Addition and Subtraction of Fractions

Introduction

Key Concepts

Understanding Fractions

A fraction consists of two parts: the numerator and the denominator. The numerator indicates how many parts are being considered, while the denominator signifies the total number of equal parts in a whole. For example, in the fraction $\frac{3}{4}$, 3 is the numerator, and 4 is the denominator, meaning three out of four equal parts.

Types of Fractions

• Proper Fractions: The numerator is less than the denominator (e.g., $\frac{2}{5}$).
• Improper Fractions: The numerator is greater than or equal to the denominator (e.g., $\frac{5}{3}$).
• Mixed Numbers: A whole number combined with a proper fraction (e.g., $1 \frac{2}{3}$).

Equivalent Fractions

Equivalent fractions are different fractions that represent the same quantity. They can be generated by multiplying or dividing both the numerator and denominator by the same non-zero number. For example, $\frac{1}{2}$ is equivalent to $\frac{2}{4}$, $\frac{3}{6}$, and $\frac{4}{8}$.

Finding Common Denominators

To add or subtract fractions, they must have the same denominator, known as a common denominator. The least common denominator (LCD) is the smallest multiple that two or more denominators share. Finding the LCD simplifies the process of adding or subtracting fractions. For instance, to add $\frac{1}{3}$ and $\frac{1}{4}$, the LCD of 3 and 4 is 12.

Steps to find the LCD:

1. List the multiples of each denominator.
2. Identify the smallest multiple common to both lists.
3. Use this common multiple as the new denominator.

In the example, the multiples of 3 are 3, 6, 9, 12, ... and the multiples of 4 are 4, 8, 12, 16, ... The LCD is 12.

Adding Fractions

To add fractions with different denominators:

1. Find the LCD of the denominators.
2. Convert each fraction to an equivalent fraction with the LCD.
3. Add the numerators while keeping the denominator the same.
4. Simplify the resulting fraction if possible.

Example: Add $\frac{2}{5}$ and $\frac{3}{4}$.

First, find the LCD of 5 and 4, which is 20.

Convert the fractions: $$ \frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20} $$ $$ \frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20} $$

Add the numerators: $$ \frac{8}{20} + \frac{15}{20} = \frac{23}{20} = 1 \frac{3}{20} $$

Subtracting Fractions

Subtracting fractions follows a similar process to addition:

1. Find the LCD of the denominators.
2. Convert each fraction to an equivalent fraction with the LCD.
3. Subtract the numerators while keeping the denominator the same.
4. Simplify the resulting fraction if possible.

Example: Subtract $\frac{5}{6}$ from $\frac{3}{4}$.

First, find the LCD of 6 and 4, which is 12.

Convert the fractions: $$ \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} $$ $$ \frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12} $$

Subtract the numerators: $$ \frac{9}{12} - \frac{10}{12} = \frac{-1}{12} = -\frac{1}{12} $$

Adding and Subtracting Mixed Numbers

When dealing with mixed numbers, it's often easier to first convert them to improper fractions.

Example: Add $1 \frac{2}{3}$ and $2 \frac{1}{4}$.

Convert to improper fractions: $$ 1 \frac{2}{3} = \frac{5}{3} $$ $$ 2 \frac{1}{4} = \frac{9}{4} $$

Find the LCD of 3 and 4, which is 12.

Convert the fractions: $$ \frac{5}{3} = \frac{5 \times 4}{3 \times 4} = \frac{20}{12} $$ $$ \frac{9}{4} = \frac{9 \times 3}{4 \times 3} = \frac{27}{12} $$

Add the numerators: $$ \frac{20}{12} + \frac{27}{12} = \frac{47}{12} = 3 \frac{11}{12} $$>

Simplifying Fractions

After adding or subtracting fractions, it's important to simplify the result to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor (GCD).

Example: Simplify $\frac{18}{24}$.

The GCD of 18 and 24 is 6.

Divide both numerator and denominator by 6: $$ \frac{18 \div 6}{24 \div 6} = \frac{3}{4} $$>

Common Mistakes to Avoid

• Forgetting to find a common denominator before adding or subtracting.
• Mistaking the LCD with the greatest common divisor (GCD).
• Incorrectly converting mixed numbers to improper fractions.
• Neglecting to simplify the final answer.

Word Problems Involving Fractions

Applying addition and subtraction of fractions to real-life scenarios enhances comprehension and utility. Consider the following example:

Example: Sarah baked $\frac{3}{4}$ of a cake on Monday and $\frac{2}{3}$ of a cake on Tuesday. How much cake did she bake in total?

Find the LCD of 4 and 3, which is 12.

Convert the fractions: $$ \frac{3}{4} = \frac{9}{12} $$> $$ \frac{2}{3} = \frac{8}{12} $$>

Add the numerators: $$ \frac{9}{12} + \frac{8}{12} = \frac{17}{12} = 1 \frac{5}{12} $$>

Sarah baked a total of $1 \frac{5}{12}$ cakes.

Visual Representations

Using visual aids like fraction bars or circles can help students better understand the concepts of adding and subtracting fractions. These tools illustrate how different fractions occupy parts of a whole and how they combine or decrease when operations are performed.

Practice Exercises

Consistent practice is key to mastering fraction operations. Here are some exercises to reinforce learning:

1. Add $\frac{1}{2}$ and $\frac{3}{4}$.
2. Subtract $\frac{5}{6}$ from $\frac{7}{8}$.
3. Add $2 \frac{1}{3}$ and $3 \frac{2}{5}$.
4. Subtract $1 \frac{1}{2}$ from $3 \frac{3}{4}$.
5. Simplify the fraction $\frac{16}{20}$.

Answers:

1. $\frac{1}{2} + \frac{3}{4} = \frac{2}{4} + \frac{3}{4} = \frac{5}{4} = 1 \frac{1}{4}$
2. $\frac{7}{8} - \frac{5}{6} = \frac{21}{24} - \frac{20}{24} = \frac{1}{24}$
3. $2 \frac{1}{3} + 3 \frac{2}{5} = \frac{7}{3} + \frac{17}{5} = \frac{35}{15} + \frac{51}{15} = \frac{86}{15} = 5 \frac{11}{15}$
4. $3 \frac{3}{4} - 1 \frac{1}{2} = \frac{15}{4} - \frac{3}{2} = \frac{15}{4} - \frac{6}{4} = \frac{9}{4} = 2 \frac{1}{4}$
5. $\frac{16}{20} = \frac{4}{5}$

Comparison Table

Summary and Key Takeaways