Adding and Subtracting Fractions

Adding and Subtracting Fractions

Fractions show up in everyday life — half a pizza, a quarter of an hour, three-eighths of a cake. When we want to know how much there is in total, or how much is left, we add and subtract fractions. On this page we'll learn how to do that in a clear, step-by-step way.

Background and Basic Definitions

A fraction has two parts:

  • Numerator — the number on top; how many parts we have.
  • Denominator — the number on the bottom; how many equal parts the whole is divided into.

To add or subtract fractions, the denominators must be the same. Once they match, we only add or subtract the numerators.

\[ \frac{a}{n} + \frac{b}{n} = \frac{a+b}{n} \qquad \frac{a}{n} - \frac{b}{n} = \frac{a-b}{n} \]

When the denominators are different, we find the least common denominator (LCD) — the smallest number divisible by both denominators — and expand each fraction accordingly.

At the end, always check whether you can simplify: divide both the numerator and denominator by the same number until you reach the simplest form.

Solution Steps

  1. Step 1 — Check whether the denominators are the same. If so, skip to Step 3.
  2. Step 2 — Find the least common denominator (LCD) of both denominators. Expand each fraction to that denominator by multiplying the numerator and denominator by the same number.
  3. Step 3 — Add or subtract only the numerators. The denominator stays the same!
  4. Step 4 — Check whether you can simplify: look for a common factor of the numerator and denominator and divide both by it.
  5. Step 5 — If the numerator is greater than the denominator, convert to a mixed number (whole number + fraction). Check that your answer makes sense.

Worked Examples

Example 1: Adding Fractions with the Same Denominator

Problem: Calculate: \( \frac{3}{7} + \frac{2}{7} \)

Solution:

  1. The denominators are equal — both 7. We add only the numerators.
  2. \( \frac{3}{7} + \frac{2}{7} = \frac{3+2}{7} = \frac{5}{7} \)
  3. Check for simplification: 5 and 7 share no common factor (both are prime). Cannot simplify.

Answer: \( \frac{5}{7} \)

Example 2: Subtracting Fractions with the Same Denominator

Problem: Calculate: \( \frac{5}{9} - \frac{2}{9} \)

Solution:

  1. The denominators are the same — 9. We subtract the numerators.
  2. \( \frac{5}{9} - \frac{2}{9} = \frac{5-2}{9} = \frac{3}{9} \)
  3. Simplify: 3 and 9 are both divisible by 3. \( \frac{3 \div 3}{9 \div 3} = \frac{1}{3} \)

Answer: \( \frac{1}{3} \)

Example 3: Adding Fractions with Different Denominators

Problem: Calculate: \( \frac{1}{4} + \frac{1}{6} \)

Solution:

  1. The denominators differ: 4 and 6. Find the LCD. Multiples of 4: 4, 8, 12… Multiples of 6: 6, 12… LCD = 12.
  2. Expand: \( \frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12} \), \( \frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12} \)
  3. Add: \( \frac{3}{12} + \frac{2}{12} = \frac{5}{12} \)
  4. 5 and 12 share no common factor — the fraction is fully simplified.

Answer: \( \frac{5}{12} \)

Example 4: Subtracting Fractions with Different Denominators

Problem: Calculate: \( \frac{3}{4} - \frac{1}{3} \)

Solution:

  1. LCD of 4 and 3: multiples of 4: 4, 8, 12. Multiples of 3: 3, 6, 9, 12. LCD = 12.
  2. Expand: \( \frac{3}{4} = \frac{9}{12} \), \( \frac{1}{3} = \frac{4}{12} \)
  3. \( \frac{9}{12} - \frac{4}{12} = \frac{5}{12} \)
  4. 5 and 12 share no common factor — the answer is final.

Answer: \( \frac{5}{12} \)

Example 5: Adding Three Fractions with Different Denominators

Problem: Calculate: \( \frac{1}{2} + \frac{1}{3} + \frac{1}{6} \)

Solution:

  1. LCD of 2, 3 and 6: 6.
  2. \( \frac{1}{2} = \frac{3}{6} \), \( \frac{1}{3} = \frac{2}{6} \), \( \frac{1}{6} = \frac{1}{6} \)
  3. \( \frac{3}{6} + \frac{2}{6} + \frac{1}{6} = \frac{6}{6} = 1 \)
  4. The numerator equals the denominator — that's exactly one whole!

Answer: \( 1 \) (one whole)

Common Mistakes

✗ Common mistake: Adding the denominators too: \( \frac{1}{3} + \frac{1}{3} = \frac{2}{6} \)

✓ The correct way: The denominator does NOT get added! When denominators are equal, they stay as they are. The correct answer is: \( \frac{1}{3} + \frac{1}{3} = \frac{2}{3} \). Think of it this way: two one-third pieces of cake together make two-thirds — not one-sixth.

✗ Common mistake: Forgetting to find a common denominator and adding directly: \( \frac{1}{2} + \frac{1}{3} = \frac{2}{5} \)

✓ The correct way: You cannot add fractions with different denominators directly. First expand both to the same denominator. The correct way: \( \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \).

✗ Common mistake: Forgetting to simplify at the end: leaving \( \frac{4}{8} \) instead of \( \frac{1}{2} \)

✓ The correct way: Always check at the end: do the numerator and denominator share a common factor? If yes — divide! The simplest form is the correct form.

Practice Tips

  • Tip — To find the LCD quickly: list multiples of the larger number (×2, ×3, ×4…) and check when the smaller number also divides evenly into the result.
  • Tip — After adding or subtracting, use a logic check: does the answer make sense? Two halves should add up to one whole — did you get 1?
  • Tip — When expanding a fraction: whatever you do to the denominator, you must also do to the numerator. Multiplied the denominator by 3? Multiply the numerator by 3 too.
  • Tip — To simplify: look for a number that divides evenly into both the numerator and denominator. Start with 2, then 3, then 5…

Summary and Key Formulas

  • When denominators are equal: \( \frac{a}{n} \pm \frac{b}{n} = \frac{a \pm b}{n} \)
  • When denominators are different: find the LCD, expand each fraction, then add or subtract the numerators.
  • At the end — always simplify if possible.
  • If the numerator is greater than the denominator, you can write the result as a mixed number.