Problem 35 - Algebra 1

Question

Evaluate the expression. $$\frac{1}{3}(9 \cdot 3)+18$$

Step-by-Step Solution

Answer

The result of evaluating the expression is 27.

Step 1: Brackets/parentheses

In the given expression there are no parentheses, so we start by solving the multiplication within the fraction: \( \frac{1}{3}(9 \cdot 3) = \frac{1}{3} \cdot 27 \).

Step 2: Multiplication and division

Next we execute the multiplication and division from left to right: \( \frac{1}{3} \cdot 27 = 9 \). Now the expression becomes \(9 + 18\).

Step 3: Addition and subtraction

Finally, we perform the addition operation: \(9 + 18 = 27\).

Key Concepts

Order of Operations

In any arithmetic expression, the order of operations is crucial in determining the correct result. The sequence, often remembered by the acronym PEMDAS, stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Let's break it down:

• First, solve expressions inside Parentheses.
• Then, handle Exponents.
• Next, perform Multiplication and Division as they appear from left to right.
• Finally, execute Addition and Subtraction from left to right.

These steps help us maintain a consistent approach to solving complex equations. In our example expression, there were no parentheses, so we moved directly to multiplication, followed by addition.

Multiplication

Multiplication is a fundamental arithmetic operation used to calculate the total of one number added to itself a specified number of times. In our example, the expression initially required us to solve the multiplication inside the term \( \frac{1}{3}(9 \cdot 3) \). Here, we first compute \( 9 \cdot 3 = 27 \). This gives us a new expression: \( \frac{1}{3} \cdot 27 \).
Multiplication is performed before addition because it follows the order of operations and comes earlier in the process. Also, notice how multiplication happens before division when they are both present due to the left-to-right rule.

Addition

Once multiplication in the arithmetic expression has been resolved, we move on to the addition step. In our example, after simplifying the multiplication, the expression became \( 9 + 18 \).
Addition combines the results of previous operations, bringing us closer to the final answer. It's an operation where numbers are added together, and each number is a term.
In this case, we add 9 and 18 together to determine the final answer: 27. Addition is straightforward when all previous operations have been correctly executed. It ensures that the expression is completely simplified.

Fractions

Fractions represent parts of a whole number and are integral in many arithmetic operations. In our example, \( \frac{1}{3}(9 \cdot 3) \), the fraction \( \frac{1}{3} \) indicates that we are taking one-third of the product inside the parentheses.
Here's how fractions relate to the order of operations:

• The fraction acts like a division, which falls after multiplication in the order of operations.
• Once the multiplication \( 9 \cdot 3 = 27 \) is done, you calculate \( \frac{1}{3} \cdot 27 \), essentially dividing 27 by 3.

Handling fractions requires careful attention to their role within the broader arithmetic expression. When used properly, they can be a powerful tool for breaking down complex problems into simpler operations.