To solve fractions with different denominators, rewrite each with a common denominator, then add or subtract the new numerators.
Fractions feel friendly until the bottoms don’t match. You see 1/3 + 1/4 and your brain goes, “Wait… now what?” The fix is steady and repeatable: make the denominators match, keep the value the same, then do the operation.
This article shows a clean routine you can use on homework, tests, or real-life math like recipes and measurements. You’ll learn two ways to find a common denominator, how to rewrite fractions fast, and how to check your work so silly slips don’t sneak in.
What “Different Denominators” Means In Plain Terms
The denominator tells you the size of each piece. Thirds and fourths are different-sized pieces, so you can’t add them directly. A common denominator is a shared piece size that both fractions can use without changing what each fraction means.
When you switch from thirds to twelfths, you aren’t changing the amount you have. You’re just naming it with smaller slices. That’s why the only safe move is to multiply the top and bottom by the same number.
Common Denominator Options At A Glance
| Situation | Good Common Denominator Choice | Why It Helps |
|---|---|---|
| One denominator is a multiple of the other (3 and 12) | Use the larger denominator (12) | Fast rewrite with one fraction unchanged |
| Small numbers that share factors (6 and 8) | Least common denominator (24) | Keeps numbers smaller while staying clean |
| Prime denominators (5 and 7) | Product (35) | No shared factors, so the product is already least |
| Several fractions (1/6, 1/4, 1/3) | LCD from prime factors (12) | Prevents the “giant denominator” problem |
| Denominators share a big factor (12 and 18) | Use LCM (36) | Less rewriting and simpler reduction later |
| You’re in a hurry and numbers are small | Use the product (a×b) | Always works; you can simplify at the end |
| Decimals or messy results appear | Stay with fraction form and use LCD | Avoids rounding and keeps exact values |
| Mixed numbers involved (2 1/3 − 1 3/4) | LCD for the fractional parts (12) | Makes borrowing predictable |
How to Solve Fractions With Different Denominators
Use this routine every time. It’s short, but it covers almost every worksheet you’ll see.
Step 1: Pick A Common Denominator
Start with the least common denominator when it’s easy to spot. If one denominator fits inside the other cleanly, grab the larger one. If not, you can build the least common multiple with prime factors, or just multiply the denominators and move on.
Step 2: Rewrite Each Fraction As An Equivalent Fraction
Multiply the numerator and denominator by the same number to hit the common denominator. This keeps the fraction’s value unchanged. If you only multiply the bottom, you change the size of the pieces but not the count, and that breaks the meaning.
Step 3: Do The Operation On The Numerators
Once denominators match, add or subtract the numerators. The denominator stays put because the piece size is now the same for both fractions.
Step 4: Simplify And Convert If Needed
Reduce the answer by dividing top and bottom by any common factor. If the result is improper, convert it to a mixed number if your class expects that style.
Solving Fractions With Different Denominators Step By Step With Real Numbers
Let’s run the routine with a few common setups. Keep your pencil moving and you’ll see the pattern lock in.
Example 1: Add Two Simple Fractions
Problem: 1/3 + 1/4
Common denominator: 12. Rewrite: 1/3 = 4/12, 1/4 = 3/12. Add numerators: 4/12 + 3/12 = 7/12. Already reduced.
Example 2: Subtract With A Borrow
Problem: 2 1/3 − 1 3/4
Common denominator for the fractions: 12. Rewrite fractional parts: 1/3 = 4/12, 3/4 = 9/12. Since 4/12 is smaller than 9/12, borrow 1 from the 2: 2 4/12 becomes 1 16/12. Now subtract: (1 16/12) − (1 9/12) = 7/12.
Example 3: Add More Than Two Fractions
Problem: 1/6 + 1/4 + 1/3
LCD: 12. Rewrite: 1/6 = 2/12, 1/4 = 3/12, 1/3 = 4/12. Add: (2+3+4)/12 = 9/12 = 3/4.
Example 4: Improper Fractions And Cleanup
Problem: 7/8 − 2/3
LCD: 24. Rewrite: 7/8 = 21/24, 2/3 = 16/24. Subtract: 21/24 − 16/24 = 5/24. Reduced.
Two Reliable Ways To Find The Least Common Denominator
You can reach a common denominator in more than one way. Pick the method that feels smooth for the numbers in front of you.
Method 1: List Multiples Until They Meet
Write a few multiples of each denominator and spot the first match. With 6 and 8: multiples of 6 are 6, 12, 18, 24… multiples of 8 are 8, 16, 24… so the LCD is 24. This is quick for small numbers, then it gets slow.
Method 2: Use Prime Factors And Build The LCM
Break each denominator into primes. Keep each prime factor at its highest power across the set, then multiply those kept factors. With 12 and 18: 12 = 2²×3, 18 = 2×3². Keep 2² and 3², so LCM = 2²×3² = 36.
If you want a clear walkthrough on factoring and least common multiples, Khan Academy’s lesson on adding and subtracting fractions with unlike denominators lines up well with this exact routine.
How To Check Your Answer Without A Calculator
A quick check keeps your confidence up and catches slips early.
Use A Reasonableness Check
Ask what the result should feel like. If you add two positive fractions, the answer should be bigger than each one. If you subtract a larger fraction from a smaller one, the answer should be negative.
Use A Backward Check
If a − b = c, then c + b should return a. With 7/8 − 2/3 = 5/24, check 5/24 + 2/3. Rewrite 2/3 as 16/24 and add: 21/24 = 7/8. Match found.
Common Mistakes That Trip People Up
Most fraction errors come from one of these patterns. Fix the habit and the problems get calmer.
Adding Denominators
1/3 + 1/4 is not 2/7. Denominators name the piece size, so you don’t add them. You match them.
Changing Only The Denominator
If you turn 1/3 into 1/12, you changed the value. You must multiply the top and bottom together so the fraction stays equivalent.
Forgetting To Reduce
Teachers often want the simplest form. After you add or subtract, take one pass to see if both numbers share a factor.
Messy Borrowing With Mixed Numbers
When you borrow 1, convert that 1 into a fraction with your common denominator. That’s the whole trick. Keep the whole-number part clean, then work in the fraction lane.
Practice Set With Answers You Can Self-Check
Try these in order. Do them on paper, then compare with the answers. If one misses, rewrite the fractions again and watch where the switch happened.
- 2/5 + 1/10
- 3/8 + 5/12
- 7/9 − 1/6
- 1 2/3 + 2 3/5
- 5/6 − 3/4
- 2/7 + 3/14 + 1/2
- 4 1/8 − 2 5/16
- 11/15 + 2/5
Answers And Error Fixes
Use this table to spot what went wrong fast. If your denominator differs, that’s fine as long as the fraction is equivalent and reduced.
| Problem | Correct Answer | Quick Fix If You Missed |
|---|---|---|
| 2/5 + 1/10 | 1/2 | Use 10 as the common denominator, then reduce 5/10 |
| 3/8 + 5/12 | 19/24 | LCD is 24; rewrite as 9/24 and 10/24 |
| 7/9 − 1/6 | 11/18 | LCD is 18; rewrite as 14/18 and 3/18 |
| 1 2/3 + 2 3/5 | 4 4/15 | Convert fractional parts to fifteenths: 10/15 and 9/15 |
| 5/6 − 3/4 | 1/12 | LCD is 12; rewrite as 10/12 and 9/12 |
| 2/7 + 3/14 + 1/2 | 13/14 | LCD is 14; rewrite as 4/14, 3/14, 7/14 |
| 4 1/8 − 2 5/16 | 1 13/16 | Borrow 1 as 16/16, then subtract 5/16 from 18/16 |
| 11/15 + 2/5 | 17/15 = 1 2/15 | Rewrite 2/5 as 6/15, then convert improper form |
Where This Skill Shows Up In School Standards
Adding and subtracting fractions with unlike denominators is a core arithmetic skill in late elementary and early middle school. If you want to see the formal wording, the Common Core standard 5.NF.A.1 describes the expectation for adding and subtracting fractions with unlike denominators.
Negative Fractions And Sign Rules
A minus sign can sit in three spots: −1/4, 1/−4, or −(1/4). Treat them as the same value, and keep the sign in the numerator to stay tidy. When you add fractions with different denominators and one is negative, the steps stay the same.
After you rewrite, combine numerators with the sign included. If the result is negative, write −7/12, not 7/−12. For subtraction, change it to adding a negative so you don’t flip the order.
Make Rewriting Faster With Early Reduction
Before you hunt for the least common denominator, see if a fraction reduces right away. Smaller numbers mean less cleanup. Keep this pattern: reduce, pick a denominator, rewrite, then reduce again. This tiny habit keeps your scratch work clean and quick.
One Clean Habit That Makes Every Problem Easier
Write the common denominator once, then keep your work stacked. When the fraction rewrites are lined up, your eyes spot mistakes sooner. This is also the fastest way to stay calm on timed work.
If you’re practicing how to solve fractions with different denominators, keep the routine fixed: choose a denominator, rewrite, operate, reduce. After a few rounds, it stops feeling like “a fraction trick” and starts feeling like normal math.
When you get stuck, go back to the meaning: same-sized pieces first, then add or subtract. That’s the whole deal for how to solve fractions with different denominators today.