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Arithmetic Operations

Fraction Addition

Adding fractions is straightforward once you understand one key rule: you can only add fractions that refer to the same-sized parts. That means the denominators must match.

Adding Fractions with the Same Denominator

When denominators are equal, simply add the numerators and keep the denominator.

⅓ + ⅓ = ⅔    (1+1=2, denominator stays 3)
2/7 + 3/7 = 5/7    (2+3=5, denominator stays 7)

Adding Fractions with Different Denominators

When denominators differ, you must find the lowest common denominator (LCD) — the smallest number that both denominators divide into exactly. Then convert each fraction to that denominator before adding.

Example: 1/3 + 1/4
  1. LCD of 3 and 4 = 12.
  2. Convert: 1/3 = 4/12 and 1/4 = 3/12.
  3. Add: 4/12 + 3/12 = 7/12.
1/3 + 1/4 = 4/12 + 3/12 = 7/12
Example: 2/5 + 3/10
  1. LCD of 5 and 10 = 10.
  2. Convert: 2/5 = 4/10.
  3. Add: 4/10 + 3/10 = 7/10.
2/5 + 3/10 = 4/10 + 3/10 = 7/10

Simplifying the Answer

Always check whether the resulting fraction can be simplified. Divide numerator and denominator by their highest common factor (HCF).

3/6 + 1/6 = 4/6 → simplify by dividing by 2 → 2/3

Adding Three Fractions

Find the LCD of all three denominators, convert all, then add the numerators.

1/2 + 1/3 + 1/6 → LCD=6 → 3/6 + 2/6 + 1/6 = 6/6 = 1

Key Takeaways

  • Same denominator: add numerators, keep denominator.
  • Different denominators: find LCD, convert, then add.
  • Always simplify (reduce) the answer if possible.
  • If the result is an improper fraction (numerator > denominator), convert to a mixed number.

Practice: Add These Fractions

Two fractions are shown below, with the same or different denominators. Give your answer as a fraction (e.g. 7/12), simplified where possible. If you get it wrong (or leave it blank), we'll find the LCD together.

Calculate the sum.