Problem 74 - Intermediate Algebra

Question

Simplify each of the numerical expressions. $$ \frac{4 \cdot 9-3 \cdot 5-3}{18-12} $$

Step-by-Step Solution

Answer

The simplified expression is 3.

Step 1: Simplify the Numerator

First, simplify the expression in the numerator by performing the operations. The expression is \(4 \cdot 9 - 3 \cdot 5 - 3\). Begin by calculating the products: \(4 \cdot 9 = 36\) and \(3 \cdot 5 = 15\). Substitute these into the expression to get \(36 - 15 - 3\). Next, perform the subtractions: \(36 - 15 = 21\) and \(21 - 3 = 18\). So, the simplified numerator is 18.

Step 2: Simplify the Denominator

Now, simplify the expression in the denominator, \(18 - 12\). Perform the subtraction: \(18 - 12 = 6\). The denominator simplifies to 6.

Step 3: Calculate the Fraction

With the simplified numerator and denominator, the expression becomes \(\frac{18}{6}\). Perform the division: \(18 \div 6 = 3\). Thus, the simplified expression is 3.

Key Concepts

Numerical Expressions

Numerical expressions are a fundamental part of algebra that involve numbers and arithmetic operations like addition, subtraction, multiplication, and division. These expressions do not contain any variables, only constants. Understanding numerical expressions is crucial as they form the basis of more complex algebraic expressions.

When dealing with numerical expressions, remember to:

• Identify all the components, such as numbers and operations.
• Follow the rules of arithmetic to simplify or evaluate them.
• Combine like terms to make calculation easier.

Let's consider the expression given: \(4 \cdot 9 - 3 \cdot 5 - 3\). To solve it, follow these steps: 1. First, compute the products, which are basic multiplication operations: \(4 \cdot 9 = 36\) and \(3 \cdot 5 = 15\). 2. Substitute these results into the expression: \(36 - 15 - 3\). 3. Perform the subtractions consecutively: \(36 - 15 = 21\) and then \(21 - 3 = 18\). Thus, the numerical expression simplifies to 18.

Order of Operations

Order of operations is an essential concept in simplifying numerical expressions correctly. Without following the right sequence, one might end up with incorrect solutions.

Often remembered by the acronym PEMDAS, which stands for:

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

Let's apply the order of operations to the numerator of the expression \(4 \cdot 9 - 3 \cdot 5 - 3\): 1. Begin with multiplication as it comes before subtraction. Calculate \(4 \cdot 9 = 36\) and \(3 \cdot 5 = 15\). 2. Once all multiplications are done, proceed with subtractions: first \(36 - 15 = 21\) and then \(21 - 3 = 18\). Applying these rules ensures the expression is simplified accurately and yields a correct solution.

Fraction Simplification

Simplifying fractions is a critical skill in algebra, which involves reducing a fraction to its simplest form by dividing both the numerator and the denominator by their greatest common factor (GCF). This process makes fractional expressions easier to understand, compare, and work with.

To simplify the fraction \(\frac{18}{6}\) in the exercise, follow these straightforward steps:

• Identify the common factor. Here, 6 is a common factor of both 18 and 6.
• Divide the numerator by 6: \(18 \div 6 = 3\).
• Also, divide the denominator by 6: \(6 \div 6 = 1\).

So, the whole fraction simplifies to \(\frac{3}{1}\), simply write this as 3. Reducing fractions to their simplest form ensures that all terms are as concise as possible, facilitating easier mathematical operations.