Adding fractions like 1/2 + 1/3 trips up almost everyone at first, because the instinct is to just add everything in sight. But fractions with different bottom numbers — unlike denominators — need one preparatory step before you can add anything: you have to give them a common denominator. Once you see why that step exists, the whole process becomes mechanical and fast.
The One Rule That Matters
You can only add the numerators when the denominators are identical. The denominators never get added to each other. The expression (a/b) + (c/d) does not equal (a + c)/(b + d). Think of it like adding apples to apples: thirds and fourths are different “fruit,” so you convert both into the same unit first.
The LCM Method: 4 Steps
- Find the least common multiple (LCM) of the denominators. This becomes your common denominator.
- Convert each fraction. Multiply the top and bottom of each fraction by the same number so the denominator becomes the LCM.
- Add the numerators. Keep the denominator as-is and add only the top numbers.
- Simplify. Reduce the answer to lowest terms, and convert an improper fraction to a mixed number if needed.
Example 1: 1/4 + 2/5
Step 1: Find the LCM of 4 and 5
List multiples until you find a match:
- Multiples of 4: 4, 8, 12, 16, 20, 24…
- Multiples of 5: 5, 10, 15, 20, 25…
The least common denominator is 20.
Step 2: Convert both fractions
- 1/4: multiply top and bottom by 5 → (1 × 5)/(4 × 5) = 5/20
- 2/5: multiply top and bottom by 4 → (2 × 4)/(5 × 4) = 8/20
Step 3: Add the numerators
5/20 + 8/20 = (5 + 8)/20 = 13/20
Step 4: Simplify
13 is prime and shares no common factor with 20, so the answer stays as is.
Final answer: 13/20
Example 2: 2/3 + 5/6
Sometimes one denominator is already a multiple of the other, which makes the LCM step instant.
Step 1: Identify the LCM of 3 and 6
Since 6 is a multiple of 3, the least common denominator is simply 6.
Step 2: Adjust the fractions
- 2/3: multiply top and bottom by 2 → (2 × 2)/(3 × 2) = 4/6
- 5/6: already has denominator 6 → stays 5/6
Step 3: Add the numerators
4/6 + 5/6 = (4 + 5)/6 = 9/6
Step 4: Simplify and convert
The GCF of 9 and 6 is 3: 9 ÷ 3 = 3, 6 ÷ 3 = 2, giving 3/2. As a mixed number: 3 ÷ 2 = 1 remainder 1, so 1 1/2.
Final answer: 3/2, or 1 1/2
Example 3: 3/8 + 1/6
Step 1: Find the LCM of 8 and 6
- Multiples of 8: 8, 16, 24, 32…
- Multiples of 6: 6, 12, 18, 24, 30…
The least common denominator is 24.
Step 2: Convert both fractions
- 3/8: multiply top and bottom by 3 → (3 × 3)/(8 × 3) = 9/24
- 1/6: multiply top and bottom by 4 → (1 × 4)/(6 × 4) = 4/24
Step 3: Add the numerators
9/24 + 4/24 = (9 + 4)/24 = 13/24
Step 4: Simplify
13 is prime and shares no factor with 24.
Final answer: 13/24
Always ask: “Do the denominators match yet?” If they do not, nothing can be added — convert first, then add only the tops.
Common Mistakes to Avoid
- Adding the denominators: 1/4 + 2/5 is not 3/9. Never add bottom numbers.
- Multiplying only the numerator: when converting to a common denominator, multiply the top and bottom by the same value, or you change the fraction’s value.
- Skipping the LCM: you can use any common denominator (like 4 × 5 = 20 in Example 1), but the smallest one keeps the numbers friendlier and the final simplification shorter.
- Forgetting to simplify: an answer like 9/6 is correct but incomplete — always reduce to lowest terms.
Quick Practice (with answers)
Try these, then check your work:
- 1/3 + 1/6
- 2/5 + 1/4
- 5/12 + 1/3
Answers: 1) 1/3 = 2/6, so 2/6 + 1/6 = 3/6 = 1/2; 2) 8/20 + 5/20 = 13/20 = 13/20; 3) 5/12 + 4/12 = 9/12 = 3/4.
The same procedure works for subtraction — only the plus sign changes. Master the LCM step and every unlike-denominator problem becomes routine.