Fraction Calculator

🍕What a fraction actually means

A fraction is a way of writing “part of a whole”: the denominator (bottom number) says how many equal pieces the whole is cut into, and the numerator (top number) says how many of those pieces you have. In 3/4, the whole is split into 4 equal parts and you hold 3 of them — three quarters of a pizza, three quarters of an hour (45 minutes), three quarters of a dollar (75 cents).

Fractions are just division written vertically: 3/4 means 3 ÷ 4 = 0.75. That is why the fraction bar is sometimes called a vinculum — it groups the numerator as a single quantity being divided. Every fraction is a rational number, and every rational number can be written as a fraction of two whole numbers.

Comparing fractions is easy once they share a denominator: 3/8 vs 5/12 → 9/24 vs 10/24, so 5/12 is bigger. Without a common denominator, cross-multiplication does the same job: compare 3×12 = 36 with 5×8 = 40 — the second fraction wins because 40 > 36. This trick works because you are effectively giving both fractions the denominator 8×12 = 96 without writing it out.

Equivalent fractions are also how you scale recipes. To triple a recipe calling for 2/3 cup: 2/3 × 3 = 6/3 = 2 cups. To make two-thirds of a recipe calling for 3/4 teaspoon: 3/4 × 2/3 = 6/12 = 1/2 teaspoon. And reducing a recipe that serves 8 down to 3 servings means multiplying every quantity by 3/8 — so 1 1/2 cups becomes 3/2 × 3/8 = 9/16 cup (about 1/2 cup plus a tablespoon).

Two fractions can look different and mean the same thing: 1/2 = 2/4 = 3/6 = 50/100. These are equivalent fractions — multiply (or divide) the top and bottom by the same number and the value never changes. This single idea powers everything else on this page: simplifying, common denominators, and comparing fractions all come down to finding the right equivalent form.

✂️Simplifying: the GCD does the heavy lifting

A fraction is in simplest form when its numerator and denominator share no common factor other than 1. To simplify, divide both by their greatest common divisor (GCD) — the largest whole number that divides both evenly. For 34/8: the GCD of 34 and 8 is 2, so 34 ÷ 2 = 17 and 8 ÷ 2 = 4, giving 17/4. For 48/60: the GCD is 12, so 48 ÷ 12 = 4 and 60 ÷ 12 = 5, giving 4/5.

The standard way to find the GCD is Euclid’s algorithm: repeatedly replace the larger number with the remainder of dividing it by the smaller, until the remainder is 0. For 48 and 60: 60 ÷ 48 leaves remainder 12; 48 ÷ 12 leaves remainder 0 — so the GCD is 12. It finishes in a handful of steps even for enormous numbers, which is why the Simplify tab above handles 30-digit inputs instantly using BigInt arithmetic. Once you are comfortable simplifying fractions by hand, the tab is a fast way to check homework or reduce messy real-world measurements.

A useful habit is simplifying before you multiply: in 4/9 × 3/8, cross-cancel first — 4 and 8 share a factor of 4 (→ 1 and 2), 3 and 9 share 3 (→ 1 and 3) — leaving 1/3 × 1/2 = 1/6. The numbers stay small and you often skip the final simplification step entirely.

Why simplify at all? Simplified fractions are easier to read, compare, and convert — and most teachers and textbooks require answers in lowest terms. The calculator above simplifies every result automatically, so 2/4 + 1/4 shows as 3/4, never as 6/8.

➕Adding and subtracting: the common-denominator rule

You can only add or subtract fractions that share a denominator — “like” fractions. 1/4 + 2/4 = 3/4: three quarter-pieces plus… well, the pieces are already the same size, so just count them. For unlike fractions like 1/2 + 1/3, first rewrite both with a common denominator: multiply each fraction’s top and bottom by the other fraction’s denominator. So 1/2 becomes 3/6 and 1/3 becomes 2/6, then add the numerators: 3/6 + 2/6 = 5/6.

The general rule: a/b + c/d = (ad + bc) / bd and a/b − c/d = (ad − bc) / bd. The calculator then simplifies the answer. Worked example: 1 1/2 + 2 3/4 → convert to improper fractions 3/2 + 11/4 → common denominator 8: 12/8 + 22/8 = 34/8 → simplify → 17/4 = 4 1/4 = 4.25.

Subtracting mixed numbers sometimes needs borrowing: 3 1/2 − 1 3/4. The fraction 1/2 is smaller than 3/4, so borrow 1 from the 3: 3 1/2 becomes 2 3/2, then 2 3/2 − 1 3/4 = (2−1) + (6/4 − 3/4) = 1 3/4. Check: 3.5 − 1.75 = 1.75 ✓. Alternatively convert to improper fractions first: 7/2 − 7/4 = 14/4 − 7/4 = 7/4 = 1 3/4 — same answer, no borrowing needed.

When denominators share factors, the least common denominator (LCD) keeps numbers smaller than the straight product. For 5/6 + 7/12 the LCD is 12 (not 72): 10/12 + 7/12 = 17/12 = 1 5/12. The LCD is the least common multiple of the denominators — 12 is the smallest number both 6 and 12 divide into.

A common mistake is adding denominators: 1/2 + 1/3 is not 2/5. Think of the pizza: half a pizza plus a third of a pizza is five sixths — clearly more than half, while 2/5 is less than half. If your answer looks smaller than one of the inputs you added, something went wrong.

✖️How to multiply fractions (and divide them): simpler than they look

Multiplication needs no common denominator — just multiply straight across: a/b × c/d = ac / bd. So 2/3 × 3/4 = 6/12 = 1/2. Notice the answer (1/2) is smaller than both inputs — multiplying by a proper fraction always shrinks the result, because you are taking “a fraction of a fraction.”

Division uses the “flip and multiply” rule: a/b ÷ c/d = a/b × d/c = ad / bc. To divide by a fraction, multiply by its reciprocal (the fraction flipped upside down). So 1/2 ÷ 1/4 = 1/2 × 4/1 = 4/2 = 2 — which makes sense: there are two quarter-pieces in a half. And 3/4 ÷ 2/3 = 3/4 × 3/2 = 9/8 = 1 1/8.

Multiplying fractions also answers “what is a fraction of a fraction?” — if you eat 1/2 of a pizza and your friend eats 1/3 of your half, they ate 1/2 × 1/3 = 1/6 of the whole pizza. This is exactly how probability chains work: the chance of two independent 1/2 events both happening is 1/2 × 1/2 = 1/4.

Complex fractions — fractions with fractions inside — are just division in disguise: (3/4)/(2/5) = 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8. And dividing mixed numbers always goes through improper form first: 2 1/2 ÷ 1 1/4 = 5/2 ÷ 5/4 = 5/2 × 4/5 = 20/10 = 2.

Division by zero is the one hard rule: you cannot divide by a fraction equal to zero, and a denominator can never be zero. The calculator shows an error instead of a result in both cases.

🧮How to do fractions: every operation, worked step by step

When I teach someone how to solve fractions, I make them run the same four operations by hand once, slowly, before I let them trust any calculator — including this one. Here is that lesson, one operation at a time. If your real question is how to do fractions on a calculator, these steps are exactly what the machine runs for you: it treats each fraction as a division and applies these rules in order.

Multiply. Start with the operation students ask about most. This page works as a multiply fractions calculator, so here is its canonical example, 2/3 × 2/3, done by hand. Step 1: multiply the numerators — 2 × 2 = 4. Step 2: multiply the denominators — 3 × 3 = 9. Step 3: simplify — 4/9 is already in lowest terms. So 2/3 × 2/3 = 4/9 = 0.444…. Nothing else is hiding inside a fraction multiplication calculator: multiply across, simplify, done.

Divide. Division of fractions is multiplication in disguise. I keep the first fraction, change the sign to ×, and flip the second fraction (its reciprocal): 3/4 ÷ 1/2 = 3/4 × 2/1 = 6/4 = 3/2 = 1 1/2. Keep, change, flip — that is the whole method, and it works because dividing by a half asks how many halves fit, which is why the answer grows.

Add. Adding is where an adding fractions calculator earns its keep, because unlike denominators need a common denominator first. Take 1/3 + 1/4. Step 1: the least common denominator of 3 and 4 is 12. Step 2: rewrite both — 1/3 = 4/12 and 1/4 = 3/12. Step 3: add the numerators and keep the denominator — 4/12 + 3/12 = 7/12. Step 4: check the simplify step — 7 and 12 share no factor, so 7/12 is final.

Subtract. Subtraction walks the same road: 5/6 − 2/3. The LCD of 6 and 3 is 6, so 2/3 becomes 4/6, and 5/6 − 4/6 = 1/6. The only extra way to go wrong is the sign: if the second fraction is larger, the answer is negative, and the calculator carries that sign through every form of the result.

Whole numbers. Finally, the whole number fraction calculator case — a whole number times a fraction. I write the whole number as a fraction over 1: 4 × 3/5 = 4/1 × 3/5 = 12/5 = 2 2/5. The same trick answers every fraction-of-a-quantity question: 2/3 of 12 is 2/3 × 12/1 = 24/3 = 8. Run each operation once like this and the tab above stops being a black box — it becomes a fast version of your own working.

🔢Mixed numbers vs improper fractions

A mixed number like 4 1/4 pairs a whole number with a proper fraction — easy for humans to picture. An improper fraction like 17/4 has a numerator larger than its denominator — easy for calculators to work with. Convert mixed → improper with (whole × denominator + numerator) / denominator: 4 1/4 = (4 × 4 + 1)/4 = 17/4. Convert back by dividing: 17 ÷ 4 = 4 remainder 1, so 17/4 = 4 1/4.

Watch the sign on negative mixed numbers: −1 1/2 means −(1 + 1/2) = −3/2, not −1 + 1/2. The minus sign belongs to the whole quantity. The calculator follows this convention: enter whole = −1, numerator = 1, denominator = 2 and it reads the value as −3/2.

For unlike denominators, cross-multiplication compares without finding a common denominator: to compare 4/7 and 5/9, compute 4×9 = 36 and 5×7 = 35 — since 36 > 35, 4/7 > 5/9. This works because both products are the numerators the fractions would have over the common denominator 63 (36/63 vs 35/63).

To compare mixed numbers, look at the whole part first: 4 1/4 beats 3 7/8 no matter what the fractions say. If the whole parts tie, compare the fractions: 2 5/6 vs 2 2/3 → 5/6 vs 4/6, so 2 5/6 is larger. Adding mixed numbers works in columns — wholes with wholes, fractions with fractions: 2 1/3 + 1 1/6 = 3 + (2/6 + 1/6) = 3 1/2. If the fraction sum reaches a whole, carry it over: 1 2/3 + 2 2/3 = 3 + 4/3 = 4 1/3.

Mixed numbers show up constantly in real life — 2 1/2 cups of flour, 5 3/4 inches, 1 1/2 hours — while improper fractions dominate algebra and higher math. Being fluent in both directions is the skill that makes fraction arithmetic feel easy.

🔁Decimals, fractions, and repeating digits

Every fraction has a decimal form — just divide. Some terminate: 3/4 = 0.75, 7/8 = 0.875, 1/20 = 0.05. Others repeat forever: 1/3 = 0.333…, 2/3 = 0.666…, 1/6 = 0.1666…, 1/7 = 0.142857142857…. The repeating part is called the repetend, and mathematicians write it with a bar or parentheses: 1/3 = 0.(3), 1/7 = 0.(142857).

Whether a fraction terminates depends only on its denominator (in simplest form): if the denominator’s prime factors are only 2s and 5s, the decimal terminates. So 7/8 terminates (8 = 2³), but 1/6 repeats (6 = 2 × 3 — that factor of 3 forces repetition) and 1/7 repeats (7 is prime, neither 2 nor 5). The Decimal ⇄ Fraction tab above detects repetends automatically and marks them, handling fraction to decimal conversion with the repeating part clearly shown.

Going the other way — decimal to fraction — is mechanical: 0.75 = 75/100 = 3/4 (divide top and bottom by 25). For a repeating decimal like 0.(3): call it x, then 10x = 3.(3), subtract: 9x = 3, so x = 1/3. Try entering 0.(3) in the converter to see it work.

Percentages are fractions in disguise: 37.5% = 37.5/100 = 375/1000 = 3/8 (divide by 125). Going the other way, 5/8 = 0.625 = 62.5%. Any percentage converts to a fraction over 100, simplified — and any fraction converts to a percentage by multiplying its decimal by 100.

Some repetends are famously long. 1/17 = 0.(0588235294117647) — a 16-digit cycle. 1/19 repeats with period 18. In general, for a prime denominator p (other than 2 or 5), the repetend length divides p−1 — which is why 1/7 has period 6 and 1/13 has period 6 as well (0.(076923)). These cycles are not random: multiply 142857 by 2, 3, 4, 5, 6 and you get rotations of the same digits (285714, 428571, 571428, 714285) — a cyclic number.

One note on the converter itself: the Decimal ⇄ Fraction tab above handles any terminating decimal you type — 0.8 becomes 4/5, 0.125 becomes 1/8, 2.25 becomes 2 1/4 — by writing the digits over a power of ten and simplifying by the GCD. Repeating decimals are the one case where you have to tell it the cycle, using the 0.(3) notation. I keep a short mental table for the common ones — 1/2 = 0.5, 1/4 = 0.25, 1/5 = 0.2, 1/8 = 0.125, 3/4 = 0.75 — because those five cover nearly every kitchen and workshop conversion I ever meet.

🏠Fractions in everyday life

Fractions run the practical world. In the kitchen, recipes scale by fractions: halving a recipe that calls for 3/4 cup of sugar gives 3/8 cup; doubling 2/3 cup of flour gives 4/3 = 1 1/3 cups. Measuring cups themselves are fractions — 1/4, 1/3, 1/2, 1 cup — and converting between them is fraction arithmetic: 2/3 cup is 10 tablespoons plus 2 teaspoons (since 1 cup = 16 tbsp, 2/3 × 16 = 32/3 = 10 2/3 tbsp).

In construction and DIY, the imperial system is fractional to the core: tape measures read in 1/16-inch increments, so 5/8 inch is 10/16 — and adding 5/8 + 7/16 means converting to 10/16 + 7/16 = 17/16 = 1 1/16 inches. Lumber dimensions, drill bits, and wrench sizes all follow the same pattern.

Time is fractions: a quarter hour is 1/4 × 60 = 15 minutes, two-thirds of an hour is 40 minutes, and 1 1/2 hours is 90 minutes. Money too: a quarter is literally 1/4 of a dollar, and stock prices were quoted in fractions (halves, quarters, eighths) until 2001. Music runs on fractions as well — a whole note divides into halves, quarters, eighths, and sixteenths, and a 3/4 time signature means three quarter-note beats per measure.

Even probability speaks fractions: a 1/6 chance of rolling a six, 1/2 of a coin landing heads, 4/52 = 1/13 of drawing an ace from a deck. Whenever you see “1 in 4”, that is the fraction 1/4 wearing different clothes.

⚠️Common fraction mistakes (and how to avoid them)

Mistake 1: adding denominators. 1/2 + 1/3 ≠ 2/5. The denominators count piece-sizes, which must match before you count pieces. Always find a common denominator first: 1/2 + 1/3 = 3/6 + 2/6 = 5/6.

Mistake 2: forgetting to simplify. 4/8 + 2/8 = 6/8 is correct but unfinished — simplify to 3/4. Get in the habit of dividing by the GCD as the last step of every problem.

Mistake 3: flipping the wrong fraction when dividing. In a/b ÷ c/d, flip the second fraction (the divisor), not the first: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6. Flipping the first gives the reciprocal of the right answer.

Mistake 4: misplacing the negative sign. −2/3, 2/−3, and −(2/3) are all the same value (−0.666…), but −1 1/2 is −3/2, not −1/2. Keep the sign with the whole mixed quantity.

Mistake 5: comparing fractions by numerator alone. 3/4 is bigger than 5/8 even though 5 > 3 — the piece-size matters. Convert to a common denominator (6/8 vs 5/8) or to decimals (0.75 vs 0.625) before comparing.

Mistake 6: treating “of” as addition. In word problems, “1/2 of 3/4″ means multiply: 1/2 × 3/4 = 3/8. “Of” almost always signals multiplication with fractions.

A mixed number is a whole number parked next to a proper fraction: 1 3/4 is one whole plus three quarters. I meet mixed numbers everywhere fractions matter — recipes, tape measures, lumber, journey times — so the Arithmetic tab above accepts them directly: fill in the Whole box beside each fraction and it treats the pair as a single value. People also search for this as a mixed fraction calculator or a mixed number fraction calculator; all three names describe the same tool and the same method.

Converting mixed → improper is the move everything else builds on. I multiply the whole number by the denominator and add the numerator: 1 3/4 = (1 × 4 + 3)/4 = 7/4. One more: 3 2/5 = (3 × 5 + 2)/5 = 17/5. Going back, I divide: 17 ÷ 5 = 3 remainder 2, so 17/5 = 3 2/5. If the conversion feels like a ritual, that is because it is one — do it every time, in the same order, and mixed numbers stop causing errors.

Adding mixed numbers, I convert first rather than juggling columns: 1 3/4 + 2 1/2 = 7/4 + 5/2 = 7/4 + 10/4 = 17/4 = 4 1/4. Multiplying, conversion is non-negotiable. The canonical example: 1 3/4 × 2. Convert: 7/4. Write the 2 as 2/1. Multiply across: 7/4 × 2/1 = 14/4. Simplify: 14/4 = 7/2 = 3 1/2. Multiplying the parts separately (wholes × wholes, fractions × fractions) is the classic trap — with two genuine mixed numbers it quietly drops the cross-terms. The tab above converts, calculates, simplifies, and shows the result as an improper fraction, a mixed number, and a decimal, so you can read it in whichever form your recipe or worksheet expects.

An improper fraction is one whose numerator is equal to or larger than its denominator: 4/3, 7/2, 17/5. “Improper” sounds like a mistake; it is not. It is simply the form calculators and algebra prefer, because the four operation rules work on it directly, with no whole-number part to carry around or borrow from. Every result the Arithmetic tab produces passes through this form on its way to the mixed number it shows you.

The conversion I use runs both directions. Improper → mixed: divide the numerator by the denominator; the quotient is the whole part and the remainder goes over the same denominator. The example I always start with: 4/3 = 1 1/3, because 4 ÷ 3 = 1 remainder 1. Bigger ones behave identically: 17/5 = 3 2/5 (17 ÷ 5 = 3 remainder 2) and 7/2 = 3 1/2 (7 ÷ 2 = 3 remainder 1). Mixed → improper reverses it: multiply the whole part by the denominator and add the numerator, so 2 5/6 = (2 × 6 + 5)/6 = 17/6.

Two edge cases I make a point of checking: a fraction like 8/4 is improper in form but equals exactly 2 — the remainder is zero, so the mixed form has no fraction part at all. And any fraction whose numerator equals its denominator, like 5/5, is exactly 1. When I sanity-check a result, I ask whether it sits on the right side of 1 for its form: improper means one or more, proper means less than one.

Adding or subtracting unlike fractions lives or dies on one step: rewriting both fractions over a common denominator. The smallest one that works — the least common denominator (LCD) — is the least common multiple of the two denominators, and it is what the Arithmetic tab finds for you behind the scenes.

My by-hand method: list multiples of the larger denominator until one is divisible by the smaller. The worked example: 2/3 + 1/4. Multiples of 4 run 4, 8, 12 — and 12 is divisible by 3, so the LCD is 12. Rewrite each fraction by multiplying top and bottom by whatever reaches 12: 2/3 = 8/12 (×4) and 1/4 = 3/12 (×3). Now the pieces are the same size, so I add the numerators and keep the denominator: 8/12 + 3/12 = 11/12.

Subtraction is identical apart from the final step: 5/6 − 1/4. The LCD of 6 and 4 is 12, so 5/6 = 10/12 and 1/4 = 3/12, giving 10/12 − 3/12 = 7/12. Could I just multiply the denominators instead (6 × 4 = 24)? Yes — the answers agree after simplifying — but the LCD keeps the numbers small enough to check in my head, and an LCD result usually arrives already simplified. That is the trade the calculator makes for you on every + and − you enter.