Fraction Calculator
Two fractions, one operation, and the answer in lowest terms. Kept exact throughout, because a third has no decimal representation and three lots of 0.333 is not one.
Answer
5/6
- Simplified
- 5/6
- Mixed number
- —
- Decimal
- 0.833333
- Percentage
- 83.33%
- Adding and subtracting need a common denominator, which is why the working goes through the product of the two before simplifying. Multiplying and dividing do not.
- Dividing by a fraction means multiplying by its reciprocal — turn the second fraction upside down and multiply. That is why dividing by a half doubles a number.
- Kept in fractions rather than decimals because a third has no decimal representation. Three thirds is exactly one; three lots of 0.333 is not.
About the fraction calculator
Adding and subtracting fractions needs a common denominator, which is the step people skip. A half plus a third is not two fifths — the halves and thirds are different sized pieces and cannot be counted together until they are made the same size. Rewritten over six, they become three sixths plus two sixths, which is five sixths.
Multiplying and dividing need no common denominator at all, which surprises people who have just learned to find one. Multiply straight across the top and the bottom. To divide, turn the second fraction upside down and multiply — which is why dividing by a half doubles a number rather than halving it.
The answer always comes back in lowest terms. Two quarters is one half; both are correct and only one is finished. Reducing means dividing top and bottom by their greatest common divisor, which is the same operation the GCF calculator does.
Working in fractions rather than decimals matters when the values do not terminate. A third is 0.333 recurring, so three of them is 0.999 recurring in decimal and exactly one in fractions. For anything that has to be exact — a recipe scaled up, a measure divided into parts — the fraction is the honest form.
What it works out
- Adding, subtracting, multiplying and dividing
- The answer in lowest terms
- Mixed number and decimal forms
- Improper fractions and negatives
The formula
a/b + c/d = (ad + cb) / bd a/b ÷ c/d = ad / bc
Adding: multiply each numerator by the other denominator, add them, and put the result over the product of the denominators. One half plus one third is (1×3 + 1×2) over (2×3), which is 5/6.
That gives a common denominator every time, though not always the lowest one. A quarter plus a quarter comes out as 8/16 by this route and reduces to 1/2, which is why the simplification step exists.
Multiplying is straight across: numerators together, denominators together. Dividing is the same thing after inverting the second fraction, because dividing by a number is multiplying by its reciprocal.
The reduction at the end divides both parts by their greatest common divisor. Euclid's algorithm finds it in a handful of steps regardless of how large the numbers are.
- Numerator
- The top number — how many pieces.
- Denominator
- The bottom number — how many pieces make a whole. Never zero.
- Reciprocal
- A fraction turned upside down. Dividing by a fraction multiplies by this.
A worked example
A half plus a third.
That works out to 5/6 .
- Simplified
- 5/6
- Mixed number
- —
- Decimal
- 0.833333
- Percentage
- 83.33%
Questions
How do I add fractions with different denominators?
Rewrite both over a common denominator first. A half plus a third becomes three sixths plus two sixths, which is five sixths. You cannot add halves to thirds any more than you can add metres to pounds.
Do I need a common denominator to multiply?
No. Multiply straight across — numerators together, denominators together. The common denominator is only needed for adding and subtracting.
How do I divide by a fraction?
Turn the second fraction upside down and multiply. A half divided by a quarter is a half times four, which is two — dividing by a fraction below one makes a number bigger.
What is a mixed number?
A whole number with a fraction beside it — 2⅓ rather than 7/3. Both are the same value; the mixed form is easier to picture and the improper one is easier to calculate with.
How do I simplify a fraction?
Divide the top and bottom by their greatest common divisor. For 8/12 that is 4, giving 2/3. This calculator always does it, so the answer is always in lowest terms.
Why not just use decimals?
Because some fractions do not terminate. A third is 0.333 recurring, so three of them is 0.999 recurring rather than exactly one. Where the answer has to be exact, fractions are the honest form.
Can the denominator be zero?
No. A denominator says how many pieces make a whole, and zero pieces cannot make anything. Division by zero is undefined and the calculator will say so.