Problem 8 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. $$ 9 \div 3 \cdot 2 $$

Step-by-Step Solution

Answer

The result of the expression \(9 \div 3 \cdot 2\) is 6.

Step 1: Perform the division operation

According to the order of operations, division comes before multiplication when they are in the same hierarchy, and we should evaluate from left to right. So, take \(9 \div 3\) which equals to 3.

Step 2: Perform the multiplication

After the division, only multiplication operation remains. Perform \(3 \cdot 2\) which equals to 6.

Key Concepts

Division

When faced with mathematical expressions that require division, it's important to approach them in a structured way. Division is taking a number and splitting it into equal parts. The main goal is to determine how many times a number (the divisor) is contained within another number (the dividend). For instance, in the expression \(9 \div 3\), 9 is divided by 3. This operation simply asks how many 3s fit into 9. In this case, it is exactly 3.

Recognizing division in an expression is crucial. It often appears side by side with other operations like multiplication, necessitating knowledge of which to perform first. In our problem, when you see \(9 \div 3 \cdot 2\), you begin with division due to the left-to-right rule. Start by dividing 9 by 3, which gives 3 as a result.

• Understand division as splitting a whole into equal parts.
• Identify the dividend and divisor in the equation.
• Apply the left-to-right rule when division is alongside multiplication in a formula.

Multiplication

Next, we delve into multiplication, which is essentially repeated addition. It involves multiplying two numbers — one being the multiplicand and the other the multiplier. In simpler terms, it's about how many times you add a number to itself. For example, when we solve \(3 \cdot 2\), we are essentially adding 3 twice, resulting in 6.

After completing any division in an expression, multiplication typically follows, as outlined in the order of operations. In \(9 \div 3 \cdot 2\), following the division, you're left with \(3 \cdot 2\). By multiplying 3 by 2, you reach the final answer of 6. This keeps the expression simplified and clear.

• See multiplication as repeated addition of a number.
• In our example, after dividing, multiply 3 by 2 to get a final result of 6.
• Ensure clarity by reducing unnecessary steps once division is complete.

Evaluate Expressions

Evaluating expressions means finding the value of a mathematical phrase that includes numbers, operations like division and multiplication, and potentially brackets or exponents. This process needs a methodical approach to ensure accuracy.

To effectively evaluate expressions such as \(9 \div 3 \cdot 2\), the order of operations (often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)) plays a pivotal role.

Following this rule set, always tackle multiplication and division as they appear from left to right in the expression. This clarity prevents mistakes and maintains the intended mathematical relationships.

• Memorize the order of operations to achieve correct results.
• Handle multiplication and division sequentially from left to right.
• Conclude with a verified solution, ensuring no steps are skipped or prematurely evaluated.