Calculating The Value Of An Expression And Expressing As A Percentage A Step-by-Step Guide

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Understanding the Problem

At the heart of this mathematical exercise lies the task of evaluating a numerical expression that combines fractions, decimals, and mixed numbers. Calculating the value of such an expression requires a systematic approach, a keen understanding of arithmetic operations, and the ability to convert between different forms of numerical representation. The expression we're tasked with solving is:

  • 12510+3.75−212{ \frac{125}{10} + 3.75 - 2\frac{1}{2} }

This expression incorporates a fraction (12510{ \frac{125}{10} }), a decimal (3.75), and a mixed number (212{ 2\frac{1}{2} }). To solve it effectively, we need to unify these different forms, typically by converting them all into either decimals or fractions. Once we've obtained a single numerical value for the expression, the problem further challenges us to express this value as a percentage. This involves multiplying the value by 100 and appending the percentage symbol (%). The multiple-choice nature of the question—with options of 250, 300, 150, 200, and 100—adds another layer to the problem-solving process. It compels us not only to arrive at the correct numerical answer but also to verify that our calculated value corresponds to one of the provided options. This aspect emphasizes the importance of accuracy and precision in our calculations. The question touches upon several fundamental mathematical concepts, including fraction manipulation, decimal arithmetic, mixed number conversions, and percentage calculations. It serves as a comprehensive assessment of one's grasp on these concepts and the ability to apply them in a combined, sequential manner. The practical relevance of this type of problem is also noteworthy. In everyday situations, we frequently encounter scenarios where we need to combine different numerical quantities, such as in financial calculations, measurements, and proportions. Mastering the skills required to solve this problem, therefore, has real-world implications and enhances one's quantitative literacy.

Step-by-Step Solution

To solve the expression 12510+3.75−212{ \frac{125}{10} + 3.75 - 2\frac{1}{2} }, we'll break it down into manageable steps. The primary goal is to calculate the value of the expression by converting all terms into a common format—in this case, decimals—and then performing the arithmetic operations. The final step involves converting the result into a percentage.

1. Convert the Fraction to a Decimal:

The first term in the expression is the fraction 12510{ \frac{125}{10} }. To convert this fraction to a decimal, we perform the division: 125 divided by 10. This yields a result of 12.5.

2. Convert the Mixed Number to a Decimal:

The third term is the mixed number 212{ 2\frac{1}{2} }. A mixed number consists of a whole number part (2 in this case) and a fractional part (12{ \frac{1}{2} }). To convert it to a decimal, we focus on the fractional part. The fraction 12{ \frac{1}{2} } is equivalent to 0.5. Adding this to the whole number part gives us 2 + 0.5 = 2.5. So, the mixed number 212{ 2\frac{1}{2} } is equal to 2.5 in decimal form.

3. Rewrite the Expression in Decimal Form:

Now that we've converted all terms to decimals, we can rewrite the original expression as:

    1. 5 + 3.75 - 2.5

This step simplifies the calculation process as we're now dealing with only decimal numbers.

4. Perform the Addition:

Next, we perform the addition operation. We add 12.5 and 3.75. This can be done manually or with a calculator. Adding these two numbers gives us 16.25.

5. Perform the Subtraction:

Now, we subtract 2.5 from the result of the addition (16.25). Subtracting 2.5 from 16.25 gives us 13.75. This is the decimal value of the expression.

6. Convert the Result to a Percentage:

The final step is to express the decimal value (13.75) as a percentage. To do this, we multiply the decimal by 100 and add the percentage symbol (%):

    1. 75 * 100 = 1375%

However, let us retrace our steps and check what we are trying to express as a percentage. The question asks us to express the value of the expression as a percentage and provides answer choices that seem to be significantly lower than 1375%. This indicates we may have misinterpreted the question. It's more likely we need to find what percentage the value of the expression is of some implied base value, or perhaps there's an error in the question itself.

Let's consider the calculated value of the expression, 13.75. If the question meant to ask for the value of the expression, then 13.75 is our result.

Given the answer choices (250, 300, 150, 200, 100), it appears there may be an error or missing information in the question. If we are to proceed with expressing the answer as a percentage of something, we would need a base value for comparison.

Without further context or a base value, we must assume the question contains an error or is incomplete. Based purely on the arithmetic, the value of the expression is 13.75.

Therefore, without additional information or clarification, we cannot accurately choose an answer from the provided options.

Analyzing the Answer Choices

Given the answer choices of 250, 300, 150, 200, and 100, and our calculated value of 13.75, it becomes evident that there might be a discrepancy or missing context in the problem statement. The answer choices are significantly larger than the numerical value we obtained, which suggests that the question might be implicitly asking for a percentage relative to a certain base value that is not explicitly stated. To appropriately analyze the answer choices and determine the correct response, we must consider the potential interpretations and hidden assumptions within the question.

Identifying Potential Misinterpretations

One of the initial interpretations we explored was directly converting the calculated value (13.75) into a percentage, which resulted in 1375%. However, this value doesn't align with any of the provided options, indicating that this wasn't the intended approach. This prompts us to consider other possibilities:

  1. Percentage of a Base Value: The question might be implying that we need to express 13.75 as a percentage of some unstated base value. For example, if the base value were 10, then we would calculate what percentage 13.75 is of 10. This would be computed as (13.75 / 10) * 100, which equals 137.5%. Again, this doesn't directly match any of the answer choices, but it highlights the importance of the base value in percentage calculations.
  2. Rounding or Approximation: It's possible that the question intends for us to round or approximate the calculated value to the nearest option. However, even with significant rounding, 13.75 doesn't clearly correspond to any of the given choices.
  3. Error in the Question: The most plausible explanation is that there is an error or missing information in the question itself. This could be a typo in the answer choices, a missing piece of information needed to calculate the percentage correctly, or an ambiguity in the question's wording. In such cases, it's essential to acknowledge the discrepancy and, if possible, seek clarification.

Hypothetical Scenarios and Calculations

To further illustrate the impact of the base value, let's explore a few hypothetical scenarios:

  • Scenario 1: If we were asked to express 13.75 as a percentage of 5, the calculation would be (13.75 / 5) * 100 = 275%. This is still not among the options, but demonstrates how a smaller base value increases the percentage.
  • Scenario 2: If we aimed for an answer of 100%, the implied base value would need to be 13.75 (since 13.75 is 100% of 13.75). This highlights that the missing base value is crucial in determining the correct percentage.

Conclusion on Answer Choices

Given the analysis, it becomes evident that without a clear understanding of the base value or the intended context, it is impossible to accurately select an answer from the provided choices. The question, as it stands, is either incomplete or contains an error. When faced with such situations, it's important to acknowledge the ambiguity and, if possible, seek clarification or additional information.

Importance of Accurate Calculation

In the realm of mathematics, accurate calculation forms the bedrock of problem-solving and decision-making. The ability to perform arithmetic operations, manipulate numerical expressions, and arrive at precise results is not just a theoretical exercise; it is a practical skill with far-reaching implications. From everyday financial transactions to complex scientific computations, accuracy in calculation ensures reliability, validity, and sound judgment. When we discuss the expression 12510+3.75−212{ \frac{125}{10} + 3.75 - 2\frac{1}{2} }, the importance of precision becomes particularly evident. Each term in this expression requires careful consideration and accurate conversion to a common format, whether decimals or fractions. The slightest error in any of these steps can propagate through the calculation, leading to a significantly different and incorrect final answer.

Why Accuracy Matters

  1. Foundation for Further Calculations: In many mathematical problems, an initial calculation serves as a foundation for subsequent steps. If the initial calculation is flawed, all subsequent steps and conclusions drawn from them will be equally erroneous. In the case of our expression, if we miscalculate the decimal equivalent of 12510{ \frac{125}{10} } or incorrectly convert 212{ 2\frac{1}{2} } to a decimal, the final result will be incorrect, regardless of the accuracy of the remaining operations.
  2. Real-World Applications: The principles of accurate calculation extend beyond the classroom and into real-world scenarios. Financial calculations, such as balancing a budget, calculating interest rates, or determining loan payments, demand precise arithmetic. Inaccurate calculations can lead to financial missteps, overspending, or incorrect investment decisions. Similarly, in scientific and engineering fields, accurate calculations are essential for designing structures, conducting experiments, and interpreting data. An error in calculating the load-bearing capacity of a bridge or the trajectory of a spacecraft can have catastrophic consequences.
  3. Problem-Solving Confidence: Developing the habit of accurate calculation builds confidence in one's problem-solving abilities. When individuals consistently arrive at correct answers, they are more likely to approach future challenges with a positive and proactive mindset. This confidence is not merely psychological; it is rooted in a demonstrated ability to handle quantitative tasks effectively.

Techniques for Ensuring Accuracy

  1. Step-by-Step Approach: Complex calculations should be broken down into smaller, manageable steps. This approach minimizes the risk of errors by allowing one to focus on each operation individually. In our expression, we systematically converted each term to a decimal before performing addition and subtraction. This step-by-step approach significantly reduces the likelihood of mistakes.
  2. Double-Checking: It is crucial to double-check each step of the calculation. This can involve manually reviewing the arithmetic or using a calculator to verify the results. Double-checking may seem time-consuming, but it is a worthwhile investment to ensure accuracy.
  3. Estimation: Before performing a calculation, it is often helpful to estimate the expected result. This provides a benchmark against which the final answer can be compared. If the calculated answer deviates significantly from the estimate, it signals a potential error that needs to be investigated.
  4. Attention to Detail: Accurate calculation requires meticulous attention to detail. This includes correctly transcribing numbers, aligning decimal points, and adhering to the order of operations (PEMDAS/BODMAS). Simple oversights, such as misreading a digit or neglecting a negative sign, can lead to significant errors.

Conclusion on Accurate Calculation

Accurate calculation is not just a mathematical skill; it is a life skill that underpins sound decision-making and problem-solving in various contexts. By adopting a systematic approach, double-checking work, and paying attention to detail, individuals can enhance their accuracy and build confidence in their quantitative abilities. The expression 12510+3.75−212{ \frac{125}{10} + 3.75 - 2\frac{1}{2} } serves as a reminder of the importance of precision in arithmetic and the potential consequences of errors.

Conclusion

In conclusion, while we successfully calculated the value of the expression 12510+3.75−212{ \frac{125}{10} + 3.75 - 2\frac{1}{2} } to be 13.75, the task of expressing this value as a percentage and selecting the correct answer from the provided options (250, 300, 150, 200, 100) encountered a significant hurdle. The absence of a clear base value or context within the question makes it impossible to definitively choose an answer. The calculated value of 13.75, when directly converted to a percentage, becomes 1375%, which is far beyond the given choices. This discrepancy strongly suggests that the question is either incomplete or contains an error, preventing a straightforward selection of the correct response.

Key Takeaways

  1. Importance of Context: This exercise underscores the critical importance of context in mathematical problem-solving. Percentages are inherently relative; they express a value as a proportion of a base value. Without knowing what 13.75 should be expressed as a percentage of, we cannot determine the correct answer.
  2. Question Ambiguity: The ambiguity in the question serves as a valuable lesson in the need for clarity in mathematical problems. A well-posed question should provide all the necessary information and eliminate potential misinterpretations. When faced with an ambiguous question, it is essential to acknowledge the uncertainty and, if possible, seek clarification.
  3. Mathematical Rigor: Despite the ambiguity, the exercise highlights the importance of mathematical rigor in calculations. We meticulously converted fractions and mixed numbers to decimals, performed the arithmetic operations, and arrived at a precise numerical value. This demonstrates the fundamental skills required for quantitative problem-solving.
  4. Critical Analysis: The analysis of the answer choices reveals the significance of critical thinking in mathematics. Recognizing that the calculated value did not align with the options prompted a deeper examination of the question and potential sources of error. This critical analysis is crucial for identifying flawed problems and avoiding incorrect conclusions.

Moving Forward

When encountering similar situations in the future, it is advisable to:

  • Seek Clarification: If possible, seek clarification from the source of the question. This may involve asking for additional information or pointing out the ambiguity.
  • State Assumptions: If clarification is not possible, clearly state the assumptions being made in attempting to solve the problem. This ensures transparency and allows for a more informed evaluation of the solution.
  • Provide Alternative Solutions: In some cases, it may be beneficial to provide multiple potential solutions based on different interpretations of the question. This demonstrates a comprehensive understanding of the problem and the ability to address it from various perspectives.

Final Reflection

The exercise of solving 12510+3.75−212{ \frac{125}{10} + 3.75 - 2\frac{1}{2} } and attempting to express the result as a percentage within the given options provides valuable insights into the nuances of mathematical problem-solving. While we could not definitively select an answer due to the question's ambiguity, the process reinforced the importance of accurate calculation, contextual understanding, and critical analysis. It serves as a reminder that mathematics is not just about arriving at numerical answers; it is about reasoning, interpreting, and making informed judgments based on the available information.