Adding Fractions with Unlike Denominators: LCM Method + Mistake Finder

The problem with unlike denominators

You cannot add 1/3 + 1/4 directly because the pieces are different sizes. A third is not the same size as a quarter, so combining one of each does not give "2 of something". You first need to rewrite both fractions using the same denominator.

The LCM method in four steps

LCM stands for Least Common Multiple. The idea: find the smallest number that both denominators divide into evenly, rewrite both fractions using that denominator, then add the numerators.

Step 1 — Find the LCM of the denominators

For 1/3 + 1/4, the denominators are 3 and 4.

Multiples of 3: 3, 6, 9, 12, 15, 18... Multiples of 4: 4, 8, 12, 16, 20...

The smallest number in both lists is 12. That is the LCM.

Step 2 — Rewrite each fraction with the LCM as the new denominator

For 1/3: multiply top and bottom by 4 (because 3 × 4 = 12). You get 4/12.

For 1/4: multiply top and bottom by 3 (because 4 × 3 = 12). You get 3/12.

Step 3 — Add the numerators

4/12 + 3/12 = 7/12.

Step 4 — Simplify if possible

7 and 12 share no common factors, so 7/12 is already in simplest form.

Answer: 1/3 + 1/4 = 7/12.

Why the LCM and not just any common multiple?

You can use any common multiple, not just the least. For 1/3 + 1/4, you could use 24 as the common denominator:

1/3 = 8/24, 1/4 = 6/24, sum = 14/24 = 7/12.

The answer is the same after simplification. The advantage of the LCM is smaller numbers, which means fewer opportunities for arithmetic errors and less simplifying at the end.

Worked example 2: Larger denominators

Add 5/6 + 3/8.

Step 1: LCM of 6 and 8.

Multiples of 6: 6, 12, 18, 24... Multiples of 8: 8, 16, 24...

LCM = 24.

Step 2: 5/6 = 20/24 (multiply by 4). 3/8 = 9/24 (multiply by 3).

Step 3: 20/24 + 9/24 = 29/24.

Step 4: 29/24 = 1 5/24. Already simplified.

Worked example 3: Three fractions

Add 1/2 + 1/3 + 1/5.

LCM of 2, 3, and 5 = 30.

1/2 = 15/30, 1/3 = 10/30, 1/5 = 6/30.

Sum: 15 + 10 + 6 = 31. Answer: 31/30 = 1 1/30.

Finding the LCM efficiently

For small numbers, listing multiples works fine. For larger numbers, use prime factorisation:

Example: LCM of 12 and 18.

• 12 = 2² × 3
• 18 = 2 × 3²

Take the highest power of each prime: 2² × 3² = 4 × 9 = 36.

LCM(12, 18) = 36.

The mistake-finder checklist

After you add fractions, run through this checklist:

1. Did you change the denominator, not just the numerator? When converting 1/3 to twelfths, you must multiply both top and bottom by 4 to get 4/12. Multiplying only the numerator gives a different value.

2. Did you multiply the numerators correctly? Double-check each conversion step.

3. Did you add only the numerators? The denominator stays the same after conversion. Do not add the denominators.

4. Did you simplify? Check whether the numerator and denominator share any common factor. If in doubt, try dividing by 2, 3, or 5.

5. Is the answer reasonable? 1/3 ≈ 0.33 and 1/4 = 0.25. Sum ≈ 0.58. Your answer 7/12 ≈ 0.583. That matches.

6. If the answer is an improper fraction, did you convert? Many questions expect a mixed number.

Common mistakes

| Mistake | Example | Why it's wrong | |---------|---------|---------------| | Adding denominators | 1/3 + 1/4 = 2/7 | You need a common denominator, not a sum | | Forgetting to adjust numerators | Using 12 as denominator but writing 1/12 + 1/12 | Each numerator must be scaled by the same factor as its denominator | | Using a non-common multiple | Rewriting as 1/9 + 1/8 | The new denominator must be a multiple of both original denominators | | Not simplifying | Leaving 14/24 instead of 7/12 | Always reduce to simplest form |

Subtracting fractions with unlike denominators

The process is identical, except you subtract the numerators in Step 3 instead of adding them.

5/6 - 3/8: LCM = 24. 20/24 - 9/24 = 11/24.

When the denominators are already the same

If both fractions already share a denominator, skip straight to adding the numerators. No LCM step needed.

3/7 + 2/7 = 5/7. Done.

For a deeper look at the full addition process, see the How to Add Fractions guide. You can also check your working with the 1/2 + 1/4 worked example.

FAQ

Can I always just multiply the two denominators? Yes, the product of the denominators is always a common multiple. But it may not be the least common multiple, so your numbers will be larger and you will need to simplify more at the end.

What if one denominator is a multiple of the other? Then the LCM is the larger denominator. For 1/4 + 5/12: the LCM is 12. Convert 1/4 to 3/12, then add.

How do I add mixed numbers with unlike denominators? Convert each mixed number to an improper fraction, find the LCM, add, then convert back.

Is there a quick mental method? For two fractions a/b + c/d, you can use the "butterfly" cross-multiply shortcut: (ad + bc) / bd. Then simplify. It is equivalent to using bd as the common denominator.

What about negative fractions? The LCM method works the same way. Just keep track of the signs when adding the adjusted numerators.