Solving Equations with Fractions: A Step-by-Step Guide

Let's delve into the intricacies of solving equations involving fractions. In mathematics, fractions can sometimes present complex scenarios. In this article, we will solve a specific problem by breaking it down into manageable steps and incorporating the given conditions to find the answer.

Problem Statement

We are given that the denominator of a fraction is 5 more than the numerator, and when we subtract 2 from both the numerator and the denominator, the fraction becomes 1/6.

Problem Setup

Let's denote the numerator of the fraction as ( x ). According to the problem, the denominator is ( x 5 ). Therefore, the fraction can be expressed as:

Original Fraction: (frac{x}{x 5})

When we subtract 2 from both the numerator and the denominator, the new fraction becomes:

Adjusted Fraction: (frac{x - 2}{x 3})

According to the problem, this new fraction equals (frac{1}{6}). Therefore, we can set up the equation:

(frac{x - 2}{x 3} frac{1}{6})

Step-by-Step Solution

To solve for ( x ), we start by cross-multiplying:

[6(x - 2) (x 3)]

Expanding both sides of the equation, we get:

[6x - 12 x 3]

Next, we isolate ( x ) by moving the ( x )-terms to one side and the constant terms to the other side:

[6x - x 3 12]

This simplifies to:

[5x 15]

Dividing both sides by 5 gives:

[x 3]

Now that we have the numerator ( x 3 ), we can find the denominator:

[x 5 3 5 8]

Thus, the original fraction is:

(frac{3}{8})

Verification

Let's verify the solution by checking the condition after subtracting 2 from both the numerator and the denominator:

(frac{3 - 2}{8 - 2} frac{1}{6})

This confirms that our solution is correct.

Conclusion

The original fraction is (frac{3}{8}).

This problem demonstrates the importance of algebraic manipulation and step-by-step solving in mathematics. Understanding how to approach and solve such equations is crucial for both academic and real-world applications.