Applying the Multiplication Property of Distributivity

Oct 13, 2025 by miodeya.com 66 views

Introduction

In mathematics, the number sentence $6 \times 7 = (6 \times 5) + (6 \times 2)$ is a classic example of the multiplication property of distributivity. This property is a fundamental concept in arithmetic and algebra, and it plays a crucial role in simplifying complex mathematical expressions. In this article, we will delve into the details of the multiplication property of distributivity, and explore how it is applied in the given number sentence.

What is the Multiplication Property of Distributivity?

The multiplication property of distributivity is a mathematical property that states that the product of a number and a sum is equal to the sum of the products of the number and each of the addends. In other words, if we have a number $a$ and two numbers $b$ and $c$ , then we can write:

$a \times (b + c) = a \times b + a \times c$

This property is also known as the distributive property of multiplication over addition.

Applying the Multiplication Property of Distributivity

Now, let's apply the multiplication property of distributivity to the number sentence $6 \times 7 = (6 \times 5) + (6 \times 2)$ . We can see that the left-hand side of the equation is a product of two numbers, $6$ and $7$ . On the right-hand side, we have a sum of two products, $6 \times 5$ and $6 \times 2$ .

Using the multiplication property of distributivity, we can rewrite the right-hand side of the equation as:

$6 \times (5 + 2) = 6 \times 5 + 6 \times 2$

Now, we can simplify the expression inside the parentheses:

$6 \times (5 + 2) = 6 \times 7$

As we can see, the left-hand side and the right-hand side of the equation are now equal. This demonstrates that the multiplication property of distributivity is indeed applicable to the number sentence $6 \times 7 = (6 \times 5) + (6 \times 2)$ .

Why is the Multiplication Property of Distributivity Important?

The multiplication property of distributivity is an essential concept in mathematics because it allows us to simplify complex mathematical expressions. By applying this property, we can break down a product of two numbers into a sum of products, which can make it easier to solve mathematical problems.

For example, consider the expression $3 \times (2 + 4)$ . Using the multiplication property of distributivity, we can rewrite this expression as:

$3 \times (2 + 4) = 3 \times 2 + 3 \times 4$

This simplifies the expression and makes it easier to evaluate.

Real-World Applications of the Multiplication Property of Distributivity

The multiplication property of distributivity has numerous real-world applications in various fields, including:

Business: In business, the multiplication property of distributivity is used to calculate the total cost of a product or service. For example, if a company sells a product for $10 and the cost of production is $5, then the profit is $5. If the company sells 10 units of the product, then the total profit is $50.
Science: In science, the multiplication property of distributivity is used to calculate the total amount of a substance that is present in a mixture. For example, if a mixture contains 20% of a substance and the total mixture is 100 grams, then the amount of the substance present is 20 grams.
Engineering: In engineering, the multiplication property of distributivity is used to calculate the total force or torque that is applied to an object. For example, if a force of 10 Newtons is applied to an object and the object is moved 5 meters, then the total work done is 50 Joules.

Conclusion

In conclusion, the number sentence $6 \times 7 = (6 \times 5) + (6 \times 2)$ is a classic example of the multiplication property of distributivity. This property is a fundamental concept in arithmetic and algebra, and it plays a crucial role in simplifying complex mathematical expressions. By applying the multiplication property of distributivity, we can break down a product of two numbers into a sum of products, which can make it easier to solve mathematical problems. The real-world applications of the multiplication property of distributivity are numerous, and it is an essential concept in various fields, including business, science, and engineering.

References

"Algebra" by Michael Artin
"Mathematics for the Nonmathematician" by Morris Kline
"The Joy of Mathematics" by Alfred S. Posamentier

Q: What is the multiplication property of distributivity?

A: The multiplication property of distributivity is a mathematical property that states that the product of a number and a sum is equal to the sum of the products of the number and each of the addends. In other words, if we have a number $a$ and two numbers $b$ and $c$ , then we can write:

$a \times (b + c) = a \times b + a \times c$

Q: How is the multiplication property of distributivity used in real-life situations?

A: The multiplication property of distributivity is used in various real-life situations, such as:

Business: To calculate the total cost of a product or service.
Science: To calculate the total amount of a substance that is present in a mixture.
Engineering: To calculate the total force or torque that is applied to an object.

Q: Can you give an example of how the multiplication property of distributivity is used in a real-life situation?

A: Consider a business scenario where a company sells a product for $10 and the cost of production is $5. If the company sells 10 units of the product, then the total profit is $50. Using the multiplication property of distributivity, we can calculate the total profit as follows:

$10 \times (10 + 0) = 10 \times 10 + 10 \times 0$ $= 100 + 0$ $= 100$

Q: How is the multiplication property of distributivity related to other mathematical concepts?

A: The multiplication property of distributivity is related to other mathematical concepts, such as:

Addition: The multiplication property of distributivity is the opposite of the addition property of multiplication.
Subtraction: The multiplication property of distributivity can be used to simplify expressions involving subtraction.
Exponents: The multiplication property of distributivity can be used to simplify expressions involving exponents.

Q: Can you give an example of how the multiplication property of distributivity is used to simplify an expression involving exponents?

A: Consider the expression $2^3 \times (2^2 + 2^1)$ . Using the multiplication property of distributivity, we can simplify this expression as follows:

$2^3 \times (2^2 + 2^1) = 2^3 \times 2^2 + 2^3 \times 2^1$ $= 2^{3+2} + 2^{3+1}$ $= 2^5 + 2^4$

Q: What are some common mistakes to avoid when using the multiplication property of distributivity?

A: Some common mistakes to avoid when using the multiplication property of distributivity include:

Not distributing the multiplication correctly: Make sure to distribute the multiplication to each term in the expression.
Not simplifying the expression correctly: Make sure to simplify the expression correctly after applying the multiplication property of distributivity.
Not checking the units correctly: Make sure to check the units correctly after applying the multiplication property of distributivity.

Q: How can I practice using the multiplication property of distributivity?

A: You can practice using the multiplication property of distributivity by:

Solving problems involving the multiplication property of distributivity: Try solving problems that involve the multiplication property of distributivity, such as those found in math textbooks or online resources.
Using online resources: Use online resources, such as math websites or apps, to practice using the multiplication property of distributivity.
Working with a tutor or teacher: Work with a tutor or teacher to practice using the multiplication property of distributivity and get feedback on your work.

Conclusion

In conclusion, the multiplication property of distributivity is a fundamental concept in mathematics that is used to simplify complex expressions. By understanding and applying this property, you can solve a wide range of mathematical problems and make sense of the world around you. Remember to practice using the multiplication property of distributivity regularly to build your skills and confidence.