Math Expression Evaluation: Find The Error In Jane's Work

Hey math enthusiasts! Today, we're diving into a common pitfall that many students encounter when evaluating mathematical expressions: the order of operations. It's a fundamental concept, but one that can easily trip you up if you're not careful. Let's take a close look at Jane's work as she tries to evaluate the expression $6+48 \div 4 \times 3$. We'll break down each step, identify where things went awry, and solidify our understanding of how to conquer these problems.

Understanding the Order of Operations

Before we jump into Jane's specific calculation, it's crucial to have a solid grasp of the order of operations. You might have heard of acronyms like PEMDAS or BODMAS. These are mnemonics designed to help us remember the correct sequence to perform mathematical operations. PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). BODMAS is similar, standing for Brackets, Orders (powers and square roots, etc.), Division and Multiplication (from left to right), and Addition and Subtraction (from left to right). The key takeaway here is that multiplication and division have the same priority, and addition and subtraction have the same priority. When operations share the same priority, we work from left to right.

This left-to-right rule for operations of equal priority is often the source of mistakes. Many people incorrectly assume that multiplication always comes before division, or addition before subtraction, regardless of their position in the expression. This can lead to drastically different and incorrect answers. In Jane's case, understanding this hierarchy is absolutely essential to pinpointing the error in her steps. We need to meticulously follow the PEMDAS (or BODMAS) rules step-by-step to see if her application of these rules is accurate. It's not just about knowing the rules, but applying them correctly within the given expression. Think of it as a carefully choreographed dance; every step must be in the right order for the whole routine to work.

Analyzing Jane's Work: Step-by-Step

Now, let's dissect Jane's solution to the expression $6+48 \div 4 \times 3$. We'll examine each of her three steps to see where the deviation from the correct procedure occurred.

Step 1: $6+48 \div 4 \times 3 = 6+48 \div 12$

In this first step, Jane seems to have focused on the $48 \div 4 \times 3$ part of the expression. She correctly identified that division and multiplication are next in line after any potential parentheses or exponents (which aren't present here). However, the crucial part is how she handled the division and multiplication together. According to the order of operations (PEMDAS/BODMAS), when we have both multiplication and division, we perform them from left to right. In the sequence $48 \div 4 \times 3$, the division $48 \div 4$ comes first because it is to the left of the multiplication. Jane performed this division and got $12$. So far, so good. But then, instead of continuing from left to right with the multiplication, she seems to have combined the division and multiplication incorrectly. It appears she performed $4 \times 3$ first, getting $12$, and then perhaps intended to divide $48$ by that result. If that were the case, the expression would become $6 + 48 \div 12$. This is exactly what Jane has written. The intent might have been to handle multiplication and division together, but the execution led to an error by not strictly adhering to the left-to-right rule. The expression should have been processed as $48 \div 4$ first, yielding $12$, and then multiplying that result by $3$. This would have given us $12 \times 3 = 36$. So, Jane's expression should have been $6 + 36$ at this stage, not $6 + 48 \div 12$. The error lies in how she grouped the operations; she should have resolved the division and then the multiplication sequentially, not combined them into a single step that altered the order.

Step 2: $6+48 \div 12 = 6+4$

This step is where the mistake from Step 1 becomes even more apparent and a new error is introduced. Jane is now working with the expression $6 + 48 \div 12$. According to the order of operations, after dealing with multiplication and division, we move on to addition and subtraction. In this expression, we have an addition and a division. Therefore, the division must be performed before the addition. Jane performed $48 \div 12$ and correctly got $4$. However, she then wrote $6+4$. This implies she performed the addition before completing the division, or that she incorrectly carried over the result from her flawed Step 1. If we were to follow her logic in Step 1 ($6 + 48 \div 12$), the next correct step would be to perform the division $48 \div 12$, which equals $4$. So, the expression would become $6 + 4$. This part, mathematically, is correct based on her previous (incorrect) result. The true error was already made in Step 1 by not calculating $48 \div 4 \times 3$ correctly. In this step, Jane did correctly execute the division of $48 \div 12$, but the context of this division stemmed from a misinterpretation of the order of operations in the previous step. Essentially, she arrived at $6 + 48 \div 12$ because of an error in Step 1, and then proceeded to correctly solve $48 \div 12$ and write $6+4$. The problem is that the original expression $6+48 \div 4 \times 3$ should have led to $6+36$ after all multiplication and division were handled, not $6+4$. So, while the calculation $48 \div 12 = 4$ is sound, its placement and the resulting expression $6+4$ are incorrect due to the initial misstep.

Step 3: $6+4 = 10$

In her final step, Jane takes the result from Step 2, which is $6+4$, and performs the addition to get $10$. Mathematically, $6+4$ does indeed equal $10$. So, if the expression $6+4$ were the correct one to evaluate, Jane's final answer would be correct. The issue, as we've established, lies in the preceding steps. Because Step 1 and Step 2 contained errors in the application of the order of operations, the intermediate result of $6+4$ was incorrect. Therefore, even though the final addition is arithmetically sound, the overall answer of $10$ is wrong because it's based on flawed calculations from earlier stages. It's like building a house with a weak foundation; the walls might go up straight, but the whole structure is compromised. Jane's calculation of $10$ is the final outcome of a chain reaction of errors in applying the fundamental rules of arithmetic.

The Correct Way to Evaluate the Expression

Let's walk through the expression $6+48 \div 4 \times 3$ using the correct order of operations (PEMDAS/BODMAS) to see where Jane went wrong and what the actual answer should be.

Our expression is: $6+48 \div 4 \times 3$

1. Parentheses/Brackets: There are none.

2. Exponents/Orders: There are none.

3. Multiplication and Division (from left to right): We encounter division ($48 \div 4$) and multiplication ($\times 3$). Since division comes first from the left, we perform that operation: $48 \div 4 = 12$ Now our expression looks like: $6 + 12 \times 3$ Next, we perform the multiplication: $12 \times 3 = 36$ Our expression now simplifies to: $6 + 36$

4. Addition and Subtraction (from left to right): We only have addition left: $6 + 36 = 42$

So, the correct answer is 42, not 10. Jane's error stemmed from misinterpreting the left-to-right rule for multiplication and division in Step 1. She incorrectly performed the $4 \times 3$ before the $48 \div 4$, or she incorrectly combined these operations.

Conclusion: Mastering the Order of Operations

Jane's evaluation of $6+48 \div 4 \times 3$ highlights a common misunderstanding of the order of operations. The critical takeaway is that multiplication and division must be performed from left to right as they appear, and similarly, addition and subtraction must be performed from left to right. In Jane's work, the mistake occurred in Step 1 when she did not correctly handle the sequence of $48 \div 4 \times 3$. Instead of calculating $48 \div 4$ first, then multiplying the result by $3$, she seems to have performed the operations in a different order, leading to an incorrect intermediate result. This error then cascaded through the subsequent steps, resulting in a final answer of $10$ instead of the correct answer, $42$. By diligently following the PEMDAS/BODMAS rules, especially the left-to-right directive for operations of equal precedence, you can confidently tackle any mathematical expression. Remember, practice makes perfect! Keep working through problems, and you'll master this important skill.

For further practice and to deepen your understanding of mathematical principles, you can explore resources from Khan Academy. They offer a vast library of exercises and lessons on order of operations and other crucial math topics.