Example Of Distributive Property Of Multiplication Over Addition

Examples of the Distributive Property of Multiplication Over Addition

The distributive property is a fundamental concept in mathematics, particularly in algebra. It states that multiplying a number by a sum is the same as multiplying the number by each term in the sum and then adding the products. This property holds true for both whole numbers, fractions, decimals, and even more complex algebraic expressions. Understanding this property is crucial for simplifying expressions and solving equations efficiently. This article will explore the distributive property with numerous examples, showcasing its application across various mathematical contexts.

Understanding the Distributive Property

The distributive property of multiplication over addition can be formally expressed as:

a(b + c) = ab + ac

Where 'a', 'b', and 'c' represent any numbers (integers, fractions, decimals, variables, etc.). The property essentially states that we can "distribute" the multiplication across the terms within the parentheses.

Let's break this down:

• a(b + c): This represents a number 'a' multiplied by the sum of two numbers, 'b' and 'c'.
• ab + ac: This represents the product of 'a' and 'b' added to the product of 'a' and 'c'.

The distributive property also works with subtraction:

a(b - c) = ab - ac

This means that multiplying a number by a difference is the same as multiplying the number by each term in the difference and then subtracting the products.

Examples with Whole Numbers

Let's start with some simple examples using whole numbers to solidify the understanding of the distributive property.

Example 1:

3(4 + 5) = ?

Using the distributive property:

3(4 + 5) = 3(4) + 3(5) = 12 + 15 = 27

We can verify this by first adding the numbers in the parentheses:

3(4 + 5) = 3(9) = 27

Both methods yield the same result, demonstrating the validity of the distributive property.

Example 2:

5(6 - 2) = ?

Using the distributive property:

5(6 - 2) = 5(6) - 5(2) = 30 - 10 = 20

Verifying the result:

5(6 - 2) = 5(4) = 20

Again, both methods produce the same answer.

Example 3:

A slightly more complex example:

2(3 + 4 + 5) = ?

Applying the distributive property:

2(3 + 4 + 5) = 2(3) + 2(4) + 2(5) = 6 + 8 + 10 = 24

Verifying:

2(3 + 4 + 5) = 2(12) = 24

Examples with Fractions and Decimals

The distributive property works seamlessly with fractions and decimals as well.

Example 4:

½(4 + 6) = ?

Using the distributive property:

½(4 + 6) = ½(4) + ½(6) = 2 + 3 = 5

Verifying:

½(4 + 6) = ½(10) = 5

Example 5:

0.2(5 + 10) = ?

Using the distributive property:

0.2(5 + 10) = 0.2(5) + 0.2(10) = 1 + 2 = 3

Verifying:

0.2(5 + 10) = 0.2(15) = 3

Examples with Variables

The power of the distributive property truly shines when dealing with algebraic expressions containing variables.

Example 6:

x(y + z) = ?

Using the distributive property:

x(y + z) = xy + xz

This cannot be simplified further unless we have specific values for x, y, and z.

Example 7:

3(2x + 5) = ?

Using the distributive property:

3(2x + 5) = 3(2x) + 3(5) = 6x + 15

Example 8:

-2(4x - 3y) = ?

Using the distributive property (remembering the rules of multiplying with negative numbers):

-2(4x - 3y) = -2(4x) - 2(-3y) = -8x + 6y

Example 9: Factoring using the Distributive Property

The distributive property isn't just for expanding expressions; it's also crucial for factoring. Factoring is the reverse process of expanding. Let's say we have the expression 4x + 8. Notice that both terms, 4x and 8, share a common factor of 4. Using the distributive property in reverse, we can factor out the 4:

4x + 8 = 4(x + 2)

More Complex Examples

Let's tackle some examples involving more complex expressions.

Example 10:

2x(3x² + 4x - 1) = ?

Using the distributive property:

2x(3x² + 4x - 1) = 2x(3x²) + 2x(4x) + 2x(-1) = 6x³ + 8x² - 2x

Example 11:

(x + 2)(x + 3) = ?

This example involves multiplying two binomials. We can use the distributive property (often referred to as the FOIL method – First, Outer, Inner, Last) to expand this expression:

(x + 2)(x + 3) = x(x + 3) + 2(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6

Example 12:

(2x - 1)(x² + 3x + 2) = ?

This is a binomial multiplied by a trinomial. We apply the distributive property systematically:

(2x - 1)(x² + 3x + 2) = 2x(x² + 3x + 2) - 1(x² + 3x + 2) = 2x³ + 6x² + 4x - x² - 3x - 2 = 2x³ + 5x² + x - 2

Real-World Applications

The distributive property isn't just a theoretical concept; it has practical applications in many real-world scenarios. For example:

• Calculating discounts: If a store offers a 20% discount on all items in a particular section, and you buy two items priced at $25 and $30, you can use the distributive property to calculate the total discount: 0.20($25 + $30) = 0.20($55) = $11.
• Calculating area: If you have a rectangular garden with one side measuring (x + 2) meters and the other side measuring 5 meters, the area is calculated as 5(x + 2) = 5x + 10 square meters.
• Financial calculations: The distributive property is frequently used in compound interest calculations and other financial formulas.

Conclusion

The distributive property is a cornerstone of algebra and arithmetic. Its applications extend far beyond simple numerical calculations, proving invaluable in simplifying complex expressions, solving equations, and understanding various real-world phenomena. By mastering this property, you equip yourself with a powerful tool for tackling mathematical challenges with greater efficiency and confidence. Practicing with diverse examples, as demonstrated in this article, is key to solidifying your understanding and building a strong foundation in mathematics. Remember to always check your work by solving the problem using alternative methods to ensure accuracy. Consistent practice will make you proficient in applying the distributive property in various contexts.