Problem 120 - Elementary and Intermediate Algebra

Question

Explain the error in the following addition. $$ \frac{4}{3}+\frac{3}{2}=\frac{4+3}{3+2}=\frac{7}{5} $$

Step-by-Step Solution

Answer

The error was adding numerators and denominators separately. Correctly, \( \frac{4}{3} + \frac{3}{2} = \frac{17}{6} \).

Step 1: Identify the Problem

Let's examine the presented addition of fractions: \( \frac{4}{3} + \frac{3}{2} = \frac{4+3}{3+2} = \frac{7}{5} \). The error lies in the incorrect way of adding fractions. The formula used is wrong because it adds numerators and denominators separately, which is not how fraction addition works.

Step 2: Review Fraction Addition Rules

Recall that to add fractions, they need to have a common denominator. To add \( \frac{a}{b} + \frac{c}{d} \), the fractions must be converted to have the same denominator, typically the least common denominator (LCD).

Step 3: Find the Least Common Denominator (LCD)

The denominators of the fractions are 3 and 2. The least common denominator (LCD) of 3 and 2 is 6. This will allow us to add the fractions correctly.

Step 4: Convert Fractions to Common Denominator

Convert \( \frac{4}{3} \) to the new denominator: Multiply both numerator and denominator by 2 to get \( \frac{8}{6} \). Similarly, convert \( \frac{3}{2} \) by multiplying both by 3 to get \( \frac{9}{6} \).

Step 5: Add the Converted Fractions

Once the fractions have been converted to have the same denominator, add the numerators and keep the common denominator: \( \frac{8}{6} + \frac{9}{6} = \frac{8+9}{6} = \frac{17}{6} \).

Step 6: Simplify the Result if Necessary

In this case, \( \frac{17}{6} \) is already in its simplest form, as 17 and 6 have no common factors other than 1.

Key Concepts

Fraction Addition Rules

When adding fractions, understanding the basic rules is crucial. Many mistakes arise from incorrect methods, such as adding numerators and denominators separately, as seen in the original error. The fundamental rule for adding fractions is that they must have the same denominator. This is because the denominator represents the same 'whole' or unit across the fractions. To ensure fractions are comparable and can be added directly, they must refer to the same division of this whole. To add fractions properly, follow these basic steps:

• Ensure both fractions have a common denominator.
• Add the numerators together, while keeping the common denominator unchanged.
• Simplify the resulting fraction if possible.

These straightforward steps will help you avoid common pitfalls, ensuring accurate fractions addition every time.

Least Common Denominator

Finding the least common denominator (LCD) is an essential part of adding fractions correctly. The LCD is the smallest number that can be divided by each denominator of the fractions without leaving a remainder. This common ground is crucial because it normalizes the fractions, making them comparable in terms of size and value. To find the LCD, you can use these steps:

• List the multiples of each denominator.
• Identify the smallest multiple that appears in both lists.

For instance, with denominators 3 and 2, the multiples of 3 (like 3, 6, 9) and 2 (like 2, 4, 6) both include 6, making it the LCD. Once you know the LCD, you adjust, or 'scale up', each fraction to this common denominator, paving the way for straightforward addition.

Simplifying Fractions

After adding fractions and arriving at a common denominator, the resulting fraction may need to be simplified. Simplifying is the process of reducing a fraction to its smallest form, where the numerator and denominator no longer share any common factors other than 1. This makes your final answer easy to read and understand. Here's how you simplify a fraction:

• Identify the greatest common factor (GCF) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCF.
• The result is the simplified fraction.

In cases where the fraction is already as simplified as possible, the fraction remains unchanged. Simplifying a fraction not only refines your final answer but also reinforces the concept of equivalent fractions, emphasizing that the simplified version is mathematically the same as the original.