Solving The Expression 14 + 2 + 3(4) - (6 1/2 - 3) A Step-by-Step Guide

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In the realm of mathematics, evaluating expressions accurately is a fundamental skill. This article delves into the step-by-step process of determining the value of the expression: 14 + 2 + 3(4) - (6 1/2 - 3). We will meticulously break down each operation, adhering to the order of operations (PEMDAS/BODMAS) to arrive at the correct solution. This comprehensive guide will not only provide the answer but also enhance your understanding of mathematical principles.

Demystifying the Order of Operations

Before we embark on solving the expression, it is crucial to grasp the concept of the order of operations. This universally accepted convention dictates the sequence in which mathematical operations must be performed to ensure consistency and accuracy. The acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) and BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) serve as handy mnemonics to remember the correct order.

  1. Parentheses/Brackets: Operations enclosed within parentheses or brackets are executed first.
  2. Exponents/Orders: Next, we address exponents or orders (powers and square roots).
  3. Multiplication and Division: Multiplication and division are performed from left to right.
  4. Addition and Subtraction: Finally, addition and subtraction are carried out from left to right.

By adhering to this order, we can systematically simplify complex expressions and arrive at the correct result. Let's now apply these principles to our given expression.

Step-by-Step Evaluation of the Expression

1. Simplifying within the Parentheses

The expression contains a set of parentheses: (6 1/2 - 3). Our first task is to simplify the operation within these parentheses. We have a mixed number, 6 1/2, which can be converted to an improper fraction for easier calculation. 6 1/2 is equivalent to (6 * 2 + 1) / 2 = 13/2.

Now, we can rewrite the expression within the parentheses as (13/2 - 3). To subtract these numbers, we need a common denominator. We can express 3 as 6/2. Therefore, the expression becomes (13/2 - 6/2) = 7/2.

So, the simplified expression within the parentheses is 7/2, which is equivalent to 3 1/2 or 3.5.

2. Addressing Multiplication

With the parentheses simplified, our expression now looks like this: 14 + 2 + 3(4) - 3.5. According to the order of operations, we must perform multiplication before addition and subtraction. We have the term 3(4), which represents 3 multiplied by 4. This gives us 3 * 4 = 12.

Now, our expression is further simplified to: 14 + 2 + 12 - 3.5.

3. Performing Addition and Subtraction (from Left to Right)

Finally, we are left with addition and subtraction operations. We perform these operations from left to right. Let's start with the first addition: 14 + 2 = 16.

Our expression now becomes: 16 + 12 - 3.5. Next, we add 16 and 12: 16 + 12 = 28.

Now, the expression is: 28 - 3.5. Subtracting 3.5 from 28 gives us: 28 - 3.5 = 24.5.

4. Converting to a Mixed Number (if necessary)

The result we obtained, 24.5, is a decimal. To express it as a mixed number, we can separate the whole number part (24) and the decimal part (0.5). 0.5 is equivalent to 1/2. Therefore, 24.5 can be expressed as the mixed number 24 1/2.

However, upon reviewing the original problem and the provided options, we realize there was an earlier error in our calculations. Let's revisit the steps to pinpoint the mistake and arrive at the correct answer.

Identifying and Correcting the Error

Upon careful review, the error occurred in the final calculation. While the steps up to 28 - 3.5 are correct, the subtraction was performed incorrectly. The correct subtraction is 28 - 3.5 = 24.5. However, this result does not match any of the provided options (A. 10 1/2, B. 15 1/2, C. 30 1/2, D. 36 1/2).

This discrepancy indicates a potential error earlier in the calculation process. Let's retrace our steps from the beginning to ensure accuracy.

Retracing and Recalculating

  1. Parentheses: (6 1/2 - 3) = (13/2 - 6/2) = 7/2 = 3.5 (Correct)
  2. Multiplication: 3(4) = 12 (Correct)
  3. Expression: 14 + 2 + 12 - 3.5
  4. Addition (Left to Right): 14 + 2 = 16 (Correct)
  5. Expression: 16 + 12 - 3.5
  6. Addition: 16 + 12 = 28 (Correct)
  7. Expression: 28 - 3.5

Here's where the initial error occurred. While the setup was correct, the final subtraction was miscalculated in the previous attempt. The correct subtraction is:

  • 28 - 3.5 = 24.5

Converting 24.5 to a Mixed Number and Comparing with Options

As we determined earlier, 24.5 is equivalent to the mixed number 24 1/2. However, this result still does not match any of the provided answer choices. This suggests a persistent error in our calculations or a potential mistake in the original problem statement or answer options.

Let's meticulously re-examine each step once more to ensure complete accuracy.

Final Verification and Correction

After another thorough review, we can confirm that all steps have been performed correctly according to the order of operations. The expression 14 + 2 + 3(4) - (6 1/2 - 3) simplifies to 24.5, which is equivalent to 24 1/2.

Since none of the provided options (A. 10 1/2, B. 15 1/2, C. 30 1/2, D. 36 1/2) match our calculated result, it is highly likely that there is an error in the answer choices provided in the original problem. The correct answer should be 24 1/2.

Conclusion

Through this detailed step-by-step analysis, we have demonstrated the process of evaluating the mathematical expression 14 + 2 + 3(4) - (6 1/2 - 3). By meticulously following the order of operations and verifying each step, we arrived at the solution 24 1/2. The discrepancy between our result and the provided options highlights the importance of careful calculation and the possibility of errors in problem statements or answer choices. This exercise underscores the significance of understanding mathematical principles and applying them rigorously to ensure accuracy.

This comprehensive guide not only provides the solution but also reinforces the importance of the order of operations and meticulous calculation in mathematics. Remember, practice and attention to detail are key to mastering mathematical expressions.