Problem 22 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(12 \cdot 5-3 \cdot 6\)

Step-by-Step Solution

Answer

The expression simplifies to 42.

Step 1: Break Down the Expression

The original expression given is \(12 \cdot 5 - 3 \cdot 6\). The first thing to do is determine which operations need to be performed first. The expression has two multiplication operations and one subtraction operation.

Step 2: Perform the Multiplications

According to the order of operations (PEMDAS/BODMAS), multiplication comes before subtraction. First, calculate \(12 \cdot 5\) which equals 60. Then, calculate \(3 \cdot 6\) which equals 18.

Step 3: Perform the Subtraction

Now, subtract the result of the second multiplication from the first. So, subtract \(18\) from \(60\): \[60 - 18 = 42\] Thus, the expression simplifies to 42.

Key Concepts

Order of Operations

When simplifying expressions in mathematics, following the correct order of operations is crucial. It ensures each expression is simplified correctly and consistently. We often use the acronym PEMDAS (or BODMAS in some countries) to help remember these steps:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In the expression \(12 \cdot 5 - 3 \cdot 6\), we first perform multiplication because it comes before subtraction according to PEMDAS.
By systematically following this order, you ensure the operations are completed correctly, preventing errors in complex calculations. Keeping this order in mind will consistently guide you in simplifying mathematical expressions successfully.

Multiplication

Multiplication is one of the four fundamental arithmetic operations, and it involves combining equal groups. In our original exercise, multiplication occurs twice: \(12 \cdot 5\) and \(3 \cdot 6\).
These multiplications need to be performed before any other operations because of their placement in the order of operations.

• To multiply \(12 \cdot 5\), imagine having 12 groups with 5 items in each. You get a total of 60 items.
• Similarly, for \(3 \cdot 6\), picture 3 groups of 6. This results in 18 total.

Multiplication simplifies groups into a single quantity making it easier to manage larger numbers in problems. Understanding multiplication is essential for converting complex expressions into simpler forms.

Subtraction

Subtraction is the process of deducting one quantity from another. After performing the multiplications in the expression, subtraction is used to finalize the simplification.
When the expression \(12 \cdot 5 - 3 \cdot 6\) is broken down, we are left with \(60 - 18\). Here, you take away 18 from 60 to find out what is left.
To visualize this:

• Imagine having 60 apples, and someone takes away 18. You are left with 42 apples.

Subtraction helps in determining the difference between numbers, which is crucial in finding solutions to algebraic expressions. It provides the final touch in simplifying an expression to its simplest form.