Adding Fractions – Step-by-Step
Adding fractions is one of the most important fraction skills. The key rule is simple: you can only add fractions that have the same denominator. Everything else follows from this.
The reason denominators must match traces back to what a fraction actually represents: a count of equal-sized pieces. You cannot directly add “3 thirds” to “2 quarters” any more than you could add 3 metres to 2 feet without converting to a shared unit first — the denominator is the “unit,” and the numerator is the count. Finding the LCM of the denominators is simply finding a common unit both fractions can be measured in. This is why ancient Egyptian mathematicians, who worked almost exclusively with unit fractions (1/n), developed elaborate tables of equivalent fractions purely to make addition possible.
Case 1 – Like Fractions (Same Denominator)
Add the numerators. Keep the denominator. Simplify if possible.
Seeing it as area: if the denominators already match, both fractions are cut from the same size pieces, so you can just push the shaded pieces together in one bar.
orange = 3/8, blue = 2/8, together = 5 shaded eighths out of 8
(3 + 2)/8 = 5/8. Already simplified. Answer: 5/8.
Case 2 – Unlike Fractions (Different Denominators)
Find the LCM of the denominators. Convert both fractions. Then add.
Why you need a common denominator, visually: 1/3 and 1/4 are cut from different-sized pieces, so their shaded areas can't be combined directly — it's like trying to add "3 thirds" to "2 quarters" without a shared unit. The fix is to re-cut BOTH fractions using a grid fine enough for both: a grid with 3 columns and 4 rows has 12 equal cells, so it can show thirds AND quarters at the same time.
Both grids have exactly the same 12 cells — only the shading changes. Once both fractions are expressed in twelfths, they can finally be added directly: 4/12 + 3/12 = 7/12, matching the LCM method below.
LCM(3,4) = 12. 1/3 = 4/12. 1/4 = 3/12. 4/12 + 3/12 = 7/12. Answer: 7/12.
LCM(5,8) = 40. 2/5 = 16/40. 3/8 = 15/40. 16/40 + 15/40 = 31/40. Answer: 31/40.
Case 3 – Mixed Numbers
Method A: Add whole parts and fraction parts separately.
Method B: Convert to improper fractions first, then add.
Method A: Wholes: 2+1=3. Fractions: 1/2+1/3. LCM=6: 3/6+2/6=5/6. Total: 3⅞.
Convert: 15/4 + 8/3. LCM=12: 45/12 + 32/12 = 77/12 = 6 5/12. Answer: 6 5/12.
Case 4 – Adding Fractions and Whole Numbers
Adding Three or More Fractions
LCM(2,3,6) = 6. 3/6 + 2/6 + 1/6 = 6/6 = 1.
Key Takeaways
- Same denominators: add numerators, keep denominator.
- Different denominators: find LCM, convert, then add.
- Always simplify the answer.
- If the result is improper, convert to a mixed number.