Problem 49 - Prealgebra

Question

Emily and Marcus are evaluating \(24 \div 2 \times 3 .\) Who is correct? Explain your reasoning. \begin{aligned}\text { Emily } & \\\24 \div 2 \times 3 &=12 \times 3\\\&=36\end{aligned}\begin{aligned}\text { Marcus } \\\24 \div 2 \times 3 &=24 \div 6 \\\&=4\end{aligned}

Step-by-Step Solution

Answer

Emily is correct; the expression evaluates to 36.

Step 1: Understand the Operation Order

The expression given is \(24 \div 2 \times 3\). According to the order of operations (PEMDAS/BODMAS), multiplication and division are performed from left to right. So, we should address the operations as they appear in the expression and not based on priority over one another.

Step 2: Perform Division First

Start with the first operation from the left, which is the division. Divide 24 by 2. \[24 \div 2 = 12\]

Step 3: Perform Multiplication Next

Now, take the result from step 2 and multiply it by 3. \[12 \times 3 = 36\]

Step 4: Confirm Who is Correct

According to the calculations performed, the expression \(24 \div 2 \times 3\) evaluates to 36. Therefore, Emily calculated correctly.

Key Concepts

Division

Division is the arithmetic operation where we split a number into equal parts. It is often represented by the symbol "\( \div \)" or a slash "\( / \)". In mathematics, division is the process of finding out how many times one number is contained within another. For example, in the expression \( 24 \div 2 \), we are trying to determine how many twos fit into 24.
Breaking this into simpler steps, division can be thought of as repeated subtraction. In our example, 24 can be formed by repeatedly subtracting 2, and we'd find this can be done 12 times which is the answer to the division.
Remember, division is handled from left to right in a mathematical expression. If paired with multiplication, it is solved in the order it appears, not based on a hierarchy between division and multiplication. This will be important to remember when we discuss the order of operations more deeply.

Multiplication

Multiplication is another fundamental mathematical operation. It represents the concept of adding a number to itself a certain number of times. For example, \( 3 \times 4 \) means adding 3, four times, i.e., \( 3 + 3 + 3 + 3 \), which equals 12.
In the context of multiple operations within an expression, multiplication is handled along with division from left to right. This means in expressions like \( 24 \div 2 \times 3 \), after performing the division, the result then gets multiplied by 3 as Emily correctly did.
When evaluating expressions with both multiplication and division, reading and executing these from left to right avoids mistakes. This approach adheres to the mathematical convention set by the order of operations rules.

PEMDAS

PEMDAS is a mnemonic that helps remember the sequence of operations to be performed in a mathematical expression. It stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition, and Subtraction (from left to right).
This order of operations ensures that anyone reading or writing a mathematical expression will reach the same result by following these priorities:

• Start with operations inside **Parentheses**.
• Handle any **Exponents** like powers or roots.
• Proceed with **Multiplication** and **Division** as they appear from left to right.
• Finally, tackle **Addition** and **Subtraction** from left to right.

Applying PEMDAS to \( 24 \div 2 \times 3 \), we execute division first to obtain 12, then multiply by 3 to get 36, reflecting Emily’s correct approach.

BODMAS

BODMAS is another rule for determining the order of operations, similar to PEMDAS, but it stands for Brackets, Orders (i.e., powers and roots), Division and Multiplication (from left to right), Addition and Subtraction (from left to right). This rule simplifies complex calculations and ensures consistency.
The key point with BODMAS is to resolve, just like PEMDAS, Division and Multiplication from left to right, as they appear. This is crucial in calculations such as \( 24 \div 2 \times 3 \), ensuring the operations are executed sequentially from left to right.
Following BODMAS, we first divide 24 by 2, resulting in 12, then multiply by 3, leading to the final answer of 36. Thus, Emily's evaluation of the expression was correct.