Add 1/2 and 1/3
Question
Find the sum: 1/2 + 1/3
Solution — Step by Step
These fractions have different denominators (2 and 3), so they are unlike fractions. We cannot add them directly. We must first convert them to like fractions using the LCM method.
Step 1: Find the LCM of the denominators.
Denominators: 2 and 3
Multiples of 2: 2, 4, 6, 8, 10, 12, … Multiples of 3: 3, 6, 9, 12, …
LCM of 2 and 3 = 6
Step 2: Convert 1/2 to an equivalent fraction with denominator 6.
2 × ? = 6 → multiply by 3
1/2 = (1 × 3)/(2 × 3) = 3/6
Step 3: Convert 1/3 to an equivalent fraction with denominator 6.
3 × ? = 6 → multiply by 2
1/3 = (1 × 2)/(3 × 2) = 2/6
Step 4: Now both fractions have the same denominator. Add them.
3/6 + 2/6 = (3 + 2)/6 = 5/6
Step 5: Simplify the result.
5/6 — HCF of 5 and 6 is 1. So 5/6 is already in its simplest form.
Step 1: Find LCM of denominators (LCM of 2 and 3 = 6) Step 2: Convert 1/2 → 3/6 Step 3: Convert 1/3 → 2/6 Step 4: Add → 3/6 + 2/6 = 5/6 Step 5: Simplify → 5/6 (already in simplest form)
Answer: 1/2 + 1/3 = 5/6
Why This Works
Adding unlike fractions directly is like trying to add 3 apples and 2 oranges — you can’t just say “5 apples.” You need a common unit first. In fractions, the “unit” is the denominator (the size of each piece).
Picture a roti. Half the roti (1/2) and one-third of the roti (1/3) are different sized pieces. To add them, we need both cut into pieces of the same size. By using LCM = 6, we imagine the roti cut into 6 equal pieces. Half the roti = 3 of those 6 pieces. One-third of the roti = 2 of those 6 pieces. Together: 3 + 2 = 5 pieces out of 6, which is 5/6.
1/2 + 1/3 = 5/6 is a result worth remembering. It appears in many different problems. Notice that 5/6 is less than 1 — which makes sense, since we added less than a half and less than a half (one-third is less than one-half), so the total should be less than 1.
Common Mistake
Wrong: 1/2 + 1/3 = 2/5
This is the most common error students make with unlike fractions — adding both numerators and both denominators separately. This gives a completely wrong answer. The denominator of the answer is NOT the sum of the denominators. Always find the LCM first, then convert, then add.
Verify the Answer
A quick sense-check: 1/2 = 0.5 and 1/3 ≈ 0.333. Their sum ≈ 0.833.
Now check 5/6 ≈ 0.833. ✓ The decimal values match, confirming our answer is correct.
If you finish a fraction addition problem and have time left, always verify using decimals. Convert both sides to decimals to make sure the left side and right side give the same approximate value. This is a great way to catch calculation errors.