4.4 Add and Subtract Fractions with Common Denominators

Fractions with common denominators are like puzzle pieces that fit together perfectly. Adding them is as simple as combining the top numbers while keeping the bottom the same. Subtracting works similarly, just take away one top number from the other.

These operations are crucial for solving real-world problems involving parts of a whole. Whether you're dividing pizza or measuring ingredients, understanding how to work with fractions that share a common denominator is a fundamental math skill.

Adding and Subtracting Fractions with Common Denominators

Addition of fractions with common denominators

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• Add numerators while keeping the common denominator the same
  • $\frac{3}{8} + \frac{1}{8} = \frac{4}{8}$ (numerators 3 and 1 added, denominator 8 remains the same)
• Combine shaded parts of visual models (pie charts, bar models) with the same total number of equal parts
  • Two pie charts, each divided into 6 equal parts, with 2 parts shaded in one and 4 parts shaded in the other, when combined, result in 6 out of 6 parts shaded, or $\frac{6}{6} = 1$
• Formula for adding fractions with common denominators: $\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}$
  • $a$ and $b$ represent the numerators, $c$ represents the common denominator
• The fraction bar (also known as vinculum) separates the numerator from the denominator

Subtraction of fractions with common denominators

• Subtract numerators while keeping the common denominator the same
  • $\frac{7}{10} - \frac{3}{10} = \frac{4}{10}$ (numerator 3 subtracted from 7, denominator 10 remains the same)
• Remove shaded parts of one fraction from another with the same denominator in visual models
  • A bar model divided into 5 equal parts, with 4 parts shaded, when 2 shaded parts are removed, results in 2 out of 5 parts shaded, or $\frac{2}{5}$
• Formula for subtracting fractions with common denominators: $\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}$
  • $a$ and $b$ represent the numerators, $c$ represents the common denominator

Word problems for fraction operations

• Recognize fractions in the context of the problem, ensuring common denominators
  • If denominators differ, find a common denominator before performing operations
• Identify the required operation (addition or subtraction) based on the problem's context
  • Keywords like "combined," "total," or "altogether" often indicate addition, while "difference," "less," or "remaining" suggest subtraction
• Apply the appropriate operation to the fractions and simplify the result
• Express the final answer in the context of the word problem
  • "John ate $\frac{2}{6}$ of a pizza, and Mary ate $\frac{1}{6}$ of the same pizza. What fraction of the pizza did they eat together?" (Solution: $\frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}$, so they ate $\frac{1}{2}$ of the pizza together)

Importance of common denominators

• Fractions represent equal parts of a whole, with the denominator indicating the total number of equal parts
• Adding or subtracting fractions requires the parts to be the same size (i.e., common denominators)
  • Combining $\frac{1}{4}$ and $\frac{1}{5}$ directly is not meaningful, as quarters and fifths are different-sized parts
• Finding a common denominator allows fractions to be expressed as equivalent fractions with the same denominator
  • To subtract $\frac{3}{4}$ from $\frac{5}{6}$, find a common denominator of 12 and rewrite the fractions as $\frac{9}{12}$ and $\frac{10}{12}$, then subtract: $\frac{10}{12} - \frac{9}{12} = \frac{1}{12}$
• Common denominators enable the addition or subtraction of like parts, ensuring a meaningful result
• The least common denominator (LCD) is the smallest common denominator that can be used for the given fractions

Types of Fractions and Related Concepts

• Improper fractions have numerators greater than or equal to their denominators (e.g., $\frac{5}{3}$)
• Mixed numbers combine a whole number and a proper fraction (e.g., 2$\frac{1}{4}$)
• A reciprocal is found by flipping the numerator and denominator of a fraction (e.g., the reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$)