How fraction arithmetic works
A fraction is just a division waiting to happen: the numerator (top) counts the parts you have, the denominator (bottom) counts how many parts make a whole. Two fractions can only be added or subtracted once they are counted in the same size of part, which is why the four operations do not all work the same way.
Adding and subtracting
Multiply each fraction by whatever turns its denominator into a shared one, add or subtract the numerators, then simplify. The safest shared denominator is simply the product of the two you started with, so 1/2 + 1/3 becomes 3/6 + 2/6 by multiplying the first fraction by 3/3 and the second by 2/2, giving 5/6.
Subtraction works identically with a minus sign in place of the plus. 2 3/4 minus 1 5/6 first turns both into improper fractions, 11/4 and 11/6, then onto a common denominator of 12: 33/12 minus 22/12, leaving 11/12.
Multiplying
Multiplying needs no shared denominator at all. Multiply the two numerators together, multiply the two denominators together, then simplify what is left.
Dividing
Dividing by a fraction means multiplying by it turned upside down, a shortcut usually remembered as "keep, change, flip": keep the first fraction as it is, change the divide sign to a multiply, and flip the second fraction over.
Flipping a fraction whose numerator is zero produces a division by zero, which has no answer, so a divisor of zero always stops the calculator with an error rather than a number.
Choosing the right tab for your problem
Use "Add, subtract, multiply, divide" whenever you have two fractions and a job to do between them: doubling a recipe that calls for 3/4 of a cup, working out how many 2/3-metre lengths fit along a 5-metre length of timber, or checking a worksheet answer for a class. Use "Simplify a fraction" when a calculation elsewhere has already handed you an ugly fraction, such as a ratio from a spreadsheet or a probability written as a raw count over a raw total, and you want the smallest equivalent numbers to write down or compare against another fraction at a glance. Use "Decimal to fraction" when the number you are holding came from a measurement, a currency amount or a reading on a device, and you need the exact fraction it represents, for instance turning a 0.375-inch drill bit size into 3/8 for a parts list.
All three tabs share the same answer panel, so switching between them mid-problem, say simplifying an odd result from the first tab, carries no extra typing.
Mixed numbers, and what a minus sign means
A mixed number such as 2 3/4 is a whole number and a fraction written together, meaning 2 + 3/4. Before any arithmetic happens it gets converted to an improper fraction (top-heavy, with the numerator bigger than the denominator) by multiplying the whole number by the denominator and adding the numerator: 2×4 + 3 = 11, so 2 3/4 is 11/4.
Negative mixed numbers put the sign on the whole thing, not on one piece of it: −1 5/6 means −(1 + 5/6), which is −11/6, not −1 + 5/6. This calculator follows that convention, so enter the minus sign on the whole-number field (or on the numerator when there is no whole number, for a fraction like −1/2) and it will apply to the entire value.
Simplifying to lowest terms
A fraction is in its lowest terms when the numerator and denominator share no common factor bigger than 1. Finding that shared factor by eye gets hard fast with larger numbers, which is where the greatest common divisor (GCD) and Euclid's algorithm come in: divide the larger number by the smaller, keep the remainder, then repeat with the smaller number and that remainder until the remainder hits zero. Whatever you divided by last is the GCD.
Take 462/1071. 1071 divided by 462 leaves a remainder of 147; 462 divided by 147 leaves 21; 147 divided by 21 leaves 0. The last non-zero remainder, 21, is the GCD, so dividing top and bottom by 21 gives 22/51, and that fraction cannot be reduced any further.
Turning a decimal into a fraction
A decimal is already a fraction in disguise: the digits after the point sit over a power of ten equal to how many digits there are. 0.125 has three decimal digits, so it is 125 over 1000; simplifying that (the GCD of 125 and 1000 is 125) gives 1/8. The same method turns 0.6 into 6/10, then 3/5 once the common factor of 2 is divided out.
This only works exactly for decimals that terminate, meaning they stop after a finite number of digits. A recurring decimal such as 0.333... goes on forever and needs a different technique (subtracting a shifted version of itself to cancel the repeating part), so this tool sticks to terminating decimals and says so plainly rather than guessing at where a repeating pattern was meant to start.
Reading fraction, mixed number, decimal and percent together
Every answer here is shown four ways: as a simplified fraction, as a mixed number when it is top-heavy, as a decimal (exact when it terminates, shown with a bracketed repeating block when it does not), and as a percentage. They are the same value in four different outfits, and which one is useful depends on what you are about to do with it: a recipe wants the fraction, a spreadsheet wants the decimal, a grade or a discount wants the percent.
Questions people ask
How do you add fractions that have different denominators?
Give them a shared denominator first. Multiplying each fraction by the other one's denominator always works, even if it is not the smallest possible common denominator: 1/2 + 1/3 becomes 3/6 + 2/6 = 5/6. Simplifying at the end tidies up anything that a smaller common denominator would have avoided.
Why do you flip and multiply to divide fractions?
Dividing by a number and multiplying by its reciprocal are the same operation. Because the reciprocal of c/d is d/c, dividing by c/d is identical to multiplying by d/c, which turns an awkward division into a straightforward multiplication.
What does it mean to simplify a fraction?
It means dividing the numerator and denominator by their greatest common divisor so no smaller whole-number version of the same fraction exists. 462/1071 and 22/51 point to exactly the same place on the number line; simplifying just renames it with the smallest possible numbers.
How do I enter a mixed number like 2 3/4?
Put 2 in the whole-number field, 3 in the numerator and 4 in the denominator. Leave the whole-number field at 0 for a plain fraction with no whole part.
Can a fraction be negative, and how do I enter one?
Yes. Put the minus sign on the whole number for a negative mixed number (−1 5/6 means −(1 + 5/6)), or on the numerator when the whole-number field is 0, for a plain negative fraction such as −1/2.
Why does the decimal sometimes show dots and brackets?
Not every fraction converts to a decimal that stops. 5/6 is 0.8333 with the 3 repeating forever, which this calculator shows as 0.8(3...) rather than rounding it and pretending the digits end.
What is the difference between simplifying and converting to a decimal?
Simplifying keeps the answer as a fraction and only changes which numbers are used to write it, such as 22/51 instead of 462/1071. Converting to a decimal changes its form entirely, into tenths, hundredths and so on, which is convenient for typing into a spreadsheet but sometimes hides a repeating pattern a fraction states exactly.
Every result here is exact integer arithmetic; nothing is rounded until the decimal display asks for it. For classroom-style working of ratios rather than fractions, try the ratio calculator on our build list.