Why use this calculator
Fractions are the arithmetic of anything that gets cut, shared or measured in parts, and they are the one topic where mental arithmetic reliably fails. The trap is that fractions do not add the way they multiply: you can multiply the tops and bottoms straight across, but you cannot add them that way, and 1/2 plus 1/3 is emphatically not 2/5.
This tool does the common-denominator step for you, simplifies the answer by its greatest common divisor, and then shows the same value three more ways so you can sanity-check it. Seeing 5/4 alongside 1.25 and 125 per cent makes an error obvious in a way that a bare fraction does not, which matters when the number is about to be cut into timber, doubled into a recipe, or marked in an exam.
How to use it
- Enter the top number of your first fraction in Numerator 1; if you are working from a measurement such as three-quarters of an inch, that is the 3.
- Enter the bottom number in Denominator 1, which is the number of equal parts the whole is divided into, so the 4 in three-quarters.
- Choose the operation you need from the dropdown, remembering that word problems asking for a fraction of something mean multiply, not add.
- Enter the second fraction the same way in Numerator 2 and Denominator 2; for a whole number such as 3, enter 3 over 1.
- Negative fractions are entered with a minus sign on the numerator.
- The result appears simplified, as a mixed number, and as a decimal and percentage.
What each result means
- Result
- The answer as a fraction in lowest terms, meaning no whole number other than one divides both the top and the bottom.
- Mixed number
- The same value split into a whole-number part and a proper fraction, which is how measurements are usually spoken and written on a plan.
- Decimal
- The fraction as a decimal to six places. Repeating decimals such as one third are shown truncated, not rounded up to a false exactness.
- Percentage
- The same value expressed as parts per hundred, which is what a spreadsheet or a discount sign would show for the identical quantity.
- Unsimplified
- The raw answer before cancelling, so you can check the common-denominator step in your own working line by line.
- Working
- The substitution actually performed, with your numbers in place of the letters, so the method can be copied into homework.
The formula
Addition
a/b + c/d = (a×d + c×b) ÷ (b×d)
Subtraction
a/b − c/d = (a×d − c×b) ÷ (b×d)
Multiplication
a/b × c/d = (a×c) ÷ (b×d)
Division
a/b ÷ c/d = (a×d) ÷ (b×c), which is multiplying by the reciprocal of the second fraction
The answer is then simplified by dividing the numerator and denominator by their greatest common divisor, found with the Euclidean algorithm.
How this number is used in practice
Fractions survive in the trades because imperial measurement is built on repeated halving. A tape measure is marked in halves, quarters, eighths, sixteenths and thirty-seconds, so a carpenter subtracting a 3/8 inch kerf from a 5 3/4 inch offcut is doing fraction arithmetic whether or not they name it that way.
The same applies to plumbing pipe sized in fractions of an inch and to sheet material sold in nominal fractional thicknesses. Converting to decimal too early is the classic mistake: 1/3 becomes 0.333, the rounding error is multiplied by a rafter count, and the last member ends up short.
In kitchens the arithmetic runs the other way. Scaling a recipe by two thirds, or halving one that calls for 3/4 cup, produces fractions with awkward denominators that no measuring cup carries. Converting the answer to a decimal and then to grams is usually the practical route, which is why this calculator shows the decimal alongside the fraction.
In the classroom fractions are the foundation that ratio, algebra and probability are built on. A student who cannot add 1/2 and 1/3 will not be able to combine algebraic fractions, simplify a rate, or handle a probability tree, because all three use the same common-denominator move. Examiners consistently report it as the single most common source of lost marks in lower secondary arithmetic.
Worked examples
3/4 + 1/2: adding two fractions with different denominators
5/4, which is the mixed number 1 1/4, or 1.25 as a decimal.
2/3 × 3/5: two thirds of three fifths
2/5, or 0.40. The unsimplified answer 6/15 cancels by a factor of 3.
7/8 − 1/3: trimming a third off a seven-eighths measurement
13/24, or 0.541667, which is about 13.76 mm of one inch.
1/2 ÷ 3/4: how many three-quarter portions fit in a half
2/3, or 0.666667. Dividing by a fraction smaller than one gives a larger answer.
When people use it
- Adding and subtracting tape-measure fractions when cutting timber, pipe or sheet material
- Scaling a recipe up or down by a fractional factor without drifting off the ratio
- Checking homework and exam working where the marked answer must be in lowest terms
- Converting fractional drill, bolt and fastener sizes to decimals or millimetres
- Combining fractional shares in a split bill, a partnership or an estate distribution
Tips
- Multiply straight across, but never add straight across. Addition needs a common denominator first.
- To divide, flip the second fraction and multiply. Dividing by a fraction below one always makes the answer bigger.
- Convert a mixed number to an improper fraction before calculating: 2 1/4 becomes 9/4, not 21/4.
- Simplify by the greatest common divisor in one step rather than cancelling repeatedly, which is where sign errors creep in.
- Keep the value as a fraction until the very last step. Rounding 1/3 to 0.33 early and then multiplying by 30 loses a tenth of a unit.
- A negative fraction carries one minus sign only. Writing minus signs on both the top and the bottom makes it positive again.
What it assumes, and where it stops being accurate
- Numerators and denominators are treated as whole numbers; decimal entries are rounded to the nearest integer before calculating.
- Values are capped at plus or minus one million each so a shared link cannot request an unbounded simplification.
- The decimal is truncated for display at six places, so a repeating value such as 1/3 is shown as 0.333333 and is not exact.
- Only two fractions are combined at a time; for a longer chain, feed each result back in as the first fraction.
- This is exact rational arithmetic, not a model of measurement tolerance. It will not tell you whether a cut is achievable to that precision.
- Mixed-number entry is not supported directly; convert to an improper fraction first.
Terms used on this page
- Numerator
- The top number of a fraction: how many of the equal parts you have.
- Denominator
- The bottom number: how many equal parts the whole has been divided into.
- Greatest common divisor
- The largest whole number that divides two integers exactly; dividing by it puts a fraction in lowest terms.
- Reciprocal
- A fraction turned upside down. Dividing by a fraction is the same as multiplying by its reciprocal.
- Improper fraction
- A fraction whose numerator is at least as large as its denominator, such as 7/4.
Questions people ask
- Why can I not just add the tops and the bottoms?
- Because the denominator names the size of the pieces, not a quantity of them. Adding 1/2 and 1/3 as 2/5 would claim that one half plus one third is less than one half, which is plainly false. You must first rewrite both fractions with the same denominator so the pieces are the same size, then add only the counts on top.
- How do I simplify a fraction to lowest terms?
- Find the greatest common divisor of the numerator and denominator, then divide both by it. For 18/24 the greatest common divisor is 6, giving 3/4. The Euclidean algorithm finds it quickly: divide the larger by the smaller, keep the remainder, and repeat until the remainder is zero. The last non-zero remainder is the divisor.
- What is the difference between an improper fraction and a mixed number?
- An improper fraction has a numerator at least as large as its denominator, such as 5/4. A mixed number splits the same value into a whole part and a proper fraction, so 5/4 becomes 1 1/4. They are the same quantity. Calculations are easier in improper form, while measurements are usually read out as mixed numbers.
- Why does dividing by a fraction make the answer bigger?
- Division asks how many of the second quantity fit inside the first. If the second quantity is smaller than one, many of them fit, so the answer exceeds the starting value. Half divided by three quarters is two thirds because a three-quarter portion is larger than the half you are dividing, so less than one of them fits.
- Can a denominator ever be zero?
- No. The denominator states how many equal parts the whole is cut into, and cutting something into zero parts is undefined rather than infinite. The calculator refuses the input instead of returning a symbol, because any answer it produced would be meaningless and would propagate silently through the rest of your working.
- Should I work in fractions or decimals on site?
- Work in whichever unit your tape measure and your plan use, and convert only once at the end. Converting a repeating fraction such as one third to a decimal early introduces a rounding error that multiplies with every repeated member, which is how a run of rafters ends up several millimetres out by the last one.
Use this with
Fraction Calculator answers one part of the question. These pick up where it stops: Percentage Calculator, Ratio Calculator, Recipe Scaler and Quadratic Equation Solver.
Where these figures come from
This calculation is jurisdiction-neutral arithmetic. No country-specific rules are applied.
- Khan Academy: Mathematics. Free worked lessons and practice sets for each topic
- NRICH: University of Cambridge. University-produced problems and explanations for school and competition mathematics
- Wolfram MathWorld. Reference definitions, derivations and special cases for standard formulas
Fraction Calculator produces an estimate for general information. It is not financial, tax, medical, legal or engineering advice. read the full disclaimer. Spotted a wrong number? Report it and it gets fixed. The form picks up this calculator and the inputs you used.