Fraction Calculator

How to use this calculator

1. Enter the numerator and denominator of both fractions.
2. Choose the operation (add, subtract, multiply, divide).
3. The answer is simplified automatically and shown as a mixed number and a decimal.

The formula

a, c

Numerators (the top numbers)

b, d

Denominators (the bottom numbers, never zero)

How it works

To add or subtract, the fractions need a common denominator: multiply each fraction's top and bottom by the other's denominator, then combine the numerators.

Multiplication is straight across (tops × tops, bottoms × bottoms). Division flips the second fraction and multiplies. Every result is then reduced by dividing top and bottom by their greatest common divisor.

The four operations, rule by rule

To add or subtract, rewrite both fractions over a common denominator, combine the numerators, and simplify. To multiply, multiply the numerators together and the denominators together — no common denominator is needed. To divide, invert the second fraction and multiply, because dividing by a number is the same as multiplying by its reciprocal.

Simplification at the end uses the greatest common divisor of the numerator and denominator. Dividing both by that number produces the unique lowest-terms form, which is the conventional way to present a fractional answer.

Why fractions beat decimals for exact work

One third written as a decimal is 0.3333…, and every truncation introduces error that compounds through a calculation. Kept as 1/3, the value is exact, and three of them sum to precisely 1. For recipes scaled by odd factors, gear ratios, or probability trees, that exactness matters.

Decimals win for comparison and for feeding into measuring instruments, which is why the calculator shows both forms of every answer.

Mixed numbers and improper fractions

A mixed number such as 2 3/4 is a whole number plus a proper fraction. Convert it to the improper fraction 11/4 before calculating by multiplying the whole part by the denominator and adding the numerator; convert back afterwards by dividing and keeping the remainder.

Improper fractions are not wrong — they are simply the more convenient form for arithmetic, and many technical fields leave answers that way deliberately.

Finding common denominators quickly

Adding and subtracting fractions requires a shared denominator. Multiplying the two denominators always works but can produce large numbers; the least common multiple keeps them small.

For 3/8 + 5/12, the least common multiple of 8 and 12 is 24, giving 9/24 + 10/24 = 19/24. Using 96 as the denominator would also be correct but would need heavier simplification afterwards.

Once denominators match, only the numerators are added or subtracted — the denominator stays as it is.

Simplifying to lowest terms

A fraction is in lowest terms when the numerator and denominator share no common factor except one. Divide both by their greatest common divisor to get there.

18/24 shares a factor of 6, so it simplifies to 3/4. Doing this at the end of every calculation keeps results readable and makes comparisons easier.

Simplifying never changes a fraction's value, only how it is written — 3/4, 6/8 and 0.75 are the same number.

Common mistakes with fractions

Adding numerators and denominators separately is the classic error: 1/2 + 1/3 is not 2/5. Only multiplication works term by term.

Forgetting to flip the second fraction when dividing is another. Dividing by 2/3 means multiplying by 3/2.

With mixed numbers, ignoring the whole part during the operation and adding it back at the end causes errors when borrowing is needed. Convert to an improper fraction first.

When fractions beat decimals

Thirds, sixths, sevenths, and ninths have no exact decimal representation. Rounding them early introduces drift that compounds through a calculation.

Recipes, timber cuts, gear ratios, sheet music, and imperial measurements are all naturally fractional; converting to decimals adds a conversion step and a rounding risk with no benefit.

Decimals win where you need to compare many values at a glance or feed results into further percentage work.

Everyday applications

Scaling a recipe up or down multiplies every fractional quantity by the same factor — two and a half times 3/4 cup is 15/8, or 1 7/8 cups.

In construction and DIY, imperial measurements are fractional by convention, so cut lists and material estimates stay in halves, quarters, eighths, and sixteenths.

In finance and shift planning, fractions of an hour and fractions of a share appear constantly, and exact fractional arithmetic avoids the small mismatches decimal rounding creates.

Glossary and verification

Numerator: the top number, counting the parts you have. Denominator: the bottom number, defining how many parts make a whole. Improper fraction: numerator larger than denominator. Mixed number: a whole number plus a proper fraction.

Verify any result by converting both sides to decimals and comparing. 3/8 + 5/12 = 19/24 checks out as 0.375 + 0.4167 = 0.7917.

For multiplication, sanity-check the size: multiplying by a fraction below one always makes a number smaller, and dividing by one always makes it larger.

Common denominators and simplification

Adding or subtracting requires a common denominator; multiplying and dividing do not. The least common multiple of the denominators keeps the numbers small and the simplification easy.

Simplify by dividing the numerator and denominator by their greatest common divisor. Twelve over eighteen becomes two over three after dividing both by six.

An improper fraction such as seventeen over five is perfectly valid arithmetic; converting to the mixed number three and two fifths is a readability choice, not a correctness one.

Dividing by a fraction

Dividing by a fraction is multiplying by its reciprocal. Three quarters divided by two thirds equals three quarters times three halves, which is nine eighths.

This is why dividing by a value smaller than one makes the result larger, which surprises people expecting division always to shrink a number.

It also underlies unit conversion: dividing by 'litres per 100 km' to get distance per litre is exactly this operation.

Everyday uses and checks

Scaling a recipe by two thirds, splitting a bill three ways, converting between imperial measures, or working out how many boards of a given length fit a run.

Check any result by converting both the inputs and the answer to decimals and repeating the arithmetic; the two should agree to several places.

Check simplification by confirming the numerator and denominator share no common factor other than one.

Fractions in measurement and trade

Imperial measurement runs on halves, quarters, eighths, and sixteenths, so tradespeople add and subtract fractions constantly. Converting to a common denominator of sixteen handles almost every workshop calculation.

Recipe scaling is the other everyday case: multiplying a fraction by a whole number scales the numerator, while dividing a recipe scales the denominator.

Drill bits, timber sizes, and fastener lengths are catalogued as fractions, so reducing to lowest terms helps you match a calculated size to a stocked one.

When a result falls between available sizes, keep the fraction rather than converting to a decimal, because the nearest stocked size is itself a fraction.

Converting between fractions and decimals

Dividing the numerator by the denominator converts any fraction to a decimal, which terminates only when the denominator's prime factors are limited to two and five.

Thirds, sevenths, and ninths therefore repeat forever, and rounding them early introduces error that accumulates through a chain of calculations.

To go the other way, write the decimal over a power of ten and reduce: 0.375 becomes 375 over 1000, which reduces to three eighths.

Keeping values as fractions until the final step preserves exactness, which matters for cutting lists, dosing, and any figure that will be multiplied repeatedly.

Simplifying reliably

Divide numerator and denominator by their greatest common divisor to reach lowest terms in one step, rather than halving repeatedly and hoping to finish.

For large numbers, the Euclidean algorithm finds that divisor quickly: replace the larger number with its remainder against the smaller until the remainder reaches zero.

An improper fraction converts to a mixed number by dividing out the whole part and keeping the remainder over the original denominator.

Always simplify before comparing two fractions, since equivalent fractions written differently are easy to mistake for different values.

When this calculator is useful

• Recipes and cooking measurements
• Carpentry and DIY measurements
• Homework checking
• Comparing proportions exactly (no decimal rounding)