Step-by-Step Guide Calculating Mathematical Expressions

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Hey guys! Today, we're diving into the exciting world of mathematical calculations. We'll be breaking down two complex expressions step by step, making sure you understand every single operation involved. So, grab your calculators and let's get started!

1) Solving the First Expression: (22.83 + 73.41) ÷ 4.8 + 24.48 ÷ (65.41 – 63.01)

In this section, we'll tackle the first expression: (22.83 + 73.41) ÷ 4.8 + 24.48 ÷ (65.41 – 63.01). This expression involves a mix of addition, subtraction, and division, so we need to follow the order of operations (PEMDAS/BODMAS) to get the correct answer. Our main keywords here are order of operations, mathematical expressions, and step-by-step calculations. Let’s break it down:

Step 1: Parentheses/Brackets

First, we need to handle the expressions inside the parentheses. This is where we begin to untangle the mathematical puzzle, focusing intently on the order of operations to ensure our calculations are precise and accurate. Let's calculate the sums and differences within the parentheses:

  • (22.83 + 73.41): This is a straightforward addition. When we add these two numbers, we get:

    1. 83 + 73.41 = 96.24

  • So, the sum of 22.83 and 73.41 is 96.24. This result becomes a key component in our next steps, and it highlights the importance of accurate addition in solving complex mathematical expressions.

  • (65.41 – 63.01): Next, we subtract 63.01 from 65.41:

    1. 41 – 63.01 = 2.4

  • The difference between 65.41 and 63.01 is 2.4. This subtraction is crucial as it simplifies a part of the expression, setting the stage for the division operations that follow. By performing this subtraction correctly, we maintain the integrity of the mathematical sequence.

    By completing these operations inside the parentheses, we've simplified the original expression significantly. This is a critical step in solving any mathematical problem that involves multiple operations, as it reduces complexity and paves the way for subsequent calculations. Now, with the parentheses handled, we can move on to the next set of operations, keeping in mind the order of operations to ensure a correct final answer.

Step 2: Division Operations

After resolving the parentheses, we move on to the division operations. Division is a fundamental arithmetic operation that helps us break down numbers into smaller parts or determine how many times one number is contained within another. In our expression, we have two division operations to tackle, each of which will contribute to the final result. These divisions need to be performed accurately to maintain the correctness of the overall calculation. Let’s dive into the specifics:

  • 96.24 ÷ 4.8: This is the first division we need to perform. Dividing 96.24 by 4.8 gives us:

    1. 24 ÷ 4.8 = 20.05

  • So, when we divide 96.24 by 4.8, the result is 20.05. This result is a critical intermediate value that we will use in the final step of our calculation. Accurate division here is paramount, as it directly impacts the accuracy of the final answer. This division helps to simplify the overall expression, moving us closer to the solution.

  • 24.48 ÷ 2.4: Next, we divide 24.48 by 2.4:

    1. 48 ÷ 2.4 = 10.2

  • The result of dividing 24.48 by 2.4 is 10.2. This is another key intermediate value that we need to consider. Correct execution of this division is essential, as it contributes significantly to the final result. By performing this division, we further reduce the complexity of the expression, making it easier to manage in the subsequent addition.

    Performing these division operations accurately is crucial because each result contributes significantly to the final answer. Division, like any other arithmetic operation, follows specific rules and requires attention to detail. By handling each division step carefully, we ensure that the final result is correct. Now that we have completed the divisions, we are ready to perform the final operation: addition.

Step 3: Addition

Now that we've handled the parentheses and division, we're left with a simple addition operation. Addition is one of the most basic arithmetic operations, but it's crucial for combining the results we've obtained so far. In this final step, we'll add the results from our two division operations to arrive at the solution to the first part of our mathematical problem. This step is a straightforward combination of the values we've carefully calculated in the preceding steps, and it's essential for reaching the final answer. Let’s perform this addition:

  • We need to add the results from our divisions:

    1. 05 + 10.2

  • Adding these two numbers together gives us:

    1. 05 + 10.2 = 30.25

  • Therefore, the final result of the first expression is 30.25. This is the culmination of all the previous steps, where we meticulously followed the order of operations to ensure an accurate answer. The addition step brings together the results of our divisions, providing the ultimate solution to the complex mathematical expression.

    So, by adding 20.05 and 10.2, we arrive at the final answer of 30.25. This result shows the value of carefully following each step and ensuring accuracy in each calculation. Addition serves as the final step in combining all the intermediate results to produce the conclusive answer. Congratulations, guys, we've solved the first expression!

2) Solving the Second Expression: (24 ÷ 1.2) – (65 ÷ 1.3) – (7.5 ÷ 2.5) – (2.5 ÷ 0.1)

Next up, we have the second expression: (24 ÷ 1.2) – (65 ÷ 1.3) – (7.5 ÷ 2.5) – (2.5 ÷ 0.1). This expression is primarily made up of division and subtraction. Our main keywords for this section are division and subtraction, arithmetic operations, and expression evaluation. Let’s break it down step by step:

Step 1: Division Operations

This expression involves several division operations that need to be tackled first. Remember, the order of operations dictates that we perform divisions before subtractions. Each division will simplify the expression, bringing us closer to the final answer. Let's carefully calculate each division:

  • (24 ÷ 1.2): Let's start by dividing 24 by 1.2:

    24 ÷ 1. 2 = 20

  • The result of dividing 24 by 1.2 is 20. This value will be used in the subsequent subtraction operations. Accurate division is crucial here, as it directly impacts the overall outcome. This division simplifies the first part of the expression, making it easier to manage.

  • (65 ÷ 1.3): Next, we divide 65 by 1.3:

    65 ÷ 1. 3 = 50

  • When we divide 65 by 1.3, we get 50. This result is another key component that we will subtract later. The precision of this division is essential for maintaining the integrity of the calculation. Each division we perform brings us closer to the final solution.

  • (7.5 ÷ 2.5): Now, let's divide 7.5 by 2.5:

    1. 5 ÷ 2.5 = 3

  • Dividing 7.5 by 2.5 gives us 3. This value is yet another piece of the puzzle that needs to be subtracted. Consistent accuracy in each division is important for a correct final result. This step further simplifies the expression, making it easier to handle in the subtraction phase.

  • (2.5 ÷ 0.1): Finally, we divide 2.5 by 0.1:

    1. 5 ÷ 0.1 = 25

  • The result of dividing 2.5 by 0.1 is 25. This is the last division operation in our expression. Completing this division sets the stage for the final subtraction steps. Accurate execution of each division ensures that the final answer is correct.

    By performing these division operations carefully, we've simplified the original expression into a series of subtractions. Each division provides a value that will be used in the subsequent subtraction steps. The importance of accurate division cannot be overstated, as each result directly influences the final answer. Now that we have completed the division, we can move on to the subtraction operations.

Step 2: Subtraction Operations

With all the division operations completed, we now move on to the subtraction operations. Subtraction is a fundamental arithmetic operation that involves finding the difference between two numbers. In this expression, we have a series of subtractions to perform. It’s important to proceed from left to right to ensure we follow the correct order of operations. Each subtraction will bring us closer to the final result. Let's perform the subtractions step by step:

  • First, we subtract 50 from 20:

    20 – 50 = -30

  • Subtracting 50 from 20 gives us -30. This result is the first step in our series of subtractions. It's crucial to handle negative numbers carefully as we proceed. Accurate subtraction here sets the foundation for the next steps.

  • Next, we subtract 3 from -30:

    -30 – 3 = -33

  • Subtracting 3 from -30 results in -33. This step continues to simplify the expression. Maintaining accuracy in each subtraction is vital for a correct final answer. By correctly performing this subtraction, we move closer to the final solution.

  • Finally, we subtract 25 from -33:

    -33 – 25 = -58

  • Subtracting 25 from -33 gives us -58. This is the final subtraction operation, and it brings us to the solution. The cumulative effect of all our subtractions leads us to this result. Each subtraction has been performed carefully to ensure the final answer is accurate.

    So, the final result of the second expression is -58. This outcome is the culmination of all the division and subtraction operations we've performed. By following each step meticulously, we've arrived at the correct answer. Subtraction, like other arithmetic operations, requires precision, and by handling each step with care, we ensure the accuracy of our result. Great job, guys! We've successfully solved the second expression.

Conclusion

Alright, guys! We've successfully calculated both expressions by breaking them down step by step. Remember, the key to solving complex mathematical expressions is to follow the order of operations and take your time. Keep practicing, and you'll become math whizzes in no time! Whether it’s dealing with parentheses, divisions, additions, or subtractions, the approach remains the same: break down the problem, handle each part methodically, and combine the results accurately. Mathematical expressions might seem daunting at first, but with a bit of practice and a systematic approach, you can solve even the most complex problems. Keep up the great work, and happy calculating!