Problem 62 - Algebra for College Students

Question

Simplify each of the numerical expressions. $$48-(14-11)(10-6)$$

Step-by-Step Solution

Answer

The simplified expression is 36.

Step 1: Evaluate Inside the Parentheses

Start by simplifying the expression inside the first set of parentheses: \(14 - 11\). This results in \(3\). The expression now looks like this: \(48 - (3)(10 - 6)\).

Step 2: Continue with Parentheses

Next, simplify the expression inside the second set of parentheses: \(10 - 6\). This results in \(4\). Now the expression is \(48 - (3)(4)\).

Step 3: Perform Multiplication

Multiply the number outside of the parentheses by the result inside: \(3 \times 4 = 12\). Therefore, the expression becomes \(48 - 12\).

Step 4: Perform Final Subtraction

Finally, subtract \(12\) from \(48\), which gives \(48 - 12 = 36\).

Key Concepts

Order of Operations

The order of operations is a fundamental rule in mathematics that determines the correct sequence in which to solve different parts of a mathematical expression. This rule is essential because it ensures consistency and accuracy.

When we talk about this concept, you might hear the acronym PEMDAS, which stands for:

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

It's vital to follow these steps precisely. In our problem, we first solve whatever is in parentheses, then handle any multiplication or division, and finally, perform addition or subtraction.

Sticking to this sequence ensures the problems are solved correctly every time. Think of it as the mathematical version of following a recipe. If you don't follow the steps in order, you might end up with an incorrect outcome.

Parentheses in Math

Parentheses are like instructions to tackle the enclosed math operations before anything else. They serve as a way to group terms or operations together and prioritize them.

In our example, there were two sets of parentheses that needed to be resolved first:

• First Parentheses - Inside (14 - 11), we solved this to get 3.

• Second Parentheses - Inside (10 - 6), which simplified to 4.

Parentheses give us a clear order and help in breaking the problem into manageable pieces. They ensure we don't miss any important steps. They are a helpful visual cue, indicating the operations inside should be completed before moving on to the rest of the problem.

Think of parentheses as the key to handling complex expressions easily by creating a path to follow.

Arithmetic Operations

Arithmetic operations are the basic operations you use to manipulate numbers, including addition, subtraction, multiplication, and division. Understanding how to perform these operations is crucial for solving any mathematical expression.

In our exercise, we highlighted several essential arithmetic operations:

• Subtraction - Occurs when simplifying what is inside the parentheses (like 14 - 11 or 10 - 6).
• Multiplication - Involves the number outside the parentheses with the result inside, which in our case was 3 times 4.
• Final Subtraction - After completing all other operations, subtracting 12 from 48 to finalize the expression, resulting in 36.

These operations, when performed in the right order, allow us to simplify complex expressions efficiently. Breaking expressions down using basic arithmetic operations simplifies the solving process and helps maintain accuracy in achieving the correct answer.