Simplifying Complex Fractions: A Step-by-Step Approach

Simplifying complex fractions is a fundamental skill in high school mathematics. In this article, we will walk through the process of simplifying the expression: (9t + 3)/(6t - 5) = (3t + 6)/(3t - 5).

Step 1: Identify the Numerator and Denominator

The first step in simplifying a complex fraction is to identify the numerator and denominator. In our case, the numerator is 9t + 3 and the denominator is 6t - 5.

Step 2: Factor the Numerator and Denominator

Next, we need to factor the numerator and denominator to find common factors that can be canceled out.

The numerator 9t + 3 can be factored as 3(3t + 1). The denominator 6t - 5 can be factored as 3(2t - 1).

Step 3: Cancel Out Common Factors

Now that we have factored the numerator and denominator, we can cancel out the common factor of 3 between them.

(9t + 3)/(6t - 5) = (3(3t + 1))/(3(2t - 1)) = (3t + 1)/(2t - 1)

Step 4: Simplify the Resulting Fraction

The final step is to simplify the resulting fraction. We can do this by dividing the numerator and denominator by their greatest common factor.

The greatest common factor of 3t + 1 and 2t - 1 is 1, so the simplified fraction is:

(3t + 1)/(2t - 1)

Therefore, the simplified expression is:

(9t + 3)/(6t - 5) = (3t + 6)/(3t - 5)

By following this step-by-step approach, you can simplify complex fractions and improve your understanding of algebraic manipulations.