Problem 2 - Introductory Algebra for College Students

Question

Simplify complex rational expression by the method of your choice. \(\frac{\frac{1}{3}+\frac{1}{4}}{\frac{1}{3}+\frac{1}{6}}\)

Step-by-Step Solution

Answer

The simplified form of the given complex rational expression is \( \frac{7}{6} \)

Step 1: Simplify the Numerator

The numerator is \( \frac{1}{3} + \frac{1}{4} \) . To simplify, find the least common denominator (LCD) of these two fractions, which is 12. Rewrite each fraction with the LCD and sum them up, \( \frac{4}{12} + \frac{3}{12} = \frac{7}{12} \). So, the simplified numerator is \( \frac{7}{12} \).

Step 2: Simplify the Denominator

The denominator is \( \frac{1}{3} + \frac{1}{6} \) . The LCD of these fractions is 6. Rewrite each fraction with the LCD and add them up, \( \frac{2}{6} + \frac{1}{6} = \frac{3}{6} \). So, the simplified denominator is \( \frac{3}{6} \), which further simplifies to \( \frac{1}{2} \).

Step 3: Divide the Numerator by the Denominator

The expression now becomes \( \frac{\frac{7}{12}}{\frac{1}{2}} \). To divide fractions, multiply the first fraction by the reciprocal of the second fraction. This gives us \( \frac{7}{12} * 2 = \frac{7}{6} \)

Key Concepts

Simplifying Fractions

Simplifying fractions is a fundamental skill in mathematics. When you simplify, you make the fraction as simple as possible, ideally reducing it to a form where the numerator and the denominator have no common factors except for 1. For instance, suppose we have a fraction \( \frac{6}{8} \). To simplify this:

• Identify the greatest common factor (GCF) of 6 and 8, which is 2.
• Divide both numerator and denominator by the GCF: \( \frac{6}{2} = 3 \) and \( \frac{8}{2} = 4 \).
• The simplified form is \( \frac{3}{4} \).

By simplifying fractions, we achieve a clearer, more manageable form, which is crucial for performing operations like addition, subtraction, multiplication, or division.

Least Common Denominator

Finding the least common denominator (LCD) is crucial when adding or subtracting fractions. The LCD, the smallest multiple that the denominators of the given fractions share, allows us to compare and perform arithmetic operations on these fractions easily. Let's consider two fractions: \( \frac{1}{3} \) and \( \frac{1}{4} \). Here's how you find the LCD:

• List the multiples of each denominator. For 3: 3, 6, 9, 12; for 4: 4, 8, 12, 16.
• Identify the smallest common multiple. Here, it's 12.
• Adjust each fraction: \( \frac{1}{3} \) becomes \( \frac{4}{12} \) and \( \frac{1}{4} \) becomes \( \frac{3}{12} \).

Utilizing the LCD allows for smoother calculations, particularly in complex rational expressions where multiple fractions are involved.

Dividing Fractions

Dividing fractions might seem daunting, but it's actually quite straightforward. Essentially, dividing by a fraction is the same as multiplying by its reciprocal. Let's say you want to divide \( \frac{2}{3} \) by \( \frac{4}{5} \). Follow these steps:

• First, find the reciprocal of the divisor \( \frac{4}{5} \), which is \( \frac{5}{4} \).
• Then, multiply the first fraction \( \frac{2}{3} \) by this reciprocal: \( \frac{2}{3} \times \frac{5}{4} \).
• Multiply the numerators: 2 \( \times \) 5 = 10, and the denominators: 3 \( \times \) 4 = 12, resulting in \( \frac{10}{12} \).
• Simplify \( \frac{10}{12} \) to its simplest form \( \frac{5}{6} \).

This process highlights the simplicity and logic of fraction operations, offering clarity in solving more complex problems.