Problem 198 - Intermediate Algebra

Question

In the following exercises, simplify. $$ \frac{9 \cdot 7-3(12-8)}{8 \cdot 7-6 \cdot 6} $$

Step-by-Step Solution

Answer

\(\frac{51}{20}\)

Step 1 - Simplify the Numerator

Start by simplifying the numerator of the fraction. The numerator is: \[9 \times 7 - 3(12 - 8)\]First, perform the multiplication and subtraction inside the parenthesis:\[12 - 8 = 4\]So the expression becomes:\[9 \times 7 - 3 \times 4\] Next, perform the multiplications:\[9 \times 7 = 63\]\[3 \times 4 = 12\]Now, subtract the results:\[63 - 12 = 51\]So the numerator simplifies to 51.

Step 2 - Simplify the Denominator

Next, simplify the denominator of the fraction. The denominator is:\[8 \times 7 - 6 \times 6\]First, perform the multiplications:\[8 \times 7 = 56\]\[6 \times 6 = 36\]Now, subtract the results:\[56 - 36 = 20\]So the denominator simplifies to 20.

Step 3 - Combine and Simplify

Now, combine the simplified numerator and denominator into a single fraction:\[\frac{51}{20}\]The fraction \(\frac{51}{20}\) is already in its simplest form, as 51 and 20 do not have any common factors other than 1.

Key Concepts

Order of Operations

When simplifying algebraic expressions, following the order of operations is crucial. The order of operations is a rule that ensures everyone gets the same result when performing mathematical operations. To remember the correct order, you can use the acronym PEMDAS:

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

In the given exercise, the numerator \(9 \times 7 - 3(12 - 8)\) requires you to first handle the operations within the parentheses: \(12 - 8 = 4\). Next, solve the multiplication: \(9 \times 7\) and \(3 \times 4\). Finally, perform the subtraction: \(63 - 12\). This sequence of steps follows PEMDAS, ensuring the expression is simplified correctly and consistently.

Simplifying Fractions

After simplifying the numerator and denominator, combining them into a fraction is the last step. To determine if the fraction can be simplified further, check for common factors between the numerator and the denominator. In the exercise, we produced the fraction \(\frac{51}{20}\). We then analyzed the numbers: 51 and 20. The greatest common factor between them is 1, which means the fraction is already in its simplest form. Here are the steps to simplify fractions when possible:

• Find the greatest common factor (GCF) of the numerator and the denominator.
• Divide both the numerator and the denominator by their GCF.
• The result is the fraction in its simplest form.

Simplifying fractions is a vital skill for ensuring clarity and accuracy in presenting results.

Basic Arithmetic Operations

This exercise involves several basic arithmetic operations: addition, subtraction, multiplication, and division. Mastering these operations is essential for simplifying algebraic expressions and many other areas in mathematics. Let's break down their roles:

• Addition and Subtraction: These operations combine or separate quantities. In our exercise, subtraction was used after multiplication to simplify the numerator.
• Multiplication: It represents repeated addition. For instance, \(9 \times 7 = 63\) is a key multiplication step in the numerator.
• Division: It splits a quantity into equal parts. Although not directly highlighted in common fractions, it underpins the fraction \(\frac{51}{20}\).

Each operation has specific properties and rules, such as the distributive property used here, making it easier to approach problem-solving methodically and accurately.