Solve The Following Problem Using Decimal Numbers And The Order Of Operations:${ \frac{19.75 + 0.25 + \left( \frac{0.5 \times 10.05}{0.001} \right) - 5025}{20} }$

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Introduction

In mathematics, solving complex expressions often requires a deep understanding of the order of operations and decimal numbers. In this article, we will explore how to solve a given problem using decimal numbers and the order of operations. We will break down the problem into smaller steps, making it easier to understand and follow along.

The Problem

The given problem is:

19.75+0.25+(0.5×10.050.001)−502520{ \frac{19.75 + 0.25 + \left( \frac{0.5 \times 10.05}{0.001} \right) - 5025}{20} }

This problem involves addition, subtraction, multiplication, and division of decimal numbers, as well as the use of parentheses to group expressions. Our goal is to simplify this expression and find the final result.

Step 1: Evaluate the Expression Inside the Parentheses

The first step is to evaluate the expression inside the parentheses:

0.5×10.050.001{ \frac{0.5 \times 10.05}{0.001} }

To do this, we need to follow the order of operations (PEMDAS):

  1. Multiply 0.5 and 10.05: 0.5 * 10.05 = 5.0255
  2. Divide the result by 0.001: 5.0255 / 0.001 = 5025.5

So, the expression inside the parentheses evaluates to 5025.5.

Step 2: Add and Subtract the Decimal Numbers

Now that we have evaluated the expression inside the parentheses, we can add and subtract the decimal numbers:

19.75+0.25+5025.5−5025{ 19.75 + 0.25 + 5025.5 - 5025 }

To do this, we need to follow the order of operations (PEMDAS):

  1. Add 19.75 and 0.25: 19.75 + 0.25 = 20
  2. Add 5025.5 to the result: 20 + 5025.5 = 5045.5
  3. Subtract 5025 from the result: 5045.5 - 5025 = 20.5

So, the result of adding and subtracting the decimal numbers is 20.5.

Step 3: Divide the Result by 20

Finally, we need to divide the result by 20:

20.520{ \frac{20.5}{20} }

To do this, we can simply divide 20.5 by 20:

20.5 / 20 = 1.025

Conclusion

In this article, we solved a complex mathematical expression using decimal numbers and the order of operations. We broke down the problem into smaller steps, making it easier to understand and follow along. By following the order of operations (PEMDAS) and using parentheses to group expressions, we were able to simplify the expression and find the final result.

Tips and Tricks

  • When working with decimal numbers, make sure to follow the order of operations (PEMDAS) to avoid errors.
  • Use parentheses to group expressions and make it easier to follow the order of operations.
  • When dividing decimal numbers, make sure to use the correct decimal places to avoid errors.

Common Mistakes

  • Failing to follow the order of operations (PEMDAS) can lead to errors in complex mathematical expressions.
  • Not using parentheses to group expressions can make it difficult to follow the order of operations.
  • Not using the correct decimal places when dividing decimal numbers can lead to errors.

Real-World Applications

Solving complex mathematical expressions using decimal numbers and the order of operations has many real-world applications, including:

  • Finance: Calculating interest rates and investment returns
  • Science: Measuring physical quantities and calculating results
  • Engineering: Designing and optimizing systems and structures

Introduction

In our previous article, we explored how to solve a complex mathematical expression using decimal numbers and the order of operations. In this article, we will answer some frequently asked questions (FAQs) related to solving complex mathematical expressions.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS is commonly used to remember the order of operations:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next (e.g., 2^3).
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: Why is it important to follow the order of operations?

A: Following the order of operations is crucial to avoid errors and ensure that mathematical expressions are evaluated correctly. If we don't follow the order of operations, we may get incorrect results or even change the meaning of the expression.

Q: How do I evaluate expressions inside parentheses?

A: To evaluate expressions inside parentheses, we need to follow the order of operations within the parentheses. We can use the same rules as above (PEMDAS) to evaluate the expressions inside the parentheses.

Q: What is the difference between addition and subtraction?

A: Addition and subtraction are both binary operations that take two numbers as input and produce a result. The main difference between addition and subtraction is that addition combines two numbers, while subtraction finds the difference between two numbers.

Q: How do I handle decimal numbers in mathematical expressions?

A: When working with decimal numbers, we need to follow the same rules as above (PEMDAS). We can use decimal numbers in mathematical expressions just like we use integers. However, we need to be careful when performing operations with decimal numbers, as small errors can lead to significant differences in the result.

Q: Can I use a calculator to solve complex mathematical expressions?

A: Yes, you can use a calculator to solve complex mathematical expressions. However, it's essential to understand the underlying mathematics and the order of operations to ensure that you're using the calculator correctly.

Q: What are some common mistakes to avoid when solving complex mathematical expressions?

A: Some common mistakes to avoid when solving complex mathematical expressions include:

  • Failing to follow the order of operations (PEMDAS)
  • Not using parentheses to group expressions
  • Not using the correct decimal places when dividing decimal numbers
  • Not checking the result for errors or inconsistencies

Q: How can I apply the skills I've learned to real-world problems?

A: The skills you've learned can be applied to a wide range of real-world problems, including:

  • Finance: Calculating interest rates and investment returns
  • Science: Measuring physical quantities and calculating results
  • Engineering: Designing and optimizing systems and structures

By understanding how to solve complex mathematical expressions using decimal numbers and the order of operations, you can apply this knowledge to a wide range of real-world problems and make informed decisions.

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to solving complex mathematical expressions. We covered topics such as the order of operations, evaluating expressions inside parentheses, and handling decimal numbers. By understanding these concepts and avoiding common mistakes, you can become proficient in solving complex mathematical expressions and apply this knowledge to real-world problems.