How to Use the Distributive Property in Maths with Step-by-Step Examples
The concept of distributive property plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios.
What Is Distributive Property?
The distributive property is a rule stating that multiplying a number by a sum (or difference) is the same as multiplying each part separately, then adding (or subtracting) the results. You’ll find this concept applied in arithmetic, algebraic expressions, and solving equations.
Key Formula for Distributive Property
Here’s the standard formula: \( a \times (b + c) = a \times b + a \times c \)
Cross-Disciplinary Usage
Distributive property is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. Students preparing for JEE or NEET will see its relevance in various questions.
Step-by-Step Illustration
Let’s see how to apply the distributive property in a simple example:
| Given | Step-by-Step Solution |
|---|---|
| \( 5 \times (4 + 7) \) |
1. Apply the distributive property: \( 5 \times 4 + 5 \times 7 \)
2. Calculate each part: \( 20 + 35 \)
3. Add the results: \( 55 \)
|
Speed Trick or Vedic Shortcut
Here’s a quick shortcut that helps solve problems faster when working with distributive property. Many students use this trick during timed exams to save crucial seconds.
Example Trick: To multiply a number mentally, break it into parts using distributive law.
Let’s find \( 7 \times 16 \):
- Break 16 into 10 + 6:
\( 7 \times (10 + 6) \) - Distribute:
\( 7 \times 10 + 7 \times 6 = 70 + 42 \) - Add:
Final answer: 112
Tricks like these aren’t just cool—they’re practical in competitive exams like NTSE, Olympiads, and even JEE. Vedantu’s live sessions include more such shortcuts to help you build speed and accuracy.
Try These Yourself
- Solve \( 8 \times (5 + 9) \) using the distributive property.
- Expand \( 3(x + 2) + 4(x + 5) \) by distribution.
- Simplify \( 6 \times (y - 3) \) applying the property.
- Use distributive property to multiply \( 12 \times 24 \) by breaking 24 as 20 + 4.
Frequent Errors and Misunderstandings
- Forgetting to distribute the multiplier to every term inside the bracket.
- Confusing distributive with commutative or associative property.
- Missing negative signs when distributing over subtraction.
- Combining unlike terms after distribution by mistake.
Relation to Other Concepts
The idea of distributive property connects closely with topics such as commutative property and associative property. Mastering this helps with understanding more advanced concepts like algebraic expansion and solving equations.
Classroom Tip
A quick way to remember distributive property is to draw “arrows” from the term outside the bracket to every term inside. Vedantu’s teachers often use arrow diagrams and color-coding to simplify learning during live classes.
We explored distributive property—from definition, formula, examples, mistakes, and connections to other subjects. Continue practicing with Vedantu to become confident in solving problems using this concept.
Useful Vedantu Links
- Distributive Properties – For more solved examples & stepwise explanations.
- Commutative Property – To clearly see how distributive differs from commutative law.
- Properties of Multiplication of Integers – To explore distributive law with integers.
- Algebraic Expressions – See distributive property’s real use in expanding and simplifying algebra.
FAQs on Distributive Property Explained: Formula, Examples, and Applications
1. What is the distributive property in Maths?
The distributive property, also known as the distributive law, states that multiplying a number by a sum is the same as multiplying each addend by that number and then adding the products. It applies to both addition and subtraction. This means a × (b + c) = (a × b) + (a × c) and a × (b - c) = (a × b) - (a × c), where 'a', 'b', and 'c' can be numbers or algebraic expressions.
2. How do you use the distributive property in multiplication?
To use the distributive property in multiplication, you multiply the number outside the parentheses by each term inside the parentheses, then add or subtract the resulting products. For example, to solve 5 × (3 + 2), you would calculate (5 × 3) + (5 × 2) = 15 + 10 = 25. The same principle applies to subtraction: 4 × (7 - 2) = (4 × 7) - (4 × 2) = 28 - 8 = 20.
3. Can the distributive property be applied to division?
The distributive property directly applies to multiplication, not division. However, you can use it indirectly when dealing with fractions or expressions that involve division. For example, to simplify (12 + 6) ÷ 3, you can rewrite it as (12 ÷ 3) + (6 ÷ 3) = 4 + 2 = 6.
4. What is a real-life example of the distributive property?
Imagine buying 3 bags of apples and each bag contains 5 red apples and 2 green apples. To find the total number of apples, you can use the distributive property: 3 × (5 + 2) = (3 × 5) + (3 × 2) = 15 + 6 = 21 apples.
5. Why is the distributive property important for exams?
The distributive property is crucial for simplifying algebraic expressions, solving equations, and performing calculations quickly and efficiently. Mastering it improves problem-solving skills and reduces the chance of calculation errors during exams.
6. How does the distributive property help with mental math and quick computation?
The distributive property provides a shortcut for mental calculations. For example, to calculate 12 x 102, you can think of it as 12 x (100 + 2) = (12 x 100) + (12 x 2) = 1200 + 24 = 1224, making the multiplication easier.
7. Is the distributive property valid for subtraction or only addition?
Yes, the distributive property is valid for both addition and subtraction. It works the same way with subtraction as it does with addition, allowing you to distribute the multiplication over the terms within parentheses.
8. How does the distributive property relate to algebraic expansion like (x + y)(a + b)?
The distributive property is fundamental to algebraic expansion. To expand (x + y)(a + b), you distribute each term in the first parenthesis to each term in the second: x(a + b) + y(a + b) = xa + xb + ya + yb. This is a repeated application of the distributive property.
9. What are the most common mistakes students make when using the distributive property?
Common mistakes include forgetting to distribute the multiplication to every term inside the parentheses or incorrectly handling signs when distributing over subtraction. Carefully checking each step and paying close attention to signs are vital to avoid errors.
10. Are there situations where the distributive property cannot be used in arithmetic?
The distributive property fundamentally deals with multiplication distributed over addition or subtraction. There are no situations where it is intrinsically inapplicable; however, its direct use is limited to expressions involving multiplication and addition/subtraction.
11. What is the difference between the distributive, associative, and commutative properties?
The distributive property involves distributing multiplication over addition or subtraction. The associative property concerns the grouping of terms in addition or multiplication (order doesn't matter). The commutative property states that the order of numbers in addition or multiplication doesn't affect the result. These properties are distinct but often used together in simplifying expressions.
12. How do I use the distributive property with variables?
Using the distributive property with variables follows the same rules as with numbers. For example, to expand 3x(2x + 5), you multiply 3x by each term inside the parenthesis: (3x * 2x) + (3x * 5) = 6x² + 15x. Remember to combine like terms after distributing when possible.
