Distributive Property | Definition and Examples
Last Updated :
23 Jul, 2025
Distributive Property in Maths refers to the distribution of the number over the other operations. It is usually called the Distributive Law of Multiplication over Addition and Subtraction as the number which is to be multiplied is distributed over addition or subtraction of the numbers.
In this article, we will learn about the distributive property in detail along with its definition, formula, and distributive property of multiplication over addition and subtraction.
What is Distributive Property in Maths?
Distributive Property is a fundamental principle in mathematics that describes how operations like multiplication and division distribute over addition and subtraction. In simpler terms, it explains how you can combine operations when simplifying expressions.
Distributive Property states that when a number is multiplied by the sum or difference of two numbers then it is equal to the sum or difference of the product of the first number with the other two numbers individually.
Distributive Property is often expressed in the following formula:
a × (b + c) = (a × b) + (a × c)
In this formula:
- a is a constant or variable that is multiplied by the sum or difference of b and c.
- b and c can be constants, variables, or expressions.
As of now, we know that Distributive Property Formula is defined over two operations i.e. addition and subtraction. Let's Learn them in detail.
How to Use Distributive Property?
Some steps to use distributive property are:
Identify Terms
- Find the term outside the parentheses (this is the term that will be distributed).
- Identify the terms inside the parentheses.
Multiply Each Term
- Multiply the outside term by each term inside the parentheses separately.
- Keep the operations (addition or subtraction) between the terms inside the parentheses.
Simplify the Expression
- Combine like terms if necessary.
- Ensure the expression is fully simplified.
Distributive Property of Multiplication
The Distributive Property of Multiplication is a fundamental property in mathematics that describes how multiplication distributes over addition. In essence, it allows you to split a multiplication operation across the terms of an addition or subtraction operation.
The Distributive Property states that for any three numbers a, b and c:
a × (b + c) = a × b + a × c
a × (b - c) = a × b - a × c
Distributive Property of Multiplication over Addition
Distributive Property Formula of Multiplication over Addition, as the name suggests, is defined over the operation of addition. This Formula is used when a number is multiplied by the sum of two numbers. The Distributive Formula of Multiplication over Addition is given as
P(Q + R) = PQ + PR
Let's understand this with the help of an example.
Example: Solve 15(6 + 5) using Distributive Property.
We have, 15(6 + 5)
Using the Distributive Law of Multiplication over Addition
15(6 + 5) = 15⨯6 + 15⨯5 = 90 + 75 = 165
Distributive Property of Multiplication over Subtraction
Distributive Law of Multiplication over Subtraction, as the name suggests, is defined over the operation of subtraction. In this case, if a number is multiplied to the difference of two numbers then it is equal to the difference of product of the numbers. The distributive Property of Multiplication over Subtraction is expressed as
P(Q - R) = PQ - PR
Let's understand this with the help of an example.
Example: Solve 6(30 - 4) using Distributive Property.
We have, 6(30 - 4)
Using the Distributive Property of Multiplication over Subtraction, we have
6(30 - 4) = 6⨯30 - 6⨯4 = 180 - 24 = 156
Verification of Distributive Property
To verify the Distributive Property, we need to check if the values on both Left Hand Side and Right Hand Side of the expression. Let's see the verification of the Distributive Property of Multiplication over Addition and the Distributive Property of Multiplication over Subtraction one by one.
Verification of Distributive Property of Multiplication over Addition
Let's verify if, 15(6 + 5) = 15 ⨯ 6 + 15 ⨯ 5.
We have LHS = 15(6 + 5)
Using BODMAS Rule we will first solve the bracket.
⇒ 15(6 + 5) = 15 ⨯ 11 = 165
Hence, we have LHS = 165
We have RHS = 15 ⨯ 6 + 15 ⨯ 5
⇒ 90 + 75 = 165
Hence, RHS = 165
Thus we see that we have LHS = RHS. Hence, 15(6 + 5) = 15 ⨯ 6 + 15 ⨯ 5. Thus Distributive Property of Multiplication over Addition is verified.
Verification of Distributive Property of Multiplication over Subtraction
Distributive Property of Multiplication over Subtraction is given as P(Q - R) = PQ - PR. Let's check if it's true or not using an example.
Let's verify if 6(30 - 9) = 6⨯30 - 6⨯9.
We have LHS = 6(30 - 9)
Solving bracket first using BODMAS Rule,
⇒ 6 ⨯ 21 = 126
We have RHS = 6⨯30 - 6⨯9
⇒ 180 - 54 = 126
Hence, we have LHS = RHS, it means 6(30 - 9) = 6⨯30 - 6⨯9. Thus Distributive Law of Multiplication over Subtraction is verified.
Distributive Property of Division
Distributive Property of Division over Addition or subtraction is based on the same pattern as the Distributive Property of Multiplication just with a minor difference that the sum or difference inside the bracket is now divided by a number instead of multiplication.
Distributive Property of Division can be expressed as:
(Q + R) ÷ P = Q ÷ P + R ÷ P
(Q - R) ÷ P = Q ÷ P - R ÷ P
The same expression can be written in terms of multiplication as
(Q + R)1/P and (Q - R)1/P
Example: Divide 76 ÷ 4 using the distributive property of division.
Given expression: 76 ÷ 4
We can write 76 as 64 + 12
So, 76 ÷ 4 = (64 + 12) ÷ 4
Now, let us distribute the division operation for each factor (64 and 12) in the bracket.
= (64 ÷ 4) + (12 ÷ 4)
= 16 + 3 = 19
Therefore, the answer is 19.
Articles Related to Distributive Property:
Solved Examples problems on Distributive Property
Example 1: Solve equation 5 (y + 8) = 120 Using Distributive Property.
Solution:
Given, 5 (y + 8) = 120
We have a set of two parentheses inside the bracket. So, distribute 5.
5 × y + 5 × 8 = 120
⇒ 5y + 40 = 120
⇒ 5y = 120 - 40 = 80
⇒ y = 80/5 = 16
Hence, y = 16.
Example 2: Solve 3x + 4(x - 6) + 17 = 28 Using Distributive Property.
Solution:
Given, 3x + 4(x - 6) + 17 = 28
We have a set of two parentheses inside the bracket. So, distribute 4.
3x + 4 × x - 4 × 6 + 17 = 28
⇒ 3x + 4x - 24 + 17 = 28
⇒ 7x - 7 = 28
⇒ 7x = 28 + 7 = 35
⇒ x = 35/7 = 5
Hence, x = 5.
Example 3: Solve the equation (a + 3b) (2a + b) Using Distributive Property.
Solution:
Given, (a + 3b) (2a + b)
From the distributive property, we have
(p + q) × r=(p × r)+(q × r)
So, (a + 3b) (2a + b) = a × (2a + b) + 3b × (2a + b)
⇒ (a + 3b) (2a + b) = 2a2 + ab + 6ab + 3b2
⇒ (a + 3b) (2a + b) = 2a2 + 7ab + 3b2
Thus, (a + 3b) (2a + b) = 2a2 + 7ab + 3b2.
Practice Questions on Distributive Property
Q1.Solve using Distributive Property 99 ⨯ 23.
Q2. Solve 24 ⨯ 96 + 24 ⨯ 4 with the help of Distributive Property.
Q3. Verify using Distributive Property: Is 39 ⨯ 101 = 39 (100 + 1)?
Q4. Solve 36 ⨯ 204 - 26 ⨯ 4 using Distributive Property.
Q5. Solve 19 ⨯ 47 + 19 ⨯ 3.
Conclusion
Distributive property is a critical algebraic property that simplifies the process of expanding and simplifying expressions. By applying the distributive property, you can break down complex expressions into simpler parts, making it easier to combine like terms and solve equations.The distributive property is indispensable for solving algebraic equations, factoring polynomials, and simplifying expressions, forming a foundation for higher-level math and practical problem-solving.
Similar Reads
Maths Mathematics, often referred to as "math" for short. It is the study of numbers, quantities, shapes, structures, patterns, and relationships. It is a fundamental subject that explores the logical reasoning and systematic approach to solving problems. Mathematics is used extensively in various fields
5 min read
Basic Arithmetic
What are Numbers?Numbers are symbols we use to count, measure, and describe things. They are everywhere in our daily lives and help us understand and organize the world.Numbers are like tools that help us:Count how many things there are (e.g., 1 apple, 3 pencils).Measure things (e.g., 5 meters, 10 kilograms).Show or
15+ min read
Arithmetic OperationsArithmetic Operations are the basic mathematical operations—Addition, Subtraction, Multiplication, and Division—used for calculations. These operations form the foundation of mathematics and are essential in daily life, such as sharing items, calculating bills, solving time and work problems, and in
9 min read
Fractions - Definition, Types and ExamplesFractions are numerical expressions used to represent parts of a whole or ratios between quantities. They consist of a numerator (the top number), indicating how many parts are considered, and a denominator (the bottom number), showing the total number of equal parts the whole is divided into. For E
7 min read
What are Decimals?Decimals are numbers that use a decimal point to separate the whole number part from the fractional part. This system helps represent values between whole numbers, making it easier to express and measure smaller quantities. Each digit after the decimal point represents a specific place value, like t
10 min read
ExponentsExponents are a way to show that a number (base) is multiplied by itself many times. It's written as a small number (called the exponent) to the top right of the base number.Think of exponents as a shortcut for repeated multiplication:23 means 2 x 2 x 2 = 8 52 means 5 x 5 = 25So instead of writing t
9 min read
PercentageIn mathematics, a percentage is a figure or ratio that signifies a fraction out of 100, i.e., A fraction whose denominator is 100 is called a Percent. In all the fractions where the denominator is 100, we can remove the denominator and put the % sign.For example, the fraction 23/100 can be written a
5 min read
Algebra
Variable in MathsA variable is like a placeholder or a box that can hold different values. In math, it's often represented by a letter, like x or y. The value of a variable can change depending on the situation. For example, if you have the equation y = 2x + 3, the value of y depends on the value of x. So, if you ch
5 min read
Polynomials| Degree | Types | Properties and ExamplesPolynomials are mathematical expressions made up of variables (often represented by letters like x, y, etc.), constants (like numbers), and exponents (which are non-negative integers). These expressions are combined using addition, subtraction, and multiplication operations.A polynomial can have one
9 min read
CoefficientA coefficient is a number that multiplies a variable in a mathematical expression. It tells you how much of that variable you have. For example, in the term 5x, the coefficient is 5 — it means 5 times the variable x.Coefficients can be positive, negative, or zero. Algebraic EquationA coefficient is
8 min read
Algebraic IdentitiesAlgebraic Identities are fundamental equations in algebra where the left-hand side of the equation is always equal to the right-hand side, regardless of the values of the variables involved. These identities play a crucial role in simplifying algebraic computations and are essential for solving vari
14 min read
Properties of Algebraic OperationsAlgebraic operations are mathematical processes that involve the manipulation of numbers, variables, and symbols to produce new results or expressions. The basic algebraic operations are:Addition ( + ): The process of combining two or more numbers to get a sum. For example, 3 + 5 = 8.Subtraction (−)
3 min read
Geometry
Lines and AnglesLines and Angles are the basic terms used in geometry. They provide a base for understanding all the concepts of geometry. We define a line as a 1-D figure that can be extended to infinity in opposite directions, whereas an angle is defined as the opening created by joining two or more lines. An ang
9 min read
Geometric Shapes in MathsGeometric shapes are mathematical figures that represent the forms of objects in the real world. These shapes have defined boundaries, angles, and surfaces, and are fundamental to understanding geometry. Geometric shapes can be categorized into two main types based on their dimensions:2D Shapes (Two
2 min read
Area and Perimeter of Shapes | Formula and ExamplesArea and Perimeter are the two fundamental properties related to 2-dimensional shapes. Defining the size of the shape and the length of its boundary. By learning about the areas of 2D shapes, we can easily determine the surface areas of 3D bodies and the perimeter helps us to calculate the length of
10 min read
Surface Areas and VolumesSurface Area and Volume are two fundamental properties of a three-dimensional (3D) shape that help us understand and measure the space they occupy and their outer surfaces.Knowing how to determine surface area and volumes can be incredibly practical and handy in cases where you want to calculate the
10 min read
Points, Lines and PlanesPoints, Lines, and Planes are basic terms used in Geometry that have a specific meaning and are used to define the basis of geometry. We define a point as a location in 3-D or 2-D space that is represented using coordinates. We define a line as a geometrical figure that is extended in both direction
14 min read
Coordinate Axes and Coordinate Planes in 3D spaceIn a plane, we know that we need two mutually perpendicular lines to locate the position of a point. These lines are called coordinate axes of the plane and the plane is usually called the Cartesian plane. But in real life, we do not have such a plane. In real life, we need some extra information su
6 min read
Trigonometry & Vector Algebra
Trigonometric RatiosThere are three sides of a triangle Hypotenuse, Adjacent, and Opposite. The ratios between these sides based on the angle between them is called Trigonometric Ratio. The six trigonometric ratios are: sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec).As give
4 min read
Trigonometric Equations | Definition, Examples & How to SolveTrigonometric equations are mathematical expressions that involve trigonometric functions (such as sine, cosine, tangent, etc.) and are set equal to a value. The goal is to find the values of the variable (usually an angle) that satisfy the equation.For example, a simple trigonometric equation might
9 min read
Trigonometric IdentitiesTrigonometric identities play an important role in simplifying expressions and solving equations involving trigonometric functions. These identities, which include relationships between angles and sides of triangles, are widely used in fields like geometry, engineering, and physics. Some important t
10 min read
Trigonometric FunctionsTrigonometric Functions, often simply called trig functions, are mathematical functions that relate the angles of a right triangle to the ratios of the lengths of its sides.Trigonometric functions are the basic functions used in trigonometry and they are used for solving various types of problems in
6 min read
Inverse Trigonometric Functions | Definition, Formula, Types and Examples Inverse trigonometric functions are the inverse functions of basic trigonometric functions. In mathematics, inverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. The inverse trigonometric functions are the inverse functions of basic trigonometric function
11 min read
Inverse Trigonometric IdentitiesInverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. These functions are the inverse functions of basic trigonometric functions, i.e., sine, cosine, tangent, cosecant, secant, and cotangent. It is used to find the angles with any trigonometric ratio. Inv
9 min read
Calculus
Introduction to Differential CalculusDifferential calculus is a branch of calculus that deals with the study of rates of change of functions and the behaviour of these functions in response to infinitesimal changes in their independent variables.Some of the prerequisites for Differential Calculus include:Independent and Dependent Varia
6 min read
Limits in CalculusIn mathematics, a limit is a fundamental concept that describes the behaviour of a function or sequence as its input approaches a particular value. Limits are used in calculus to define derivatives, continuity, and integrals, and they are defined as the approaching value of the function with the inp
12 min read
Continuity of FunctionsContinuity of functions is an important unit of Calculus as it forms the base and it helps us further to prove whether a function is differentiable or not. A continuous function is a function which when drawn on a paper does not have a break. The continuity can also be proved using the concept of li
13 min read
DifferentiationDifferentiation in mathematics refers to the process of finding the derivative of a function, which involves determining the rate of change of a function with respect to its variables.In simple terms, it is a way of finding how things change. Imagine you're driving a car and looking at how your spee
2 min read
Differentiability of a Function | Class 12 MathsContinuity or continuous which means, "a function is continuous at its domain if its graph is a curve without breaks or jumps". A function is continuous at a point in its domain if its graph does not have breaks or jumps in the immediate neighborhood of the point. Continuity at a Point: A function f
11 min read
IntegrationIntegration, in simple terms, is a way to add up small pieces to find the total of something, especially when those pieces are changing or not uniform.Imagine you have a car driving along a road, and its speed changes over time. At some moments, it's going faster; at other moments, it's slower. If y
3 min read
Probability and Statistics
Basic Concepts of ProbabilityProbability is defined as the likelihood of the occurrence of any event. It is expressed as a number between 0 and 1, where 0 is the probability of an impossible event and 1 is the probability of a sure event.Concepts of Probability are used in various real life scenarios : Stock Market : Investors
7 min read
Bayes' TheoremBayes' Theorem is a mathematical formula used to determine the conditional probability of an event based on prior knowledge and new evidence. It adjusts probabilities when new information comes in and helps make better decisions in uncertain situations.Bayes' Theorem helps us update probabilities ba
13 min read
Probability Distribution - Function, Formula, TableA probability distribution is a mathematical function or rule that describes how the probabilities of different outcomes are assigned to the possible values of a random variable. It provides a way of modeling the likelihood of each outcome in a random experiment.While a Frequency Distribution shows
13 min read
Descriptive StatisticStatistics is the foundation of data science. Descriptive statistics are simple tools that help us understand and summarize data. They show the basic features of a dataset, like the average, highest and lowest values and how spread out the numbers are. It's the first step in making sense of informat
5 min read
What is Inferential Statistics?Inferential statistics is an important tool that allows us to make predictions and conclusions about a population based on sample data. Unlike descriptive statistics, which only summarize data, inferential statistics let us test hypotheses, make estimates, and measure the uncertainty about our predi
7 min read
Measures of Central Tendency in StatisticsCentral tendencies in statistics are numerical values that represent the middle or typical value of a dataset. Also known as averages, they provide a summary of the entire data, making it easier to understand the overall pattern or behavior. These values are useful because they capture the essence o
11 min read
Set TheorySet theory is a branch of mathematics that deals with collections of objects, called sets. A set is simply a collection of distinct elements, such as numbers, letters, or even everyday objects, that share a common property or rule.Example of SetsSome examples of sets include:A set of fruits: {apple,
3 min read
Practice