Fraction Calculator

What this calculator answers

It gives the exact answer to a fraction problem: the sum, difference, product or quotient of fractions, mixed numbers, whole numbers and decimals, in simplest form, as a mixed number and as a decimal with any repeating digits marked. Four more modes simplify one fraction, put a list of fractions in order from least to greatest, turn a decimal (repeating ones too) into a fraction, and turn a fraction into a decimal by long division. Nothing is rounded along the way, however long the numbers get.

How to use it

Pick a Calculation, type the numbers, and the answer updates as you type. Each number goes in one box, in any of these forms:

A line under each box says how it was read, for example “Read as 17/6” under 2 5/6, so a typing slip shows before you rely on the answer. To combine three numbers, fill in Third fraction and pick its operation. The Try buttons load examples worked through on this page, and each of the other modes opens on the example worked below for it. In Compare or order fractions, type the whole list in one box, separated by commas. Save for comparison keeps up to three answers side by side.

Worked example: 2 5/6 + 1 3/10

1. Write the mixed numbers as improper fractions: 2 5/6 = (2 × 6 + 5)/6 = 17/6 and 1 3/10 = (1 × 10 + 3)/10 = 13/10.
2. Find the least common denominator. From 6 = 2 × 3 and 10 = 2 × 5, take each prime factor the most times it appears in either: 2 × 3 × 5 = 30.
3. Rewrite both fractions over 30: 17/6 = 85/30 and 13/10 = 39/30.
4. Add the numerators and keep the denominator: (85 + 39)/30 = 124/30.
5. Simplify: 124 and 30 share only the factor 2, so 124/30 = 62/15.
6. As a mixed number, 62 ÷ 15 = 4 remainder 2, so the answer is 4 2/15.
7. As a decimal, 62 ÷ 15 = 4.1333…, where the 3 repeats forever: 4.1(3).

Multiplying the denominators instead (6 × 10 = 60) also works: 170/60 + 78/60 = 248/60, which needs dividing by 4 to reach the same 62/15. The least common denominator only saves that extra reducing.

Under the answer, a table repeats the calculation with the other three operations on the same two numbers: 2 5/6 − 1 3/10 = 23/15 (1 8/15), 2 5/6 × 1 3/10 = 221/60 (3 41/60) and 2 5/6 ÷ 1 3/10 = 85/39 (2 7/39).

How to add and subtract fractions with different denominators

Rewrite the fractions so they share a denominator, the least common multiple of the denominators, then add or subtract the numerators:

$$\frac{a}{b} \pm \frac{c}{d} = \frac{a \times (L \div b) \pm c \times (L \div d)}{L}, \qquad L = \operatorname{lcm}(b, d)$$

• $a$ and $c$ are the numerators, $b$ and $d$ the denominators (neither 0).
• $L$ is the least common denominator. Each fraction’s top and bottom are multiplied by the same number, $L \div b$ or $L \div d$, which does not change its value.

The denominators are never added: 1/2 + 1/3 is 3/6 + 2/6 = 5/6, not 2/5.

How to multiply and divide fractions

Multiply the numerators together and the denominators together. To divide, multiply by the reciprocal of the second fraction, the fraction turned upside down:

$$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \qquad \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$

For 3/4 ÷ 1 1/2, first write 1 1/2 as 3/2. Its reciprocal is 2/3, so the problem becomes 3/4 × 2/3 = (3 × 2)/(4 × 3). The 3 on top cancels the 3 below, and the 2 on top goes into the 4 below, leaving 1/2. The flip works because multiplying by 2/3 undoes multiplying by 3/2: their product is 1. Only the second fraction, the one you divide by, is flipped.

Calculating with mixed numbers

A mixed number is a whole number and a proper fraction, like 4 2/15. Change mixed numbers to improper fractions (top larger than bottom) before calculating: multiply the whole number by the denominator and add the numerator, so w n/d = (w × d + n)/d. To go back, divide: the quotient is the whole part and the remainder is the new numerator.

Mixed numbers can’t be multiplied part by part. 2 1/2 × 2 1/2 is not 4 1/4; as improper fractions it is 5/2 × 5/2 = 25/4 = 6 1/4.

Negative fractions and negative mixed numbers

A minus sign in front of a mixed number applies to the whole number: -2 1/3 means -(2 + 1/3) = -7/3, not -2 + 1/3 = -5/3. The minus of a plain fraction can sit on the numerator, the denominator or in front, since -3/4, (-3)/4 and 3/(-4) are equal; the calculator reads 3/-4 as -3/4 but asks for the sign at the front of a mixed number.

For −2 1/3 × 3/4: -7/3 × 3/4 = -(7 × 3)/(3 × 4), the 3s cancel, and the answer is -7/4 = -1 3/4 = -1.75. One negative number makes the answer negative; two make it positive.

Three numbers and the order of operations

Multiplication and division come before addition and subtraction; otherwise work from left to right. When all three numbers are added or subtracted, one common denominator serves them all: 1/2 + 1/3 − 1/4 = 6/12 + 4/12 − 3/12 = 7/12. With a × in the middle the order matters: 1/2 + 1/3 × 3/4 is 1/2 + 1/4 = 3/4, while (1/2 + 1/3) × 3/4 would be 5/8. The calculator doesn’t group with parentheses, so for that second meaning work out 1/2 + 1/3 first and use its answer.

How to simplify a fraction

Divide the numerator and the denominator by their greatest common factor. For 84/126, the prime factors are 84 = 2 × 2 × 3 × 7 and 126 = 2 × 3 × 3 × 7; they share 2 × 3 × 7 = 42, and 84 ÷ 42 = 2, 126 ÷ 42 = 3, so 84/126 = 2/3. Dividing by smaller common factors in turn (by 2, then 7, then 3) ends at the same 2/3. A fraction whose top and bottom share no factor but 1 is already in simplest form.

How to compare and order fractions

Rewrite the fractions over a common denominator and compare the numerators: once every part is the same size, more parts make a larger number. For 3/4, 5/8 and 2/3, the least common denominator of 4, 8 and 3 is 24, so they become 18/24, 15/24 and 16/24. From least to greatest that is 15/24 < 16/24 < 18/24, or 5/8 < 2/3 < 3/4. The decimals agree: 0.625 < 0.(6) < 0.75.

For two fractions, cross-multiplying is a shortcut: compare each numerator times the other denominator. For 5/8 and 2/3, 5 × 3 = 15 and 2 × 8 = 16, and 15 < 16, so 5/8 < 2/3. It is the same comparison made over the denominator 8 × 3 = 24.

Between negative numbers, the one closer to 0 is the larger: -1/2 < -1/3, because -3/6 < -2/6.

Converting a decimal to a fraction (including repeating decimals)

A decimal that ends is a whole number over a power of 10: 0.375 has three decimal places, so it is 375/1000, and dividing both by 125 gives 3/8.

A repeating decimal needs one more step. For S = 0.1666…, where only the 6 repeats:

1. The repeating block, 6, is 1 digit long, so multiply by 10: 10 × S = 1.666…
2. Subtract S. The endless 6s line up and cancel: 10 × S − S = 1.666… − 0.1666… = 1.5, so 9 × S = 1.5.
3. S = 1.5/9 = 15/90, which simplifies to 1/6.

Textbooks mark the repeating digits with a bar; here, put them in parentheses, 0.1(6), or type enough of them for the pattern to show, 0.1666…. An entry like 0.12… is ambiguous (does the 2 repeat, or the 12?), so the calculator asks. A decimal typed without either, such as 0.3333, is taken exactly as typed: 3333/10000.

For rulers and recipes, this mode also lists the nearest halves, thirds, quarters and so on down to sixty-fourths. 0.3 inch, for example, is closest to 5/16 inch among the sixteenths (0.3125, 0.0125 over) and to 19/64 among the sixty-fourths (0.296875, 0.003125 under).

Converting a fraction to a decimal

Divide the numerator by the denominator. If a remainder of 0 comes up, the decimal ends: 3/8 = 0.375. If a remainder comes back that appeared before, the same digits come around again forever. For 5/7 the remainders run 5, 1, 3, 2, 6, 4 and then 5 again, so 5/7 = 0.714285714285… = 0.(714285), which rounds to 0.7143 at four decimal places, or about 71.43% as a percent. The long-division table in this mode shows each step.

Reading the result

• The headline is the answer in simplest form: a fraction, or a decimal in the Fraction to decimal mode. Below it are the mixed number, the decimal and the percent. Digits in parentheses repeat forever; lines that say “to 4 decimal places” are rounded.
• In Compare or order fractions, the headline lists your numbers as typed, from least to greatest, joined by < (or = for equal numbers). The tiles give the largest, the smallest and the difference between them.
• Copy buttons copy the fraction, the mixed number or the full decimal on their own.
• How this was calculated shows every step with your numbers, drawn as stacked fractions.
• Fraction bars appear for adding, subtracting, simplifying, comparing and converting when the denominators are small (up to 24 parts per whole, or 16 when a number is above 2) and the numbers are between 0 and 3. Each bar is one whole cut into equal parts, so you can see that 1/2 = 6/12.
• The table depends on the mode: all four operations on your two numbers, your numbers in order over their common denominator, the nearest fractions of a decimal, or the long division of a fraction. Each can be downloaded as a CSV file.

Assumptions and limitations

• The arithmetic is exact. Each box holds up to 200 characters, and numbers that long stay exact.
• Up to three numbers in a calculation and 12 in a comparison, with no parentheses for grouping, powers or roots; for those, use a scientific calculator.
• In a comparison, commas separate the numbers, so type 1234 there rather than 1,234.
• Repeating digits are found when the first repeat ends within 1,000 decimal places; past that, the decimal is cut off with ”…”. The screen shows up to 40 decimal digits, and Copy decimal copies the rest.
• Rounded lines round a final 5 away from zero, so 0.125 to two places is 0.13.
• Fraction bars are not drawn for multiplication, division, negative numbers, large denominators or numbers above 3.

Common mistakes

• Adding the denominators. 1/2 + 1/3 is 5/6. Only the numerators are added, once the denominators match.
• Judging a fraction by its denominator. A larger denominator means smaller parts, so 1/8 is less than 1/6 (3/24 < 4/24).
• Reading -2 1/3 as -2 + 1/3. The minus sign applies to the whole mixed number, so it is -7/3, about -2.33, not -5/3.
• Multiplying mixed numbers part by part. Convert to improper fractions first: 2 1/2 × 2 1/2 = 6 1/4.
• Flipping the wrong fraction when dividing. Flip the one you divide by, the second, and keep the first as it is.
• Typing 0.33 for one third. 0.33 is exactly 33/100. Type 0.(3) for the repeating decimal that equals 1/3.
• Leaving the answer unsimplified. 124/30 and 62/15 are equal, but most answer keys expect the simplest form.