Simplifying 5.75 * 8.08 + 5.75 * 4.13 - 9 * 5.75 A Step-by-Step Guide

In this comprehensive guide, we will walk you through the process of simplifying the expression 5.75 * 8.08 + 5.75 * 4.13 - 9 * 5.75. This problem involves basic arithmetic operations, including multiplication and subtraction. By following the steps outlined below, you'll gain a clear understanding of how to solve similar mathematical expressions efficiently and accurately.

Understanding the Expression

Before diving into the calculations, let's break down the expression: 5.75 * 8.08 + 5.75 * 4.13 - 9 * 5.75. This expression consists of three terms:

1. 5. 75 * 8.08
2. 75 * 4.13
3. • 5.75

Each term involves multiplication, and they are connected by addition and subtraction operations. The key to simplifying this expression lies in recognizing the common factor among the terms and applying the distributive property. This approach will significantly reduce the complexity of the calculations and minimize the chances of making errors.

Identifying Common Factors

In this expression, the number 5.75 appears in all three terms. This common factor is crucial for simplifying the expression efficiently. By factoring out 5.75, we can rewrite the expression in a more manageable form. This technique is based on the distributive property of multiplication over addition and subtraction, which states that a(b + c) = ab + ac and a(b - c) = ab - ac. Recognizing common factors is a fundamental skill in algebra and arithmetic, enabling you to streamline calculations and solve problems more effectively.

Applying the Distributive Property

The distributive property allows us to factor out the common factor of 5.75 from the expression. This means we can rewrite the expression as: 5.75 * (8.08 + 4.13 - 9). By doing this, we've transformed the original expression into a simpler form that involves a single multiplication operation. The numbers inside the parentheses can be added and subtracted first, which reduces the overall complexity of the calculation. The distributive property is a powerful tool in mathematics, and understanding how to apply it can greatly simplify complex expressions and equations.

Step-by-Step Simplification

Now, let's proceed with the step-by-step simplification of the expression. We will focus on performing the operations inside the parentheses first, followed by the final multiplication.

Step 1: Combine Numbers Inside Parentheses

The first step is to add and subtract the numbers inside the parentheses: 8.08 + 4.13 - 9. Let's start by adding 8.08 and 4.13. 8. 08 + 4. 13 equals 12.21. Now, subtract 9 from 12.21. 12. 21 - 9 equals 3.21. So, the expression inside the parentheses simplifies to 3.21. Breaking down the calculation into smaller steps makes it easier to follow and reduces the likelihood of errors. This methodical approach is essential for solving mathematical problems accurately.

Step 2: Perform the Multiplication

Now that we've simplified the expression inside the parentheses to 3.21, the next step is to multiply this result by the common factor, 5.75. This means we need to calculate 5.75 * 3.21. To perform this multiplication, you can use a calculator or do it manually. Multiplying 5.75 by 3.21 gives us 18.4575. This is the final step in simplifying the expression. Double-checking your work ensures the accuracy of your result, especially in more complex calculations.

Step 3: Final Result

After performing the multiplication, we find that 5.75 * 3.21 = 18.4575. Therefore, the simplified value of the original expression 5.75 * 8.08 + 5.75 * 4.13 - 9 * 5.75 is 18.4575. This final answer is the culmination of the simplification process, which involved identifying common factors, applying the distributive property, and performing basic arithmetic operations. Presenting the final result clearly and concisely is important for effective communication of mathematical solutions.

Alternative Methods

While factoring out the common term 5.75 is the most efficient method for simplifying the given expression, there are alternative approaches you could take. Let's explore a couple of these methods to provide a broader understanding of how such problems can be tackled.

Method 1: Direct Calculation

One straightforward approach is to perform each multiplication separately and then combine the results. This involves calculating 5.75 * 8.08, 5.75 * 4.13, and 9 * 5.75 individually, and then adding and subtracting the products as indicated in the original expression. While this method does not take advantage of the distributive property, it can be useful for understanding the basic operations involved and verifying the final answer. Each multiplication can be performed using a calculator or by hand. However, this method can be more time-consuming and may increase the chances of making arithmetic errors, especially with decimal numbers.

Performing Individual Multiplications

First, calculate 5.75 * 8.08. This equals 46.46. Next, calculate 5.75 * 4.13, which equals 23.7475. Then, calculate 9 * 5.75, which equals 51.75. Now we have three values that need to be combined according to the original expression: 46.46, 23.7475, and 51.75. This step involves simple multiplication but requires careful attention to detail to avoid errors.

Combining the Results

Now, combine the results: 46.46 + 23.7475 - 51.75. Start by adding 46.46 and 23.7475, which gives 70.2075. Then, subtract 51.75 from 70.2075, resulting in 18.4575. This final result is the same as what we obtained using the distributive property method, confirming the accuracy of both approaches. This method highlights the importance of order of operations and the potential for arithmetic errors when dealing with multiple calculations.

Method 2: Using a Calculator Directly

The most direct method is to enter the entire expression into a calculator. This approach is quick and efficient, especially when dealing with complex expressions or decimal numbers. Modern calculators are designed to follow the correct order of operations, so you can input the expression exactly as it is written: 5.75 * 8.08 + 5.75 * 4.13 - 9 * 5.75. This method minimizes the risk of manual calculation errors and provides an immediate answer. However, it's still important to understand the underlying mathematical principles and double-check the result to ensure accuracy.

Entering the Expression

When using a calculator, carefully enter each number and operation in the correct order. Pay close attention to the placement of parentheses, if any, and the signs of the numbers. Most scientific calculators will correctly interpret the expression and provide the answer. Double-checking the input can help prevent errors caused by mistyping or misreading the expression. This step emphasizes the importance of careful data entry and the potential for human error even when using technology.

Verifying the Result

After entering the expression, the calculator should display the result, which should be 18.4575. This matches the result obtained using the distributive property and the direct calculation method, reinforcing the correctness of the answer. Verifying the result is a crucial step in problem-solving, ensuring that the solution is accurate and reliable. This approach underscores the value of using multiple methods to confirm answers and build confidence in your mathematical skills.

Conclusion

In conclusion, simplifying the expression 5.75 * 8.08 + 5.75 * 4.13 - 9 * 5.75 can be done efficiently by factoring out the common term 5.75 and applying the distributive property. This method simplifies the calculations and reduces the chance of errors. We've also explored alternative methods, such as direct calculation and using a calculator, which provide different perspectives on solving the problem. Understanding these various approaches enhances your problem-solving skills and provides a solid foundation for tackling more complex mathematical challenges. Whether you prefer to factor out common terms, perform calculations step by step, or use a calculator, the key is to approach each problem systematically and double-check your work to ensure accuracy. By mastering these techniques, you can confidently simplify and solve a wide range of mathematical expressions.