Simplify (5v-25)/(5v)

Solution:

Simplification Process:

Step 1:

Extract the common factor of $5$ from the numerator $5v$.

$$\frac{5(v) - 25}{5v}$$

Step 2:

Extract the common factor of $5$ from $-25$.

$$\frac{5(v) - 5 \cdot 5}{5v}$$

Step 3:

Take out the factor of $5$ from the entire numerator.

$$\frac{5(v - 5)}{5v}$$

Step 4:

Eliminate the common factors.

Step 4.1:

Identify the common factor of $5$ in the denominator.

$$\frac{5(v - 5)}{5(v)}$$

Step 4.2:

Remove the common factor of $5$.

$$\frac{\cancel{5}(v - 5)}{\cancel{5}v}$$

Step 4.3:

Finalize the simplified expression.

$$\frac{v - 5}{v}$$

Knowledge Notes:

To simplify a rational expression like $\frac{5v-25}{5v}$, we follow these steps:

1. Factorization: This involves breaking down the expressions into their simplest factors or components. In this case, we factor out the greatest common factor (GCF) from the numerator and denominator if possible.

2. Common Factors: After factorization, we look for common factors in the numerator and the denominator. Common factors are terms that appear in both the numerator and the denominator and can be cancelled out.

3. Cancellation: This is the process of simplifying the expression by dividing out the common factors from the numerator and the denominator. It's important to only cancel factors, not terms that are added or subtracted.

4. Rewriting: After cancellation, we rewrite what remains to present the simplified form of the original expression.

In the context of algebra, these steps are crucial for simplifying expressions, solving equations, and working with functions. Understanding how to manipulate and simplify algebraic expressions is foundational for higher-level mathematics.