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L.1 Add fractions with unlike denominators

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What does it mean to add fractions with unlike denominators?

Fractions with unlike denominators have different bottom numbers. To add them, we must first make the denominators the same before adding the numerators.

Example:
  • \( \frac{1}{4} + \frac{1}{3} \) → the denominators are different (4 and 3)
  • Find a common denominator, such as 12
  • \( \frac{1}{4} = \frac{3}{12} \) and \( \frac{1}{3} = \frac{4}{12} \)
  • Now add: \( \frac{3}{12} + \frac{4}{12} = \frac{7}{12} \)
Note

Always make sure the denominators are the same before adding or subtracting fractions.

How to find a common denominator

A common denominator is a number that both denominators divide into evenly. The simplest one to use is the least common denominator (LCD).

Steps:
  • List the multiples of each denominator.
  • Find the least common multiple (LCM).
  • Use that number as the new denominator for both fractions.
  • Adjust the numerators to keep the fractions equivalent.

Example: \( \frac{2}{5} + \frac{1}{10} \)

  • Multiples of 5: 5, 10, 15, 20...
  • Multiples of 10: 10, 20, 30...
  • LCD = 10
  • \( \frac{2}{5} = \frac{4}{10} \), so \( \frac{4}{10} + \frac{1}{10} = \frac{5}{10} = \frac{1}{2} \)
Note

Using the least common denominator makes adding fractions simpler and keeps numbers smaller.

Adding fractions step by step

To add fractions with unlike denominators, follow a clear sequence to get the correct answer.

Steps:
  • 1. Find the least common denominator (LCD).
  • 2. Rewrite each fraction as an equivalent fraction with the LCD.
  • 3. Add the numerators and keep the same denominator.
  • 4. Simplify the result if possible.

Example: \( \frac{3}{8} + \frac{1}{6} \)

  • Multiples of 8: 8, 16, 24
  • Multiples of 6: 6, 12, 18, 24
  • LCD = 24
  • \( \frac{3}{8} = \frac{9}{24} \), \( \frac{1}{6} = \frac{4}{24} \)
  • Add: \( \frac{9}{24} + \frac{4}{24} = \frac{13}{24} \)
Note

Keep the denominator the same after adding. Only the numerators are added together.

Checking your answer and simplifying

After adding, always check if your answer can be simplified. Simplifying means finding an equivalent fraction with smaller numbers.

Example:
  • \( \frac{2}{6} + \frac{1}{6} = \frac{3}{6} \)
  • \( \frac{3}{6} \) can be simplified to \( \frac{1}{2} \)
Note

To simplify, divide both the numerator and denominator by their greatest common factor (GCF).

Real-world connections

Adding fractions helps in many everyday situations, such as measuring ingredients or calculating distances.

Example:
  • You have \( \frac{1}{2} \) cup of milk and add \( \frac{1}{4} \) cup more.
  • Find the common denominator: 4
  • \( \frac{1}{2} = \frac{2}{4} \), so \( \frac{2}{4} + \frac{1}{4} = \frac{3}{4} \)
  • You now have \( \frac{3}{4} \) cup of milk.
Note

Fraction addition is useful whenever you combine parts of a whole in real life.

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