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Adding and Subtracting Fractions

Master operations with like and unlike denominators using least common denominator strategies.

Lesson 3 of 10 Fractions, Decimals & Percentages Beginner ⏱ 9 min read
🔥 Why This Matters

Adding fractions comes up every time you combine partial amounts — splitting a restaurant bill, adding work hours logged as half-days, or mixing paint in fractional cups. Get the denominator wrong and your budget is off, your time sheet is wrong, or your paint color is ruined. This is the most used fraction operation in everyday adult life.

🎯 What You'll Learn
  • Add and subtract fractions that share the same denominator
  • Find the Least Common Denominator (LCD) and build equivalent fractions to add unlike denominators
  • Add and subtract mixed numbers by converting to improper fractions first
📖 Key Vocabulary
Like DenominatorsFractions that share the same bottom number: \(\frac{1}{4}\) and \(\frac{3}{4}\). Unlike DenominatorsFractions with different bottom numbers: \(\frac{1}{3}\) and \(\frac{1}{4}\). LCDLeast Common Denominator — the smallest number that both denominators divide into evenly. Equivalent FractionA fraction with a different numerator/denominator but the same value: \(\frac{1}{2} = \frac{2}{4}\).
Key Concept

You can only add or subtract fractions that have the same denominator. With like denominators, add the numerators and keep the denominator:

\[ \frac{a}{c} + \frac{b}{c} = \frac{a + b}{c} \]

With unlike denominators, you must first rewrite both fractions with the LCD as the new denominator, then add normally.

Finding the LCD: list multiples of each denominator until you find the first one they share. Example — denominators 3 and 4: multiples of 3 are 3,6,9,12… multiples of 4 are 4,8,12… LCD = 12.

Visualizing 1/4 + 2/4 = 3/4

Adding Like Fractions — 1/4 + 2/4

+
=

\(\frac{1}{4} + \frac{2}{4} = \frac{3}{4}\) — same denominator means the pieces are the same size.

Adding Unlike Fractions: Step by Step

To add \(\frac{1}{3} + \frac{1}{4}\):

  1. Find LCD of 3 and 4 → 12.
  2. Convert: \(\frac{1}{3} = \frac{4}{12}\) (multiply top and bottom by 4) and \(\frac{1}{4} = \frac{3}{12}\) (multiply by 3).
  3. Add: \(\frac{4}{12} + \frac{3}{12} = \frac{7}{12}\).
  4. Simplify if needed — GCF(7,12)=1, already simplified.
\[ \frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12} \]
Worked Example 1 — Basic: Like Denominators

Calculate \(\frac{5}{8} - \frac{3}{8}\).

Same denominator — subtract numerators, keep denominator:

\[ \frac{5}{8} - \frac{3}{8} = \frac{5-3}{8} = \frac{2}{8} = \frac{1}{4} \]

Simplify: GCF(2,8)=2 → \(\frac{1}{4}\).

Worked Example 2 — Intermediate: Unlike Denominators

Calculate \(\frac{2}{5} + \frac{3}{4}\).

  1. LCD(5, 4) = 20.
  2. \(\frac{2}{5} = \frac{8}{20}\), \(\frac{3}{4} = \frac{15}{20}\).
  3. \(\frac{8}{20} + \frac{15}{20} = \frac{23}{20}\).
  4. Convert to mixed: 23 ÷ 20 = 1 remainder 3 → \(1\frac{3}{20}\).
\[ \frac{2}{5} + \frac{3}{4} = 1\frac{3}{20} \]
Worked Example 3 — Real World: Splitting a Workday

Project manager Elena tracked her time: she spent \(\frac{2}{3}\) of the morning on client calls and \(\frac{1}{4}\) of the morning on email. What fraction of her morning was spent on these tasks?

  1. LCD(3, 4) = 12.
  2. \(\frac{2}{3} = \frac{8}{12}\), \(\frac{1}{4} = \frac{3}{12}\).
  3. \(\frac{8}{12} + \frac{3}{12} = \frac{11}{12}\) of her morning.

Elena used 11/12 of her morning — leaving only 1/12 (about 5 minutes of a 1-hour block) unaccounted for.

✏️ Quick Check

Test yourself:

  1. Calculate \(\frac{3}{7} + \frac{2}{7}\).
  2. Calculate \(\frac{1}{2} + \frac{1}{3}\). What is the LCD?
  3. Calculate \(\frac{5}{6} - \frac{1}{4}\).
▶ Show Answers
  1. \(\frac{5}{7}\) — same denominator, add numerators.
  2. LCD = 6. \(\frac{3}{6} + \frac{2}{6} = \frac{5}{6}\).
  3. LCD=12. \(\frac{10}{12} - \frac{3}{12} = \frac{7}{12}\).
⚠️ Common Mistakes
  • Adding denominators: ❌ \(\frac{1}{3} + \frac{1}{4} = \frac{2}{7}\) — completely wrong. ✅ Find the LCD first, then add only the numerators.
  • Forgetting to simplify the result: \(\frac{2}{8}\) should always be reduced to \(\frac{1}{4}\). Always check if GCF > 1 after adding.
  • Wrong LCD: Using a common denominator that isn't the least still works but creates larger numbers to simplify later. The LCD keeps arithmetic cleaner.
✅ Key Takeaways
  • Same denominator = add/subtract numerators only, keep the denominator.
  • Unlike denominators require the LCD — convert both fractions to the same denominator first.
  • Build equivalent fractions by multiplying top and bottom by the same number.
  • Always simplify your answer after adding or subtracting.
💼 Career Connection — Construction & Trades

Carpenters and plumbers add fractional measurements all day: a board 2½ inches wide plus a gap of ⅜ inch must total exactly 2⅞ inches for a proper fit. Using a common denominator — eighths — makes the addition fast and error-free. A fraction error in framing means a wall that doesn't align, costing hours of rework.

Try it with the Calculator

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Fraction Arithmetic
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