Why fractions must share a denominator before you can add them
A fraction is a count of equal parts, and the denominator names the size of the part. Two thirds means two pieces each of size one third; one quarter means one piece of size one quarter. You cannot add two of one size to one of another size and get a meaningful count, for the same reason you cannot add two feet to one metre and write “three”. First you convert to a shared unit.
For fractions the shared unit is a common denominator: a part size that both original part sizes divide into evenly. Thirds and quarters both divide twelfths, so both fractions can be rewritten in twelfths — 2/3 becomes 8/12, and 1/4 becomes 3/12. Now the parts are the same size and you simply count them: eight plus three is eleven twelfths.
Any common denominator works. The product of the two denominators is always one, which is why the schoolbook shortcut a/b + c/d = (ad + cb)/(bd) is valid. The least common denominator is the smallest that works, and it is worth finding because it keeps the numbers small and usually leaves less reducing to do at the end. If you want it on its own, the least common multiple calculator computes it directly from any set of denominators.
Subtraction is the same operation with one sign changed. Once both fractions sit over the same denominator, you subtract the numerators and leave the denominator untouched. The denominator never changes during addition or subtraction — a mistake worth watching for, because it does change during multiplication.
The formula, term by term
The general rule has three moving parts: find the least common denominator, scale each numerator by the same factor its own denominator was scaled by, then reduce.
Finding L. The least common denominator is the least common multiple of the two denominators, and the fastest way to it is L = b·d / gcd(b, d). The greatest common divisor comes from the Euclidean algorithm: replace the larger number by its remainder on division by the smaller, repeat until one of them is zero, and the survivor is the gcd. For 4 and 6: 6 mod 4 = 2, then 4 mod 2 = 0, so gcd = 2 and L = 4·6/2 = 12.
Scaling the numerators. Multiplying a fraction's numerator and denominator by the same non-zero number leaves its value unchanged — that is the definition of equivalent fractions. So if b must grow to L, it grows by the factor L/b, and a must grow by exactly that same factor. Getting this factor right for one fraction and wrong for the other is the single most common source of wrong answers.
Reducing. The combined numerator and L often share a factor. Dividing both by gcd(numerator, L) puts the answer in lowest terms in one step, rather than in several passes of “can I halve it again?”. The simplifying fractions calculator does this step on any fraction you already have.
Mixed numbers first. A mixed number such as 3 1/2 is a sum in disguise: 3 + 1/2. Convert it to a single improper fraction, (3·2 + 1)/2 = 7/2, before doing anything else. Working the whole parts and fraction parts separately is possible but it goes wrong on subtraction the moment you have to borrow, so convert first every time.
Worked example: 2/3 + 1/4, then 3 1/2 − 1 5/6
First problem: 2/3 + 1/4.
- Both are already proper fractions, so there is nothing to convert. a = 2, b = 3, c = 1, d = 4.
- Find the gcd of the denominators. 4 mod 3 = 1; 3 mod 1 = 0. So gcd(3, 4) = 1.
- Find the LCD. L = 3 × 4 / 1 = 12.
- Scale the first fraction. 12 ÷ 3 = 4, so 2/3 = (2×4)/12 = 8/12.
- Scale the second. 12 ÷ 4 = 3, so 1/4 = (1×3)/12 = 3/12.
- Add the numerators. 8 + 3 = 11, over 12.
- Reduce. gcd(11, 12) = 1, so 11/12 is already in lowest terms. As a decimal, 11 ÷ 12 = 0.916666…
Second problem: 3 1/2 − 1 5/6. This one needs borrowing if you work it the long way, which is exactly why converting first pays.
- Convert both mixed numbers. 3 1/2 = (3×2 + 1)/2 = 7/2. And 1 5/6 = (1×6 + 5)/6 = 11/6.
- Find the gcd. 6 mod 2 = 0, so gcd(2, 6) = 2.
- Find the LCD. L = 2 × 6 / 2 = 6. Note that this is smaller than the product 12 — the shortcut formula would have worked but left you reducing twice as much.
- Scale. 6 ÷ 2 = 3, so 7/2 = 21/6. The second fraction already has denominator 6, so it stays 11/6.
- Subtract the numerators. 21 − 11 = 10, over 6.
- Reduce. gcd(10, 6) = 2, so 10/6 = 5/3.
- Write as a mixed number. 5 ÷ 3 = 1 remainder 2, so the answer is 1 2/3, or 1.66666… as a decimal.
Check it roughly: 3.5 − 1.833 ≈ 1.667. The exact answer and the decimal estimate agree, which is the sanity check to run every time.
How to read the answer: lowest terms, mixed form and sign
An answer is in lowest terms when the numerator and denominator share no common factor above 1. Teachers, standardised tests and engineering drawings all expect this form, and an unreduced answer is usually marked wrong even though it is numerically equal. If the calculator reports a greatest common factor above 1 in its steps, that is the number your own working needed to divide out.
Whether to leave the result improper or convert it to a mixed number depends on what you are doing. Algebra keeps improper fractions, because 5/3 slots into further algebra cleanly and 1 2/3 does not — juxtaposition means multiplication in algebra, so “1 2/3” is ambiguous there. Measurement and cooking prefer mixed numbers, because 1 2/3 cups is something you can act on. Both forms are shown; pick the one your context wants.
Sign needs care with mixed numbers. Written mathematics treats −2 3/4 as −(2 + 3/4) = −11/4, with the minus sign applying to the whole quantity, not just the whole part. This calculator follows that convention: when you enter a whole part, the sign comes from the whole part and the numerator is read as a magnitude. Set the whole part to 0 if you want the numerator to carry its own sign.
The decimal value is there as a check, not as the answer. Many fractions have no exact decimal — 1/3 and 5/6 both repeat forever — so the decimal shown is rounded and the fraction is the exact result. If you need to know whether a particular fraction terminates or repeats, the fraction to decimal calculator reports the period.
Least common denominators for the denominators you meet most
| 2 | 3 | 4 | 5 | 6 | 8 | 10 | 12 | |
|---|---|---|---|---|---|---|---|---|
| 2 | 2 | 6 | 4 | 10 | 6 | 8 | 10 | 12 |
| 3 | 6 | 3 | 12 | 15 | 6 | 24 | 30 | 12 |
| 4 | 4 | 12 | 4 | 20 | 12 | 8 | 20 | 12 |
| 5 | 10 | 15 | 20 | 5 | 30 | 40 | 10 | 60 |
| 6 | 6 | 6 | 12 | 30 | 6 | 24 | 30 | 12 |
| 8 | 8 | 24 | 8 | 40 | 24 | 8 | 40 | 24 |
| 10 | 10 | 30 | 20 | 10 | 30 | 40 | 10 | 60 |
| 12 | 12 | 12 | 12 | 60 | 12 | 24 | 60 | 12 |
Notice how often the LCD is smaller than the product: lcm(4, 6) = 12, not 24, and lcm(8, 12) = 24, not 96. That gap is the reducing work you save.
Mistakes that produce a wrong fraction sum
- Adding the denominators. 1/2 + 1/3 is not 2/5. Check it against decimals: 0.5 + 0.333 = 0.833, but 2/5 = 0.4. The denominator is a unit label, and you do not add labels.
- Scaling the numerator by the wrong factor. If the denominator was multiplied by 4 to reach the LCD, the numerator must be multiplied by 4 as well — not by the LCD, and not by the other denominator.
- Scaling only one of the two fractions. Both have to reach the same denominator before a single numerator is combined.
- Working mixed numbers piecewise on a subtraction. 3 1/2 − 1 5/6 cannot be done as (3−1) and (1/2−5/6) without borrowing, because 1/2 is smaller than 5/6. Converting to 7/2 and 11/6 first removes the trap entirely.
- Leaving the answer unreduced. 10/6 is the right value but the wrong form. Divide by gcd(10, 6) = 2 to get 5/3.
- Dropping the sign on a negative mixed number. −2 3/4 is −11/4, not −2 + 3/4 = −1.25.
- Rounding to decimals mid-problem. Turning 1/3 into 0.33 and adding introduces an error that grows with every further step. Keep fractions exact until the last line.
The cross-multiplication shortcut, and when it hurts
a/b ± c/d = (a·d ± c·b)/(b·d) always gives a correct value, and it needs no gcd. It is the right tool for a quick mental check or for algebra with symbolic denominators.
It costs you at the reducing stage. For 5/12 + 7/18 the product denominator is 216, giving 90/216 + 84/216 = 174/216, which then needs dividing by 6. The least common denominator is 36, giving 15/36 + 14/36 = 29/36 directly. Same answer; a lot less arithmetic and far fewer chances to slip.
Where fraction addition sits among the other operations
Addition and subtraction are the operations that need a common denominator. Multiplication and division do not: to multiply you combine numerators and denominators straight across, and to divide you invert the second fraction and multiply. Students who have just learned to find LCDs often apply them to every operation out of habit, which wastes work and, on division, gives a wrong answer. The dividing fractions calculator shows the reciprocal rule step by step.
For more than two terms, the method extends without change: take the least common multiple of all the denominators at once, rewrite every term over it, then combine. Doing it pairwise gives the same answer but can inflate the intermediate numbers.
Fractions and decimals are two notations for the same numbers, and neither is universally better. Fractions are exact for any ratio of integers, which is why they survive in music, in imperial measurement, in odds, and in symbolic algebra. Decimals compare and sort at a glance and feed directly into instruments and spreadsheets. Convert when the destination demands it, using the decimal to fraction calculator in the other direction, and remember that a repeating decimal always came from a fraction whose reduced denominator has a prime factor other than 2 or 5.
If you are adding measurements rather than abstract numbers — sixteenths on a tape measure, thirds of a cup — convert everything to the finest denominator present before you start. That denominator is usually the LCD already, and it keeps every intermediate number a whole count of the smallest unit you can actually measure.
Key terms
- Numerator and denominator
- The top and bottom of a fraction. The denominator names how many equal parts one whole is cut into; the numerator counts how many of those parts you have.
- Common denominator
- Any number that both original denominators divide into evenly. Every pair of fractions has infinitely many; their product is always one of them.
- Least common denominator (LCD)
- The smallest common denominator, equal to the least common multiple of the denominators. It keeps the arithmetic small and minimises later reducing.
- Equivalent fractions
- Fractions with the same value but different terms, such as 2/3 and 8/12. You produce one by multiplying numerator and denominator by the same non-zero number.
- Lowest terms
- A fraction whose numerator and denominator have a greatest common divisor of 1. Also called simplest form or fully reduced.
- Improper fraction
- A fraction whose numerator is at least as large as its denominator, such as 5/3. Not an error — just a form. Its mixed-number counterpart is 1 2/3.