Problem 33 - Beginning and Intermediate Algebra

Question

The value of an expression was found incorrectly as follows. $$\begin{aligned}8+2 \cdot 3 \\\=& 10 \cdot 3 \\\=& 30\end{aligned}$$ Find the correct value of the expression.

Step-by-Step Solution

Answer

14

Step 1: Identify the Expression

The given expression is:\[ 8 + 2 \cdot 3 \]

Step 2: Remember the Order of Operations

According to the order of operations (PEMDAS/BODMAS), we need to perform multiplication and division before addition and subtraction.

Step 3: Perform the Multiplication First

Identify the multiplication in the expression. Here, we have:\[ 2 \cdot 3 = 6 \]

Step 4: Perform the Addition

Now add the result of the multiplication to 8:\[ 8 + 6 = 14 \]

Step 5: State the Correct Value

Thus, the correct value of the expression \( 8 + 2 \cdot 3 \) is 14.

Key Concepts

PEMDAS

PEMDAS is an essential concept for solving mathematical expressions accurately. The term \textbf{PEMDAS} stands for Parentheses, Exponents, Multiplication and Division (left to right), and Addition and Subtraction (left to right). This order tells us the sequence of operations to follow when evaluating expressions. Working with PEMDAS helps to avoid mistakes and ensures you arrive at the right solution.
For example, in the given problem \[ 8 + 2 \times 3 \], it's crucial to perform the multiplication before the addition due to the order of operations.
In summary, always remember:

• First handle any operations inside Parentheses.
• Then calculate Exponents, if any.
• Next, perform Multiplication and Division from left to right.
• Finally, do Addition and Subtraction from left to right.

This way, we follow the correct process to get the right answer.

Multiplication

Multiplication is the key step to handle before addition when following the PEMDAS rule. Ignore addition or subtraction until all multiplications and divisions are completed first. Consider the given problem \[ 8 + 2 \times 3 \]. Before doing the addition of 8, focus on the multiplication \[ 2 \times 3 \], which equals 6.
So now, the expression becomes \[ 8 + 6 \].
Understanding multiplication's role in problem-solving ensures you don't mix up the operations and helps maintain accuracy in your calculations.

Addition

Addition comes into play after multiplication within the PEMDAS framework. Once the multiplication is done, you proceed with the addition part of the expression. By correctly following PEMDAS, we make sure every operation is performed sequentially and accurately.
Given our expression \[ 8 + 6 \], after completing the multiplication part \[ 2 \times 3 \], we simply add the results. So, \[ 8 + 6 \] equals 14.
Knowing addition's role in ordered operations helps in solving expressions step-by-step, thereby avoiding confusion and ensuring the right outcome for our expressions.