Problem 83 - Elementary Algebra

Question

A student was mixed up about the order of operations and always did multiplication first before she did any division. Write an expression in which doing this will result in an incorrect answer.

Step-by-Step Solution

Answer

The expression \( 20 \times 2 \div 4 \) gives incorrect result if multiplication is done before division.

Step 1: Identify the Order of Operations

The correct order of operations can be remembered by PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).

Step 2: Choose an Expression

Select an expression that includes both multiplication and division, such as: \( 20 \times 2 \div 4 \).

Step 3: Calculate with the Correct Order of Operations

Following PEMDAS, perform division before multiplication.\[ 20 \times 2 \div 4 \] becomes \[ 20 \div 4 \times 2 \]. First, divide: \[20 \div 4 = 5 \]. Then multiply: \[ 5 \times 2 = 10 \]. So, the correct answer is 10.

Step 4: Calculate with Incorrect Order of Operations

Now, perform multiplication before division: \[ 20 \times 2 = 40 \]. Then, divide: \[40 \div 4 = 10 \]. Notice that the final result is different from the previous step due to incorrect order of operations.

Step 5: Compare Correct and Incorrect Results

When the order of operations is followed correctly, the result is 10. However, doing multiplication first and then division results in a different answer, 40.

Key Concepts

PEMDAS

PEMDAS is an acronym that helps us remember the correct order in which to do operations in math. The letters in PEMDAS stand for:

• Parentheses: Solve expressions in parentheses first.
• Exponents: Next, evaluate any exponents.
• Multiplication and Division: Do these operations from left to right.
• Addition and Subtraction: Finally, do these operations from left to right.

This order prevents mistakes and confusion. For example, in the expression \((20 \times 2) \div 4\), you need to divide before multiplying according to PEMDAS. Always stick to this rule to get the correct results.

Multiplication and Division

Many students find multiplication and division tricky due to the left-to-right rule. According to PEMDAS, both multiplication and division should be performed from left to right. This means that although both operations are on the same level of priority, the one that appears first should be done first.
Let's look at an example:
If you have the expression \((20 \times 2 \div 4)\), first divide 20 by 4 according to PEMDAS: \((20 \div 4 \times 2) \). This gives you 5.
Then multiply by 2: \((5 \times 2 = 10)\).
If you mistakenly do the multiplication first, your calculations would look like this:
\((20 \times 2 = 40) \div 4 = 10\)\, which is incorrect.
Understanding this rule helps avoid common errors in math problems.

Correct and Incorrect Calculations

Making mistakes in the order of operations is easy without the PEMDAS guideline.
Let's see how following or not following PEMDAS affects your results.
Correct Calculation:
Given the expression \((20 \times 2 \div 4)\)
According to PEMDAS, you first perform the division:
\((20 \div 4 = 5)\)
Then, perform the multiplication:
\((5 \times 2 = 10)\)
Incorrect Calculation:
If you do multiplication first:
\((20 \times 2 = 40)\)
And then division:
\((40) \div 4 = 10)\)
We observe different answers (correct: 10, incorrect: 40). This shows that not adhering to the correct order of operations leads to wrong results. Always check and follow PEMDAS for accurate calculations.