Do I have to use the least common denominator?

No. Any common denominator works — including the simple product of the two denominators — as long as the result is reduced at the end. The LCD only keeps the intermediate numbers smaller. This calculator uses the product of the denominators and then reduces with the greatest common divisor, which always matches the LCD method's answer exactly.

How do I add a whole number to a fraction?

Write the whole number over a denominator of one and proceed as usual: 2 + 3/4 is 2/1 + 3/4 = 8/4 + 3/4 = 11/4, the mixed number 2 3/4. The same trick makes subtraction work: 2 − 3/4 = 8/4 − 3/4 = 5/4 = 1 1/4.

What happens if a subtraction gives a negative answer?

The answer is simply a negative fraction. The calculator evaluates 1/2 − 3/4 as exactly −1/4, carrying the sign on the numerator and keeping the denominator positive — the standard canonical form for a reduced fraction.

Why does the calculator's answer sometimes have a smaller denominator than mine?

Because it fully reduces every result. If you computed 5/6 + 3/4 over the common denominator 24 and got 38/24, that value is correct but unreduced — dividing numerator and denominator by their greatest common divisor of 2 gives 19/12, the engine's answer, equal to the mixed number 1 7/12.

Guides

How do you add and subtract fractions with different denominators?

By FractionLab · Published June 10, 2026 · Updated June 10, 2026

Rewrite both fractions over a common denominator, add or subtract only the numerators, then reduce: 1/3 + 1/4 becomes 4/12 + 3/12 = 7/12, exactly — and any common denominator works, because reducing with the greatest common divisor always lands on the same simplified answer.

Why unlike denominators block addition

A denominator names the size of the pieces; a numerator counts them. Adding 1/3 and 1/4 directly would mean adding counts of different-sized pieces — thirds and quarters — which is why the most common mistake, adding straight across to get 2/7, fails: 2/7 is about 0.2857, smaller than the 1/3 you started with, so it cannot be the sum of 1/3 and something positive. The fix is to re-describe both quantities in pieces of one shared size before counting anything.

For 1/3 + 1/4, twelfths work: multiply 1/3 by 4/4 to get 4/12, and 1/4 by 3/3 to get 3/12. Both fractions are unchanged in value — each was multiplied by a form of one — but both now count twelfths, so the numerators add directly: 4/12 + 3/12 = 7/12. This site's engine computes the same sum in exact integer arithmetic and returns exactly 7/12, with a six-decimal display value of 0.583333.

Least common denominator or product of denominators — both work

School methods usually hunt for the least common denominator (LCD), the smallest number both denominators divide. This calculator's engine takes the simpler route: it always uses the product of the two denominators, then fully reduces the result. The two roads meet at the same answer. Take 1/6 + 1/4. The LCD is 12: 2/12 + 3/12 = 5/12. The product method uses 24: 4/24 + 6/24 = 10/24, and reducing 10/24 by their greatest common divisor of 2 gives the same 5/12 — which is exactly what the engine returns, displaying ≈0.416667.

The lesson generalizes: the choice of common denominator is a matter of convenience, not correctness. The LCD keeps intermediate numbers small, which is kind to pencil-and-paper work; the product-of-denominators route never requires finding a least common multiple, which is kind to algorithms. As long as the final step reduces the fraction — dividing numerator and denominator by their greatest common divisor, found with Euclid's algorithm — every valid common denominator produces the identical simplified result.

Subtraction is the same walk with a minus sign

Nothing new is needed for subtraction: rewrite over a common denominator, subtract the numerators, reduce. For 3/4 − 1/6 the engine returns exactly 7/12 (≈0.583333): over twelfths, 9/12 − 2/12 = 7/12. For 5/6 − 3/8 it returns 11/24 (≈0.458333): the product denominator is 48, giving 40/48 − 18/48 = 22/48, which reduces by a factor of 2 to 11/24.

Order matters, and negative answers are legitimate fractions. The engine evaluates 1/2 − 3/4 as exactly −1/4, carrying the sign on the numerator with the denominator kept positive. Mixed-sign addition works the same way: −1/2 + 1/3 returns exactly −1/6 (≈−0.166667). The convention of normalizing the sign onto the numerator is what keeps every reduced answer in a single canonical form.

When the sum passes one, read it as a mixed number

Sums of proper fractions often exceed one, and the improper-fraction and mixed-number forms are the same value in two costumes. The engine computes 5/6 + 3/4 as exactly 19/12: over twelfths, 10/12 + 9/12. It reports the improper fraction 19/12, the mixed number 1 7/12 (divide 19 by 12: whole part 1, remainder 7), and the decimal ≈1.583333. Likewise 2/3 + 5/6 lands on 9/6, which reduces to 3/2 — reported as 1 1/2, decimal exactly 1.5.

Reducing before reading the mixed form matters. 9/6 read directly would give 1 3/6, which is correct but unsimplified; reducing first by the greatest common divisor 3 gives 3/2 and the clean mixed form 1 1/2. The calculator always reduces first, which is why its mixed-number line never carries a reducible proper part.

The three mistakes that cause most wrong answers

First, adding denominators: 1/3 + 1/4 is not 2/7. Denominators are piece sizes, not quantities, and they never add. Second, scaling a denominator without scaling its numerator: turning 1/3 into 1/12 on the way to twelfths silently divides the value by four — every rewrite must multiply numerator and denominator by the same factor, because only multiplying by a form of one preserves value. Third, skipping the final reduction: leaving 10/24 unreduced is not exactly wrong, but it hides the answer's identity — 10/24, 5/12, and ≈0.416667 are the same number, and the reduced form is the one every textbook and answer key expects.

A reliable self-check is the decimal line. Convert your answer and the original fractions to decimals and confirm the arithmetic roughly holds: 1/3 ≈ 0.333 plus 1/4 = 0.25 should land near 0.583, which 7/12 does. The check cannot prove an exact answer right, but it catches every piece-size blunder instantly, because adding denominators or dropping a scale factor moves the decimal value far from the truth.

Questions

Do I have to use the least common denominator?: No. Any common denominator works — including the simple product of the two denominators — as long as the result is reduced at the end. The LCD only keeps the intermediate numbers smaller. This calculator uses the product of the denominators and then reduces with the greatest common divisor, which always matches the LCD method's answer exactly.
How do I add a whole number to a fraction?: Write the whole number over a denominator of one and proceed as usual: 2 + 3/4 is 2/1 + 3/4 = 8/4 + 3/4 = 11/4, the mixed number 2 3/4. The same trick makes subtraction work: 2 − 3/4 = 8/4 − 3/4 = 5/4 = 1 1/4.
What happens if a subtraction gives a negative answer?: The answer is simply a negative fraction. The calculator evaluates 1/2 − 3/4 as exactly −1/4, carrying the sign on the numerator and keeping the denominator positive — the standard canonical form for a reduced fraction.
Why does the calculator's answer sometimes have a smaller denominator than mine?: Because it fully reduces every result. If you computed 5/6 + 3/4 over the common denominator 24 and got 38/24, that value is correct but unreduced — dividing numerator and denominator by their greatest common divisor of 2 gives 19/12, the engine's answer, equal to the mixed number 1 7/12.