Simplifying Fractions and Rational Expressions: A Comprehensive Guide

Introduction

Fractions and rational expressions are fundamental components of mathematics, used in various fields including engineering, physics, and economics. Properly simplifying these expressions ensures accuracy and clarity. This article will guide you through the process of simplifying the given expression 3/4 1/3 2/6, specifically using the J programming language's rational fraction notation.

Understanding the Expression

Let's start with the given expression: 3/4 1/3 2/6. When dealing with fractions, we first need to find a common denominator to ensure that we can perform operations such as addition and subtraction.

Finding a Common Denominator

The denominators of our fractions are 4, 3, and 6. The least common denominator (LCD) is the smallest number that all the denominators divide into without a remainder. In this case, the LCD is 12.

Step-by-Step Conversion

Convert 3/4 to have the LCD as the denominator: 3/4 becomes 3 × 3 / 4 × 3 9/12 Convert 1/3 to have the LCD as the denominator: 1/3 becomes 1 × 4 / 3 × 4 4/12 Convert 2/6 to have the LCD as the denominator: 2/6 becomes 2 × 2 / 6 × 2 4/12

Now that all the fractions have the same denominator, we can add them together:

Adding the Fractions

9/12 4/12 4/12 (9 4 4) / 12 17/12

The simplified result is 17/12, which is an improper fraction. This can be left as is or converted to a mixed number: 1 5/12.

Alternative Method: Simplifying Before Addition

My first move would be to simplify the fractions as much as possible. While I can't do anything with the first two, I can with the third. Note that the numerator and denominator are both even, which means that they are both multiples of 2. So, we can divide both by 2:

Step 1: Simplify the Fraction

2/6 simplified is 1/3.

Step 2: Add the Simplified Fractions

Now our expression is 3/4 1/3 1/3. Since two of our fractions have the same denominator, we can add them:

1/3 1/3 (1 1) / 3 2/3

Step 3: Combine the Results

Now our expression is 3/4 2/3. The simplest way of ensuring all fractions have the same denominator is to multiply the denominators together to create a new denominator. In this case, we have 4 × 3 12.

For the first fraction, 3/4, we have 12 divided by 4, which equals 3, so the new fraction is:

3 × 3 / 4 × 3 9/12

For the second, 2/3, we have 12 divided by 3, which equals 4, so the new fraction is:

2 × 4 / 3 × 4 8/12

Step 4: Add the New Fractions

Now that both fractions have the same denominator, we can add them:

9/12 8/12 (9 8) / 12 17/12

Therefore, the simplified result is 17/12.

Conclusion

Simplifying fractions and rational expressions is crucial for maintaining accuracy in mathematical operations. Whether you're using a common denominator or simplifying before adding, understanding the steps is key to solving complex problems efficiently.