Solving Algebraic Fractions and Understanding the Concept

Algebraic fractions are an important part of mathematics that allow us to solve complex equations and expressions. The problem at hand involves finding the original fraction when both the numerator and denominator are increased by a certain value. Let's explore how to solve such a problem step-by-step.

Problem Statement

The problem is as follows: The denominator of the fraction is 2 more than the numerator. If both the numerator and denominator are increased by 1, the resulting fraction is 1/2. What is the original fraction?

The Given Information and Formulation

We are given the following information:

The original fraction can be represented as N/D, where the denominator is 2 more than the numerator. Mathematically, this can be written as N/N3, where N3 N 2. When the numerator and denominator are increased by 1, the fraction becomes 1/2.

Solving the Problem

Let's denote the numerator as n and the denominator as n3, such that n3 n 2. We are also given that when both the numerator and denominator are increased by 1, the resulting fraction is 1/2.

Step-by-step Solution

Define the relationships and set up the equation: Let the original fraction be n/(n 2). Increase both the numerator and the denominator by 1: (n 1)/(n 3) 1/2. Solve for n: Multiply both sides by 2(n 3) to eliminate the fraction:

2(n 1) n 3

Expand and simplify:

2n 2 n 3

Subtract n from both sides:

n 2 3

Solve for n:

n 3 - 2

n 1

Thus, the original numerator is 1, and the original denominator is n 2 1 2 3. The original fraction is 1/3.

Conclusion

The original fraction, when both the numerator and denominator are increased by 1, results in a fraction that simplifies to 1/2. In this case, the original fraction is 1/3.

Additional Practice

Understanding algebraic fractions and solving such equations is essential in many mathematical fields, including calculus, algebra, and even real-world applications in engineering and physics. Practice similar problems to further enhance your skills:

The denominator of the fraction is 3 more than the numerator. If both the numerator and the denominator are increased by 1, the resulting fraction is 1/2. Find the original fraction. The numerator of the fraction is twice the denominator. If both the numerator and the denominator are increased by 3, the resulting fraction is 3/4. Find the original fraction.

Remember, the key to solving such problems is to set up the equations correctly and solve for the unknowns step-by-step.