Adding and Subtracting Fractions

Key Concepts

Adding and subtracting fractions involves understanding the following key concepts:

• Common Denominators
• Finding the Least Common Denominator (LCD)
• Adding Fractions
• Subtracting Fractions

Common Denominators

Fractions can only be added or subtracted if they have the same denominator. The denominator is the bottom number of a fraction, and it represents the total number of equal parts in a whole.

Example: The fractions 1/4 and 3/4 have the same denominator (4), so they can be added or subtracted directly.

Finding the Least Common Denominator (LCD)

If the fractions do not have the same denominator, you need to find a common denominator. The least common denominator (LCD) is the smallest number that both denominators can divide into evenly.

Example: To find the LCD of 1/3 and 1/6, list the multiples of each denominator:

Multiples of 3: 3, 6, 9, 12, ...

Multiples of 6: 6, 12, 18, ...

The smallest common multiple is 6, so the LCD is 6.

Adding Fractions

To add fractions with the same denominator, add the numerators (top numbers) and keep the denominator the same.

Example: Add 1/4 and 3/4.

1/4 + 3/4 = (1 + 3)/4 = 4/4 = 1

To add fractions with different denominators, first find the LCD, then convert each fraction to have the LCD as the denominator, and finally add the numerators.

Example: Add 1/3 and 1/6.

LCD is 6. Convert 1/3 to 2/6 (multiply numerator and denominator by 2).

1/3 + 1/6 = 2/6 + 1/6 = (2 + 1)/6 = 3/6 = 1/2

Subtracting Fractions

To subtract fractions with the same denominator, subtract the numerators and keep the denominator the same.

Example: Subtract 3/4 from 5/4.

5/4 - 3/4 = (5 - 3)/4 = 2/4 = 1/2

To subtract fractions with different denominators, first find the LCD, then convert each fraction to have the LCD as the denominator, and finally subtract the numerators.

Example: Subtract 1/3 from 1/2.

LCD is 6. Convert 1/3 to 2/6 (multiply numerator and denominator by 2), and 1/2 to 3/6 (multiply numerator and denominator by 3).

1/2 - 1/3 = 3/6 - 2/6 = (3 - 2)/6 = 1/6

Examples and Analogies

Imagine you have two pizzas. One pizza is cut into 4 equal slices (1/4 each), and the other is cut into 8 equal slices (1/8 each). If you want to add the slices together, you need to cut the first pizza into 8 slices (2/8 each) to match the second pizza. Now you can add the slices: 2/8 + 1/8 = 3/8.

Similarly, if you have 5/6 of a cake and you eat 1/6, you are left with 4/6 of the cake. This is because 5/6 - 1/6 = 4/6 = 2/3.

Practical Application

Understanding how to add and subtract fractions is essential for everyday tasks such as cooking, where recipes often involve fractions of ingredients, and for more complex mathematical problems in higher grades.