Murdude kalkulaator

A fraction calculator that shows its work

Enter any two fractions — or chain up to four — and this free fraction calculator adds, subtracts, multiplies and divides them with the full step-by-step solution: the least common denominator for additions, the straight-across products for multiplication, the 'keep, change, flip' reciprocal for division, and the GCD used to simplify at the end. The kind of working math teachers want to see — always visible, never paywalled.

Mixed numbers are first-class: every fraction gets an optional whole-number box, so 2 ¾ + 1 ⅖ just works (the conversion to improper fractions appears as step one). Whole numbers, improper fractions and negatives are all fine in any slot, and results display three ways at once — simplified fraction, mixed number, and decimal with a percent equivalent. The tool updates live as you type, so you see the answer and the working at the same moment.

Simplify fractions and see the GCD

The Simplify tab reduces any fraction to lowest terms and shows how: it finds the greatest common divisor of the numerator and denominator and divides both by it — 8/32 → GCD 8 → 1/4. If the GCD is 1, you'll know the fraction was already in simplest form. It's the fastest way to check homework or confirm an answer before writing it down.

Exact arithmetic is guaranteed throughout: the math runs on arbitrary-precision integers, so big numerators like 1234/5678 don't silently lose precision the way floating-point calculators can. You'll never get a slightly-off decimal where an exact fraction was needed — the result is always mathematically exact, whether you're simplifying, multiplying, or chaining four fractions together.

Decimal to fraction (and back) — repeating decimals included

The Decimal ⇄ Fraction tab converts both directions with the working shown. Type 0.625 and you'll see it written as 625/1000 and reduced by its GCD to 5/8; type 1/3 the other way and long division reveals the repeating decimal 0.(3) — repeats are detected and marked explicitly rather than truncated, so 1/7 shows its full 0.(142857) cycle.

Woodworkers and DIYers get a dedicated feature: a round to the nearest 1/16, 1/32 or 1/64 option that turns any decimal measurement into the closest tape-measure fraction — 0.7 inches ≈ 11/16. Enter a decimal dimension from a digital caliper and get the nearest standard fraction instantly. The working shows which two standard fractions bracket your measurement and which is closer.

Built for homework, kitchens and workshops

Halving a recipe that calls for ¾ cup, splitting 2 ½ hours across three tasks, checking a child's worksheet on adding fractions with different denominators, or converting a drill-bit size to an imperial fraction — fraction math shows up in everyday life. A calculator that explains itself beats one that just spits an answer, because you can see whether you set up the problem correctly.

Everything updates live as you type; no submit buttons, no page reloads. A one-click copy button grabs the result together with its decimal and percent forms. Like every UtiloKit tool, it's 100% client-side — no account, no ads cluttering the math output, and it works offline once the page has loaded.

How this compares to Mathway, Symbolab, and other fraction calculators

Mathway is the most widely used math calculator online, but it charges $9.99/month to see step-by-step solutions — you only get the final answer for free. Symbolab is similar: it shows steps but requires a $2.49/week subscription to access them. Cymath shows some steps on the free tier but locks detailed breakdowns behind Cymath Plus at $9.99/month. Calculator.net's fraction calculator is free and shows steps, but handles only two fractions at a time and doesn't support decimal ⇄ fraction conversion.

This calculator shows every step — LCD, GCD, reciprocal, simplification, mixed-number conversion, repeating decimal detection — at no cost, with no account, and for up to four chained fractions. Your inputs never leave your browser. It's the fraction calculator Mathway and Symbolab should be for free users, but aren't.

The four fraction operations, step by step

Addition and subtraction both require a common denominator before the numerators can be combined. To add 2/3 + 3/4, find the least common denominator of 3 and 4, which is 12. Rewrite each fraction: 2/3 = 8/12 (multiply top and bottom by 4) and 3/4 = 9/12 (multiply by 3). Now add the numerators: 8/12 + 9/12 = 17/12, an improper fraction equal to 1 5/12. Subtraction follows the same LCD step — only the numerator operation changes from addition to subtraction. The critical skill is finding the LCD efficiently; for small denominators mental arithmetic works, but for large or multiple denominators the prime factorization method is more reliable: factor each denominator, take the highest power of every prime, multiply them together.

Multiplication is the most direct of the four operations: multiply the numerators together and the denominators together, then simplify. 2/3 × 3/4 = (2 × 3) / (3 × 4) = 6/12 = 1/2. No common denominator is required. A useful shortcut is cross-cancellation — dividing a numerator and a non-matching denominator by their common factor before multiplying: in 2/3 × 3/4, the 3 in the numerator of the second fraction and the 3 in the denominator of the first share a factor of 3, so they cancel to give 2/1 × 1/4 = 2/4 = 1/2, avoiding larger intermediate numbers. Division reduces to multiplication via the reciprocal (also called "keep, change, flip"): 2/3 ÷ 3/4 becomes 2/3 × 4/3 = 8/9. The rule works because dividing by a fraction is the same as asking how many of it fit into the dividend — multiplying by its reciprocal gives exactly that count.

When mixed numbers appear in any operation, the standard approach converts them to improper fractions first: multiply the whole part by the denominator and add the numerator. 1 2/3 becomes (1 × 3 + 2)/3 = 5/3. Once everything is in improper-fraction form, apply the relevant operation normally and convert back at the end. For addition with mixed numbers it is also possible to add the whole parts and fractional parts separately, but this method breaks down when the fractional parts sum to more than 1, making the improper-fraction route more reliable in general.

Types of fractions and how to convert between them

Fractions come in several forms, and recognizing each makes arithmetic cleaner. A proper fraction has a numerator smaller than its denominator — its value is between 0 and 1 (examples: 3/4, 2/7, 5/9). An improper fraction has a numerator greater than or equal to its denominator, so its value is 1 or more (examples: 7/4, 9/9, 15/8). A mixed number combines a whole part and a proper fractional part (1 3/4, 3 1/2). Any improper fraction can be expressed as a mixed number and vice versa: 7/4 means 7 ÷ 4 = 1 remainder 3, so it equals 1 3/4. To reverse the process, multiply the whole number by the denominator and add the numerator: 1 × 4 + 3 = 7, giving 7/4. Both forms represent the same value — the choice between them is one of convenience and context.

Equivalent fractions are different-looking fractions that represent the same value. 1/2, 2/4, 3/6, 4/8, and 50/100 are all equivalent — each is obtained by multiplying or dividing both the numerator and denominator by the same non-zero integer. Finding equivalent fractions with a target denominator is exactly what the LCD step of addition and subtraction does: 2/3 must become something/12, so multiply both parts by 4 to get 8/12. Simplifying (also called reducing) goes the other direction: divide both terms by their greatest common factor. For 12/18, the GCF is 6, and 12 ÷ 6 = 2, 18 ÷ 6 = 3, giving 2/3 in lowest terms. A fraction is fully reduced when the numerator and denominator share no common factor other than 1.

Unit fractions have a numerator of exactly 1 — 1/2, 1/3, 1/4, 1/7. They are the building blocks of ancient fraction systems (more on that below) and appear in modern contexts such as probability (a fair coin shows heads with probability 1/2) and rates (one task per 4 hours = 1/4 task per hour). Negative fractions behave like their positive counterparts: the sign is conventionally placed in front of the fraction, so −3/4 is the standard form of (−3)/4 or 3/(−4). In arithmetic, a fraction with two negative signs in numerator and denominator is positive, matching the rule for signed integers.

Fractions in everyday life: cooking, construction, finance, and music

Cooking is probably the most frequent everyday encounter with fractions. Recipes specify quantities in fractions of standard cups or spoons — 2/3 cup of flour, 1/4 teaspoon of salt — and scaling them up or down means multiplying those fractions. Halving a recipe that calls for 2/3 cup means computing 2/3 × 1/2 = 2/6 = 1/3 cup. Scaling 1.5× requires 2/3 × 3/2 = 6/6 = 1 cup exactly. The fraction calculator handles multi-step scaling in a single computation: enter the original quantity, multiply by the scaling ratio (as a fraction), and read the result as a mixed number that maps to real measuring-cup markings.

Construction and carpentry still use the imperial system, where all standard measurements are given in fractions of an inch — 3/16", 7/32", 9/64". Adding two cut lengths of 3/8" and 5/16" requires a common denominator of 16: 6/16 + 5/16 = 11/16". Subtracting a kerf (saw-blade width, often 1/8") from a measured length calls for the same technique. Bolt diameters, screw gauges, pipe fittings, and sheet-metal gauges are all specified in fractions; a carpenter who cannot add and subtract them is forced to convert to decimals, round, and risk accumulated error. Financial contexts also use fractional notation: an APR of 6.5% expressed as a fraction is 13/200, which simplifies calculation of per-period interest rates. Bond yields, currency exchange rates, and loan fee ratios are sometimes expressed as fractions to make proportional comparisons clearer.

Music theory encodes rhythm as fractions: the time signature 3/4 means three quarter-note beats per bar (the numerator counts the beats; the denominator names the note value). The signature 6/8 gives six eighth-note pulses grouped into two sets of three, creating a different feel despite having the same total duration as 3/4. Subdividing a quarter note into triplets means fitting three equally spaced notes into one beat — each note lasts 1/3 of a beat. Note durations themselves are a geometric series of fractions: a whole note = 1, half note = 1/2, quarter = 1/4, eighth = 1/8, sixteenth = 1/16. The total duration of a bar in any time signature must always sum to exactly the value specified by the fraction, making fraction arithmetic a core skill in rhythmic composition and transcription.

Fractions vs decimals vs percentages — and why fractions are exact

Any rational number can be expressed as a fraction, a decimal, or a percentage, and the choice of representation affects both precision and readability. Fractions are the natural form for exact rational values: 1/3 written as a decimal becomes 0.333… — a non-terminating repeating sequence that must be truncated at some point, introducing a rounding error. As a fraction, 1/3 is exact. Similarly 2/7 = 0.(285714) repeating, and 1/6 = 0.1(6) repeating. A decimal terminates only if the denominator's prime factorization contains no primes other than 2 and 5 (the prime factors of 10). Otherwise, the decimal repeats. Whenever an exact answer is required — in algebra, number theory, or any context where accumulated rounding error matters — fractions are the correct representation.

Decimals are preferred for measurement and computing: a digital caliper reads 23.175 mm, not a fraction, and floating-point arithmetic is faster in hardware than rational arithmetic. Percentages are used for communication and comparison because the denominator is always implicitly 100, making magnitudes easy to compare: a 12% discount is immediately more intuitive than 3/25. Converting between the three forms follows straightforward rules: fraction to decimal — divide the numerator by the denominator; decimal to fraction — write the decimal over the appropriate power of 10 then reduce (0.48 = 48/100 = 12/25); anything to percentage — multiply the decimal (or fraction value) by 100. The fraction calculator's Decimal ⇄ Fraction tab automates both directions with working shown.

A critical subtlety arises in computing: binary floating-point cannot represent most decimal fractions exactly, because 10 is not a power of 2. In binary, 0.1 is the infinitely repeating pattern 0.0001100110011…, and 0.2 is similarly non-terminating. When JavaScript or Python computes 0.1 + 0.2, both values are stored as the nearest representable IEEE 754 double-precision float, and the sum lands at 0.30000000000000004 rather than 0.3 — a difference of about 5.5 × 10⁻¹⁷. This is not a bug in those languages; it is a fundamental property of representing decimal fractions in base-2 hardware. For this reason, financial software never stores money as floating-point: it uses integer arithmetic (storing amounts in cents, then dividing for display) or dedicated rational arithmetic libraries that keep exact numerator/denominator pairs throughout all calculations. The fraction calculator on this page follows the same principle — all arithmetic runs on exact integers, so there is no floating-point error at any stage.