Fraction Calculator
Add, subtract, multiply or divide two fractions or mixed numbers. The answer comes back in lowest terms, as a mixed number and as a decimal — with the common denominator, the arithmetic and the reduction all written out.
Two fractions, one operation
The whole part of a mixed number. Leave it at 0 for a plain fraction; a minus sign here makes the whole quantity negative.
How many pieces. It may be larger than the denominator — 7/4 is a perfectly good fraction.
How many pieces the whole was cut into.
Adding and subtracting need a common denominator; multiplying and dividing do not.
The whole part of the second mixed number, 0 if it has none.
How many pieces in the second fraction.
How many pieces the second whole was cut into.
1 1/2 + 2/3 = 13/6, which is 2.166667 as a decimal.
- In lowest terms
- 13/6
- As a mixed number
- 2 1/6
- As a decimal
- 2.166667
- First fraction
- 3/2
- Second fraction
- 2/3
- Before reducing
- 13/6
- The least common denominator of 2 and 3 is 6, so the two fractions become 9/6 and 4/6.
- 9 + 4 = 13, over the common denominator 6.
- 13 and 6 share no factor above 1, so the answer is already in lowest terms.
- Every step above is whole-number arithmetic. The decimal is worked out last, from the fraction — never the other way round.
One improper fraction each, then four rules
Every operation starts the same way: each mixed number is collapsed into a single numerator over a single denominator, so there is one quantity rather than two pieces to keep track of.
w n∕d → (w×d + n)∕d • a∕b ± c∕d over L = lcm(b, d) • a∕b × c∕d = ac∕bd • a∕b ÷ c∕d = ad∕bc- w, n, d
- The whole part, numerator and denominator you type
- L
- The least common denominator — the smallest number both bottoms divide into
- gcd
- The greatest common divisor, found by Euclid’s algorithm, that reduces the answer
Addition and subtraction route through the least common denominator rather than the simpler product of the two bottoms. Both give the same value, but the product leaves more to reduce: 1/6 + 1/4 over 24 is 10/24, which then has to come back down to 5/12, while over the least common denominator of 12 it arrives as 5/12 in one step.
Nothing is divided until the very last line. The two numerators are integers, the common denominator is an integer, every product and sum along the way is an integer, and the reduction is integer division by the greatest common divisor. The decimal is worked out at the end, from the finished fraction — which is the opposite of how a calculator that converts to decimals first has to work, and the reason its answers drift.
1 1/2 + 2/3, all the way through
First, collapse the mixed number: 1 1/2 is (1 × 2) + 1 = 3 over 2. The second fraction, 2/3, is already improper enough to work with.
Next, find a shared unit. The least common denominator of 2 and 3 is 6, so 3/2 becomes 9/6 (multiply top and bottom by 3) and 2/3 becomes 4/6 (multiply top and bottom by 2). Now both are counts of sixths, so they can be added: 9 + 4 = 13, giving 13/6.
Finally, reduce. Euclid’s algorithm on 13 and 6 goes 13 mod 6 = 1, then 6 mod 1 = 0, so the greatest common divisor is 1 — 13/6 is already in lowest terms. As a mixed number that is 2 1/6, because 6 goes into 13 twice with 1 left over, and as a decimal it is 2.166666…, a number that has no exact decimal form at all.
A case that does reduce: 5/6 × 3/10. Multiplying straight across gives 15/60. The greatest common divisor of 15 and 60 is 15, so the answer is 1/4 — a fraction whose bottom appears in neither input, which is exactly why the reduction step is not optional.
What this calculator assumes
- Every box holds a whole number. If your value is already a decimal, convert it first with the decimal to fraction calculator, which handles repeating decimals too.
- Denominators are at least 1. A denominator of zero does not name a very small piece; it names no size of piece at all, so it is refused rather than answered.
- A minus sign in a whole-number box applies to the entire mixed number: −2 3/4 is −11/4. When there is no whole part, the minus goes on the numerator instead.
- Whole parts run to ±1,000 and numerators and denominators to ±10,000. Those bounds are not arbitrary — they are the largest that keep every intermediate product inside the range where a computer holds integers exactly, which is what the promise of an exact answer actually rests on.
- The decimal is the only approximate figure on the page, shown to twelve significant figures. The fraction above it is the answer; the decimal is a convenience.
- Two operands at a time. For a longer chain, put the answer back in as the first fraction — the improper form is given so you can do exactly that without losing anything.
Fraction FAQ
Why use fractions at all when I could just use decimals?
Because most fractions have no decimal form. A computer stores decimals as binary fractions, which can hold 1/2, 3/8 and 5/16 exactly and cannot hold 1/3, 1/6 or even 1/10 at all. Ask any spreadsheet for 0.1 + 0.2 and it answers 0.30000000000000004; add a sixth to itself six times and you land on 0.9999999999999999 rather than 1. Those errors are small, but they compound, and they are invisible until something downstream compares two numbers that should have matched. A third is a third here, forever, because nothing is ever divided until the final decimal line.
How do I turn a mixed number into an improper fraction?
Multiply the whole part by the denominator, add the numerator, and keep the same denominator. 1 1/2 becomes (1 × 2) + 1 = 3 over 2. 2 3/4 becomes (2 × 4) + 3 = 11 over 4. Every operation on this page does that first, because addition and multiplication have no rule for a number written in two pieces — the whole part and the fraction have to be a single quantity before anything else can happen.
Why do adding and subtracting need a common denominator when multiplying does not?
Because a numerator only counts something once you know what one piece is worth. Adding 1/2 to 1/3 is adding halves to thirds, which is like adding inches to centimetres — the counts are not in the same unit, so the sum is meaningless until both are rewritten in sixths. Multiplication is a different question: two thirds of three quarters does not ask you to combine counts, it asks for a part of a part, and multiplying the tops and the bottoms answers that directly. That is why 1/2 × 1/3 is straightforward and 1/2 + 1/3 needs a step first.
Why is dividing by a fraction the same as multiplying by its reciprocal?
Division asks how many of the divisor fit into the dividend. How many quarters fit into 3? Twelve — which is 3 × 4, the reciprocal of 1/4. The general rule falls out of the same idea: dividing by 2/3 shrinks by 2 and grows by 3, and multiplying by 3/2 does exactly that in one move. It also explains why dividing by a fraction smaller than 1 makes the answer bigger, which is the single most counterintuitive thing in school arithmetic.
How do I know a fraction is already in lowest terms?
When the top and bottom share no whole-number factor above 1. The reliable way to check is Euclid's algorithm: divide the larger by the smaller, keep the remainder, repeat until the remainder is zero, and the last non-zero value is the greatest common divisor. For 13 and 6 it reaches 1 immediately, so 13/6 cannot be reduced. For 375 and 1000 it reaches 125, which is why 375/1000 is really 3/8. The calculator shows the divisor it found, so the reduction is never a step you have to take on trust.
What does a negative sign on a mixed number apply to?
The whole quantity. −2 3/4 means −(2 + 3/4) = −11/4, not −2 + 3/4 = −1.25. This trips people up because the minus sign sits next to the 2 and looks like it belongs only to it. Put a minus in the whole-number box here and the entire mixed number goes negative, which is the convention every textbook and every number line uses.
Sources and review notes
- Common Core State Standards for Mathematics, 5.NF — adding and subtracting fractions with unlike denominators by producing an equivalent pair
- IEEE Std 754-2019 — the binary floating-point format that cannot represent 1/3, 1/6 or 1/10, and why this engine avoids it
- Goldberg, D. (1991), “What Every Computer Scientist Should Know About Floating-Point Arithmetic”, ACM Computing Surveys 23(1)
The arithmetic here has no data behind it to go stale: four operations on integers and a greatest common divisor that has been in Euclid’s Elements, Book VII, since roughly 300 BC. The standards above are cited for the conventions and for the floating-point behaviour this tool exists to sidestep, not for the calculation.