Problem 3 - Prealgebra

Question

Find the value of each expression. $$2 \times 9 \div 3$$

Step-by-Step Solution

Answer

The value of the expression is 6.

Step 1: Use the Order of Operations

According to the order of operations (PEMDAS/BODMAS), we need to address any multiplication or division, working from left to right. For this expression, there are no parentheses or exponents, just multiplication and division.

Step 2: Perform the Multiplication

First, compute the multiplication portion of the expression: \(2 \times 9 = 18\). So, the expression now simplifies to \(18 \div 3\).

Step 3: Perform the Division

Next, divide the result from the multiplication by 3: \(18 \div 3 = 6\). This gives us the final value of the expression.

Key Concepts

Multiplication

Multiplication is one of the basic arithmetic operations in math, alongside addition, subtraction, and division. It’s essentially a shortcut for repeated addition. When you multiply two numbers, you are essentially adding one of the numbers to itself as many times as the value of the other number.
For example, when you calculate \(2 \times 9\), you are summing the number 2 nine times:

• \(2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 18\)

In expressions involving multiple operations like the one given, multiplication needs to be performed before addition or subtraction, following the order of operations rule (PEMDAS/BODMAS).
Multiplication is also commutative, meaning that the order of the numbers doesn’t change the result, so \(2 \times 9\) is the same as \(9 \times 2\). This property simplifies calculations and is fundamental in solving more complex expressions.

Division

Division is another fundamental operation, and it can be thought of as the opposite of multiplication. In simple terms, division helps us find out how many times one number is contained within another.
When considering \(18 \div 3\), you're essentially asked to determine how many 3s can be found in 18:

• \(3 + 3 + 3 + 3 + 3 + 3 = 18\)

Here, there are six 3s in 18, which gives us a result of 6 for the operation. Another way to view division is in terms of sharing or grouping. If you have 18 items and need to divide them into 3 groups, each group would have 6 items. Just like multiplication, division must be performed before addition or subtraction when solving expressions under the order of operations rule.

PEMDAS/BODMAS

Understanding PEMDAS (or BODMAS) is crucial for correctly solving expressions involving multiple types of operations.
Both acronyms help us remember the order of operations:

• P/B: Parentheses/Brackets
• E/O: Exponents/Orders (i.e., powers and square roots, etc.)
• MD/DM: Multiplication and Division (from left to right)
• AS: Addition and Subtraction (from left to right)

In expressions where no parentheses or exponents present, like \(2 \times 9 \div 3\), the focus shifts directly to multiplication and division. These two operations are of equal precedence, so they are carried out from left to right by default.
So, you first multiply (\(2 \times 9\)), getting 18, and then divide (\(18 \div 3\)), resulting in a final answer of 6. Remembering PEMDAS/BODMAS ensures calculations are consistent and mathematically correct.