Distributive Property Formula with Step by Step Solved Examples
Distributive Property Basics
All the numbers that are used in Mathematical calculations and have a specific value is called the real numbers. All the real numbers obey certain laws or have a few properties. These properties are shown when basic Mathematical operations are performed on these numbers. The most fundamental properties of numbers include closure property, commutative property, associative property, identity property, inverse property and distributive property. Distributive property is one of the most popular and very frequently used properties of numbers in Mathematical calculations. This section briefs about a few distributive property examples for better understanding.
What is Distributive Property?
Distributive property is one of the fundamental properties of numbers and basic operations of Mathematics. This property refers to the distribution of multiplication over addition or subtraction. Hence it is most popularly known as the distributive property of multiplication over addition or subtraction. Distribution in English refers to the breaking down into parts. What is distributive property is math can be analysed as a property used to simplify complex algebraic expressions to arrive at the solution easily. By using the distributive property of multiplication over addition or subtraction, the expressions can be written as the sum or difference of the two numbers.
How to use the Distributive Property of Multiplication Over Addition/Subtraction?
The distributive property of multiplication over addition allows the user to multiply a value of the sum by multiplying the addends individually. The individual products are then added to get the final answer. For instance, if ‘m’, ‘n’ and ‘p’ are the three numbers, then distributive property states that
m (n + p) = m n + m p
The answer of the product of a number and the addition of two other numbers can be obtained as the sum of the individual products obtained when each number inside the parentheses is multiplied by the number outside.
Any algebraic expression may contain the terms this form with several variables.
Distributive Property of Multiplication Over Addition:
While multiplying a number or a term with the sum of the other two numbers or variables, the number or the term outside the parentheses is distributed over the individual addends within the parentheses. The individual products obtained by multiplying the outer term withe inner terms are added to arrive at the final answer.
If the variables ‘p’, ‘q’ and ‘r’ represent any real numbers or the terms in an algebraic expression, then distributive property of multiplication over addition states that:
p x (q + r) = (p x q) + (p x r)
Distributive Property of Multiplication Over Subtraction:
Distributive property of multiplication over subtraction is the same as that incase of addition. However, the only change is that the difference of the individual products is found to arrive at the final answer. It does not make a great difference with the distributive property of multiplication over addition or subtraction. The only rule is that the term outside the bracket is distributed over the terms inside the bracket using multiplication operation and the basic operation indicated between the terms within the parentheses is performed between the individual products.
Distributive Property Examples
Let us consider that there are three friends sitting on the same bench. Imagine that each individual student has 4 black pencils and 5 blue pencils. Just think about “How do we calculate the total number of pencils with them all together?”. This can be done in two ways.
Method 1:
Identify the total number of pencils with each individual person and then multiply it with the total number of friends. This method follows the order of operations.
No. of pencils with each individual = 4 + 5 = 9
Total no. of pencils = 9 x 3 = 27
Method 2:
We can first find the total number of blue pencils and the total number of black pencils. Then, to find the total number of pencils, the number of black pencils and blue pencils are added. Here, the number of individuals is distributed over the number of blue and black pencils.
Total number of pencils
= No. of friends x No. of black pencils with each + No. of friends x No. of blue pencils with each
= (3 x 4) + (3 x 5) = 12 + 15 = 27
Fun Facts
The answer remains unaltered in case the distributive property example is used or the order of operations is followed.
Algebraic expressions can also be evaluated by following the PEMDAS rule which gives the order of operations. It is not mandatory to use the distributive property in all cases.
FAQs on Understanding the Distributive Property in Algebra
1. What is the distributive property in maths?
The distributive property states that multiplying a number by a sum or difference is the same as multiplying the number by each term inside the brackets and then adding or subtracting the results. In algebra, it is written as a(b + c) = ab + ac and a(b − c) = ab − ac.
- It connects multiplication with addition and subtraction.
- It is used to expand algebraic expressions.
- It works for numbers, variables, and algebraic expressions.
2. What is the formula for the distributive property?
The formula for the distributive property is a(b + c) = ab + ac and a(b − c) = ab − ac. Here:
- a is the factor outside the brackets.
- b and c are the terms inside the brackets.
- The multiplication is distributed to each term inside the parentheses.
3. How do you use the distributive property step by step?
To use the distributive property, multiply the outside factor by each term inside the brackets and then combine the results. Example: Expand 3(x + 4).
- Step 1: Multiply 3 by x → 3x
- Step 2: Multiply 3 by 4 → 12
- Step 3: Add the products → 3x + 12
4. Can you give an example of the distributive property with numbers?
An example of the distributive property with numbers is 5(2 + 3) = 25. Using the rule:
- Step 1: Multiply 5 × 2 = 10
- Step 2: Multiply 5 × 3 = 15
- Step 3: Add 10 + 15 = 25
5. Does the distributive property work with subtraction?
Yes, the distributive property works with subtraction and follows the formula a(b − c) = ab − ac. Example: Expand 4(x − 5).
- Multiply 4 × x = 4x
- Multiply 4 × (−5) = −20
- Final answer: 4x − 20
6. What is the difference between distributive property and factoring?
The distributive property expands an expression, while factoring reverses the process by taking out a common factor. For example:
- Distributive form: 3(x + 2) = 3x + 6
- Factored form: 3x + 6 = 3(x + 2)
7. Why is the distributive property important in algebra?
The distributive property is important in algebra because it allows you to expand expressions, simplify equations, and solve problems efficiently. It helps in:
- Removing parentheses
- Combining like terms
- Solving linear equations
- Factoring polynomials
8. What are common mistakes when using the distributive property?
A common mistake in the distributive property is forgetting to multiply every term inside the brackets. Key errors include:
- Multiplying only the first term and ignoring the second
- Forgetting to distribute a negative sign
- Sign errors when subtracting
9. How do you use the distributive property with variables on both sides?
To use the distributive property with variables on both sides, expand each side first and then solve the equation. Example: Solve 2(x + 3) = 4x − 2.
- Step 1: Distribute → 2x + 6 = 4x − 2
- Step 2: Subtract 2x from both sides → 6 = 2x − 2
- Step 3: Add 2 to both sides → 8 = 2x
- Step 4: Divide by 2 → x = 4
10. Is the distributive property only used in algebra?
No, the distributive property is used in both arithmetic and algebra. In arithmetic, it helps simplify calculations like 7 × 19 = 7(20 − 1).
- 7 × 20 = 140
- 7 × 1 = 7
- 140 − 7 = 133
