Distribute The Multiplication Across The Subtraction Problem:$\[ 10 \cdot (5 - 12) \\]

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Introduction

In mathematics, the distributive property is a fundamental concept that allows us to simplify complex expressions by distributing a single operation across multiple terms. In this article, we will explore how to distribute the multiplication across a subtraction problem, and provide step-by-step examples to illustrate the concept.

Understanding the Distributive Property

The distributive property states that for any real numbers a, b, and c:

a(b + c) = ab + ac

This property can be applied to both addition and subtraction, and is a crucial tool for simplifying complex expressions.

Distributing Multiplication Across a Subtraction Problem

When we encounter a problem that involves distributing multiplication across a subtraction, we can use the distributive property to simplify the expression. Let's consider the following example:

10⋅(5−12){ 10 \cdot (5 - 12) }

To distribute the multiplication across the subtraction, we need to follow these steps:

  1. Apply the distributive property: We can rewrite the expression as:

10⋅5−10⋅12{ 10 \cdot 5 - 10 \cdot 12 }

  1. Simplify the expression: Now, we can simplify the expression by multiplying 10 with 5 and 12:

50−120{ 50 - 120 }

  1. Perform the subtraction: Finally, we can perform the subtraction to get the final result:

−70{ -70 }

Step-by-Step Example

Let's consider another example to illustrate the concept:

15⋅(8−3){ 15 \cdot (8 - 3) }

To distribute the multiplication across the subtraction, we can follow the same steps:

  1. Apply the distributive property: We can rewrite the expression as:

15⋅8−15⋅3{ 15 \cdot 8 - 15 \cdot 3 }

  1. Simplify the expression: Now, we can simplify the expression by multiplying 15 with 8 and 3:

120−45{ 120 - 45 }

  1. Perform the subtraction: Finally, we can perform the subtraction to get the final result:

75{ 75 }

Real-World Applications

The distributive property has numerous real-world applications in various fields, including:

  • Algebra: The distributive property is used extensively in algebra to simplify complex expressions and solve equations.
  • Geometry: The distributive property is used to simplify expressions involving lengths and areas of shapes.
  • Physics: The distributive property is used to simplify expressions involving forces and energies.

Conclusion

In conclusion, distributing the multiplication across a subtraction problem is a fundamental concept in mathematics that can be applied to simplify complex expressions. By following the steps outlined in this article, you can master the distributive property and apply it to a wide range of problems. Whether you're a student or a professional, the distributive property is an essential tool that can help you simplify complex expressions and solve problems with ease.

Common Mistakes to Avoid

When distributing the multiplication across a subtraction problem, there are several common mistakes to avoid:

  • Forgetting to apply the distributive property: Make sure to apply the distributive property to simplify the expression.
  • Not simplifying the expression: Make sure to simplify the expression by multiplying the numbers.
  • Not performing the subtraction: Make sure to perform the subtraction to get the final result.

Practice Problems

To practice distributing the multiplication across a subtraction problem, try the following problems:

  • 20â‹…(7−2){ 20 \cdot (7 - 2) }
  • 30â‹…(9−4){ 30 \cdot (9 - 4) }
  • 40â‹…(6−1){ 40 \cdot (6 - 1) }

Answer Key

  • 20â‹…(7−2)=20â‹…5=100{ 20 \cdot (7 - 2) = 20 \cdot 5 = 100 }
  • 30â‹…(9−4)=30â‹…5=150{ 30 \cdot (9 - 4) = 30 \cdot 5 = 150 }
  • 40â‹…(6−1)=40â‹…5=200{ 40 \cdot (6 - 1) = 40 \cdot 5 = 200 }

Introduction

In our previous article, we explored how to distribute the multiplication across a subtraction problem using the distributive property. In this article, we will answer some frequently asked questions (FAQs) related to this concept.

Q: What is the distributive property?

A: The distributive property is a fundamental concept in mathematics that allows us to simplify complex expressions by distributing a single operation across multiple terms. It states that for any real numbers a, b, and c:

a(b + c) = ab + ac

Q: How do I apply the distributive property to a subtraction problem?

A: To apply the distributive property to a subtraction problem, follow these steps:

  1. Apply the distributive property: Rewrite the expression as:

a(b - c) = ab - ac

  1. Simplify the expression: Multiply the numbers.

  2. Perform the subtraction: Subtract the second term from the first term.

Q: What if the problem involves a negative number?

A: If the problem involves a negative number, you can still apply the distributive property. For example:

10⋅(−5−2){ 10 \cdot (-5 - 2) }

To solve this problem, follow the same steps:

  1. Apply the distributive property: Rewrite the expression as:

10⋅(−5)−10⋅2{ 10 \cdot (-5) - 10 \cdot 2 }

  1. Simplify the expression: Multiply the numbers.

  2. Perform the subtraction: Subtract the second term from the first term.

Q: Can I apply the distributive property to a problem with multiple subtractions?

A: Yes, you can apply the distributive property to a problem with multiple subtractions. For example:

10⋅(5−2−3){ 10 \cdot (5 - 2 - 3) }

To solve this problem, follow the same steps:

  1. Apply the distributive property: Rewrite the expression as:

10⋅(5−2)−10⋅3{ 10 \cdot (5 - 2) - 10 \cdot 3 }

  1. Simplify the expression: Multiply the numbers.

  2. Perform the subtraction: Subtract the second term from the first term.

Q: What if the problem involves a fraction?

A: If the problem involves a fraction, you can still apply the distributive property. For example:

1/2⋅(3−2){ 1/2 \cdot (3 - 2) }

To solve this problem, follow the same steps:

  1. Apply the distributive property: Rewrite the expression as:

1/2⋅3−1/2⋅2{ 1/2 \cdot 3 - 1/2 \cdot 2 }

  1. Simplify the expression: Multiply the numbers.

  2. Perform the subtraction: Subtract the second term from the first term.

Q: Can I apply the distributive property to a problem with multiple fractions?

A: Yes, you can apply the distributive property to a problem with multiple fractions. For example:

1/2⋅(3−2/3){ 1/2 \cdot (3 - 2/3) }

To solve this problem, follow the same steps:

  1. Apply the distributive property: Rewrite the expression as:

1/2⋅3−1/2⋅2/3{ 1/2 \cdot 3 - 1/2 \cdot 2/3 }

  1. Simplify the expression: Multiply the numbers.

  2. Perform the subtraction: Subtract the second term from the first term.

Conclusion

In conclusion, the distributive property is a powerful tool that can be used to simplify complex expressions involving subtractions. By following the steps outlined in this article and practicing the problems, you can master the distributive property and apply it to a wide range of problems.

Practice Problems

To practice distributing the multiplication across a subtraction problem, try the following problems:

  • 20â‹…(7−2){ 20 \cdot (7 - 2) }
  • 30â‹…(9−4){ 30 \cdot (9 - 4) }
  • 40â‹…(6−1){ 40 \cdot (6 - 1) }
  • 1/2â‹…(3−2){ 1/2 \cdot (3 - 2) }
  • 1/2â‹…(3−2/3){ 1/2 \cdot (3 - 2/3) }

Answer Key

  • 20â‹…(7−2)=20â‹…5=100{ 20 \cdot (7 - 2) = 20 \cdot 5 = 100 }
  • 30â‹…(9−4)=30â‹…5=150{ 30 \cdot (9 - 4) = 30 \cdot 5 = 150 }
  • 40â‹…(6−1)=40â‹…5=200{ 40 \cdot (6 - 1) = 40 \cdot 5 = 200 }
  • 1/2â‹…(3−2)=1/2â‹…1=1/2{ 1/2 \cdot (3 - 2) = 1/2 \cdot 1 = 1/2 }
  • 1/2â‹…(3−2/3)=1/2â‹…7/3=7/6{ 1/2 \cdot (3 - 2/3) = 1/2 \cdot 7/3 = 7/6 }

By following the steps outlined in this article and practicing the problems, you can master the distributive property and apply it to a wide range of problems.