Problem 85 - Algebra 1

Question

Simplify the expression. $$\frac{8 x}{3} \cdot \frac{1}{x}$$

Step-by-Step Solution

Answer

The simplified form of the expression \( \frac{8 x}{3} \cdot \frac{1}{x} \) is \( \frac{8}{3} \)

Step 1: Identify the common factors

Firstly, identify the similar factors in the numerator and denominator. Here, 'x' is a common factor in the numerator of the first fraction and the denominator of the second fraction.

Step 2: Cancel out the common factor

Next, cancel out the common factor 'x' from the numerator of the first fraction and the denominator of the second fraction. We are allowed to do that because multiplying by 1 doesn't affect the value of the expression.

Step 3: Simplify the expression

After cancelling out the common factor, The expression simplifies to \( \frac{8}{3} \). Thus, \( \frac{8 x}{3} \cdot \frac{1}{x} \) simplifies to \( \frac{8}{3} \).

Key Concepts

Understanding Fractions

Fractions are a way of representing a part of a whole. They consist of two main components: the numerator and the denominator. The number on top, known as the numerator, indicates how many parts we are considering. The number on the bottom, called the denominator, shows into how many parts the whole is divided. For example, in the fraction \( \frac{3}{4} \), 3 is the numerator and 4 is the denominator.
Fractions can represent values greater than one when the numerator is larger than the denominator. They are widely used in math to express parts of numbers that are not whole.

• Numerator: Number of parts we have.
• Denominator: Total parts that make up the whole.

When simplifying fractions, we look for possibilities to cancel out numbers or variables common to both the numerator and the denominator, which leads us to our next key concept.

Common Factors in Simplification

A common factor is a number or variable that divides exactly into two or more numbers or expressions. In the context of simplifying fractions, common factors play a crucial role in reducing the expression to its simplest form.
Finding common factors involves identifying values that are present both in the numerators and denominators.

• Look for shared numbers or variables.
• These common elements simplify calculations by reducing the fractions.

In our original exercise, the variable \( x \) is a common factor. It appears in the numerator of the first fraction and the denominator of the second fraction. Recognizing these common factors is essential because it helps in cancelling out terms, leading to a simpler expression.

Cancelling Out Terms

Cancelling out terms is a method to simplify expressions by eliminating factors found both in the numerator and the denominator. This is possible because dividing a number by itself equals one, and multiplying by one does not change the original value of a number.
For example, if both the numerator and denominator have a factor of \( x \), we can remove this factor from both parts, effectively reducing the complexity of the fraction without changing its value.

• Find a common factor in the numerator and denominator.
• Cancel out these factors to simplify the overall fraction.

In our example \( \frac{8x}{3} \cdot \frac{1}{x} \), the \( x \) in the numerator and the \( x \) in the denominator cancel each other out, resulting in the simplified expression \( \frac{8}{3} \). This technique is a powerful tool in algebra to simplify expressions quickly and efficiently.