOIL means first outer inner last.
     
  • Teachers can use the area models or rectangle diagrams to help students visualize how multiplication distributes over addition. This allows students to work well in any distributive property worksheet.
     
  • Parents can give small mental math tasks that use the distributive property. For example, \(7 × 19 = 7 (20 - 1) = 140 - 7 = 133\). 
     
  • Students can verify their answers using a distributive property calculator, which helps them understand where they might have made mistakes while practicing.

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.

Real-World Applications of the Distributive Property

It is a fundamental property used to simplify complex calculations. 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
     
  • In construction and architecture, distributive property is used to estimate the cost of materials for different sections of buildings.
     
  • For salary calculations, the distributive property is used to calculate the monthly salaries with allowances and bonuses.

Download Worksheets

Problem 1

Sarah wants to distribute 5 apples to each of her 3 friends. How many apples does she give in total?

Okay, lets begin

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 \(5 \times 3 = 15\)

Well explained 👍

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?

Okay, lets begin

The number of pencils he has 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 = \(4 \times (6 - 2) \)


\(= (4 \times 6) - (4 \times 2)\)


\(= 24 - 8 \)


= 16


 So, Jason is left with 16 pencils.

Well explained 👍

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?

Okay, lets begin

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 = \(12 (9 + 5) \)


\(= (12 \times 9) + (12 \times 5)\)


\(= 108 + 60\)

= 168

Well explained 👍

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?

Okay, lets begin

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 = \(6 (4 +3) \)


\(= (6 \times 4) + (6 \times 3)\)


\(= 24 + 18 \)

= 42

Well explained 👍

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?

Okay, lets begin

The number of bottles shipped is 119.

Explanation

Number of bottles per carton = 7


Total cartons \(= 11 + 6 = 17\)


Using the distributive property:


\(7 \times (11 + 6)\)

\(= (7 \times 11) + (7 \times 6)\)

\(= 77 + 42\)

= 119
 

Well explained 👍

FAQs on the Distributive Property

1.What is the distributive property of 3 × 6?

The distributive property of \(3 \times 6\) is \(3(3 + 3)\). That is \(3(3 + 3) = (3 \times 3) + (3 \times 3) = 9 + 9 = 18\).

2.What is the distributive property of 39 × 5?

The distributive property of \(39 \times 5\) is \(5(30 + 9)\). That is \(5(30 + 9) = (5 \times 30) + (5 \times 9) = 150 + 45 = 195.\).

3.What is the distributive property of 15 and 45?

The distributive property of 15 and 45 is \(15(40 + 5)\). That is \(15(40 + 5) = (15 \times 40) + (15 \times 5) = 600 + 75 = 675.\).

4.What is the greatest common factor of 27 and 36?

The factors of 27 are 1, 3, 9, and 27. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. Here the common factors are 1, 3, and 9, so the greatest common factor is 9.

5.What is the distributive property of 13 × 6?

The distributive property of \(13 \times 6 \) is \(6(10 + 3)\). That is \(6(10 + 3) = (6 \times 10) + (6 \times 3) = 60 + 18 = 78.\).

6.How can I help my child practice it at home?

Encourage them to break down problems step by step. Use real-life examples, like distributing items into boxes. Start with simple numbers, then gradually increase the difficulty.

7.How will learning distributive property help my child in real life?

Distributive property improves problem-solving, mental math, and logical thinking. Children can use it for budgeting, splitting items, calculating discounts, or any situation where they multiply across groups.

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.