Add & Subtract Fractions
Add and subtract fractions with like and unlike denominators: rename with the LCD first, then work with the numerators — and always simplify the answer.
1 The golden rule of fraction addition
You can only add or subtract fractions when the pieces are the same size — that means a common denominator first.
Like denominators: easy
38 + 28 = 58
Same-size pieces: just add the numerators. The denominator does not change.
Unlike denominators: rename first
12 + 13 = 36 + 26 = 56
Rename both with the least common denominator (6), then add the tops.
Finish the job
Simplify the answer, and if it is improper (top ≥ bottom), convert it to a mixed number: 74 = 134.
2 See it: 1/2 + 1/3 as sixths
Halves and thirds do not match — rename both as sixths, then add.
1/2 = 3/6 and 1/3 = 2/6; 3 sixths + 2 sixths = 5/6.
3 Worked examples
Rename first, add or subtract the tops, then simplify.
Example 1 Like denominators: 3/8 + 2/8
- Thinking: same denominator 8 → same-size pieces.
- Add the numerators: 3 + 2 = 5. Keep the denominator.
- 5/8 is already in simplest form.
3/8 + 2/8 = 5/8
Example 2 Like denominators, subtraction: 5/6 − 4/6
- Thinking: same denominator 6 → subtract the numerators: 5 − 4 = 1.
- Keep the denominator: 1/6.
5/6 − 4/6 = 1/6
Example 3 Unlike denominators: 1/2 + 1/3
- Thinking: the LCD of 2 and 3 is 6.
- Rename: 1/2 = 3/6; 1/3 = 2/6.
- Add: 3/6 + 2/6 = 5/6. Already simplest.
1/2 + 1/3 = 5/6
Example 4 Improper answer: 3/4 + 1/2
- Thinking: LCD of 4 and 2 is 4: 3/4 + 2/4 = 5/4.
- 5/4 is improper (5 ≥ 4) → convert: 5 ÷ 4 = 1 R 1.
- So 5/4 = 1 1/4.
3/4 + 1/2 = 1 1/4
4 Common mistakes
Two traps catch almost everyone.
Mistake 1: adding the denominators
Wrong
12 + 13 = 25 — “add tops, add bottoms.”Right
12 + 13 = 56: rename with the LCD first, then add the tops.Fix: sanity check: 2/5 is SMALLER than 1/2, but adding a positive number cannot shrink the total.
Mistake 2: forgetting to rename both
Wrong
12 + 13 = 36 + 13 — only the first fraction was renamed.Right
Rename both: 3/6 + 2/6 = 5/6.Fix: after renaming, both denominators must match before you touch the numerators.
5 Key vocabulary
- Like denominators
- Fractions sharing the same bottom number.
- Unlike denominators
- Fractions with different bottom numbers; rename before adding.
- Least common denominator (LCD)
- The smallest bottom both fractions can share.
- Improper fraction
- A fraction with numerator ≥ denominator, like 5/4.
- Mixed number
- A whole plus a fraction, like 1 1/4.
6 Quick check
Try these, then reveal the answers.
1. What is 2/6 + 3/6?
Answer
5/6 (add the tops; keep the 6).2. What is 3/4 − 1/2?
Answer
1/4 (3/4 − 2/4 = 1/4).3. What is 1/3 + 1/4?
Answer
7/12 (4/12 + 3/12 = 7/12).Key points to remember
- Add or subtract only when the pieces match — rename with the LCD first.
- Add/subtract the numerators; the denominator stays.
- Simplify the answer; convert improper results to mixed numbers.
- Never add the denominators.
- Estimate first: 1/2 + 1/3 should land near 1, not near 2/5.