Problem 197 - Intermediate Algebra

Question

In the following exercises, simplify. $$ \frac{7 \cdot 4-2(8-5)}{9 \cdot 3-3 \cdot 5} $$

Step-by-Step Solution

Answer

\[ \frac{11}{6} \]

Step 1: Simplify the numerator

First, simplify the expression in the numerator: \[ 7 \times 4 - 2(8 - 5) \] Compute inside the parentheses: \[ 8 - 5 = 3 \] So the numerator becomes: \[ 7 \times 4 - 2 \times 3 \] Next, multiply the numbers: \[ 28 - 6 = 22 \]

Step 2: Simplify the denominator

Next, simplify the expression in the denominator: \[ 9 \times 3 - 3 \times 5 \] Perform the multiplications: \[ 27 - 15 \] Then subtract the results: \[ 27 - 15 = 12 \]

Step 3: Divide the simplified numerator by the simplified denominator

Now that we have simplified the numerator and the denominator, combine them to get the final result: \[ \frac{22}{12} \] Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (which is 2): \[ \frac{22 \div 2}{12 \div 2} = \frac{11}{6} \]

Key Concepts

Algebraic Simplification

Algebraic simplification involves reducing an algebraic expression to its simplest form. This often requires applying basic arithmetic operations and algebraic rules.
For example, in our exercise, we need to simplify the expression in both the numerator and the denominator separately before combining them. This approach makes it easier to handle the operations step-by-step.
First, look for operations that can be simplified, like parentheses or multiplication. Simplify inside inner parentheses first, then move outward, applying the operations as you go. This method helps to avoid mistakes and makes the problem more manageable.

Fractions

Fractions represent parts of a whole and can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD).
In our example, after simplifying both parts of the fraction, we ended up with \( \frac{22}{12} \). To simplify this, we found the GCD of 22 and 12, which is 2, and then divided both by 2:

• Numerator: \( 22 \div 2 = 11 \)
• Denominator: \( 12 \div 2 = 6 \)

So, \( \frac{22}{12} \) simplified becomes \( \frac{11}{6} \).
Remember, a fraction is in its simplest form when the numerator and denominator have no common factors other than 1.

Math Problem Solving

Effective math problem solving involves breaking down the problem into smaller, more manageable steps.
In our exercise, we tackled the problem by simplifying the numerator and the denominator separately.
Here are some steps to simplify problem-solving:

• Understand the problem: Read it carefully to understand what is being asked.
• Plan a strategy: Decide the best way to approach the problem.
• Execute the plan: Carry out your strategy step-by-step.
• Review: Check your work to make sure it's correct.

Following these steps ensures you don't miss any important details and helps organize your work.

Order of Operations

The order of operations is a rule used to clarify which procedures should be performed first when evaluating a mathematical expression.
The commonly remembered acronym is PEMDAS:

• P: Parentheses – Simplify inside them first.
• E: Exponents – Next, take care of any exponents.
• M/D: Multiplication and Division – Perform these operations from left to right.
• A/S: Addition and Subtraction – Finally, handle these from left to right.

In our example, we first dealt with the parentheses inside the numerator. Then, we performed the multiplications and finally did the subtraction. Applying the order of operations correctly ensures you get the right answer every time.