Fraction Calculator

A fraction calculator takes the busywork out of working with fractions: it finds the common denominator, does the arithmetic, reduces the answer, and shows it three ways at once - as a simplified fraction, as a mixed number, and as a decimal. Type 1 1/2 + 2/3 and you instantly get 13/6, the mixed number 2 1/6, and the decimal 2.166667, plus the steps so you can see exactly how the answer was reached. It handles two or three fractions, mixed numbers, and negatives.

What a fraction is

A fraction represents a part of a whole, written as a numerator over a denominator. The denominator says how many equal pieces the whole is divided into; the numerator says how many of those pieces you have. In 3/4, the whole is split into 4 parts and you have 3 of them. When the numerator is smaller than the denominator it is a proper fraction; when it is equal or larger it is an improper fraction (like 7/4) that can also be written as a mixed number (1 3/4).

The four operations, as formulas

Every operation in this calculator follows one of these four rules, where a/b and c/d are the two fractions:

After applying the right rule, you reduce the result by dividing the numerator and denominator by their greatest common divisor (GCD). That is the whole method in four lines.

A worked example, step by step

Take 1 1/2 + 2/3:

1. Convert the mixed number: 1 1/2 = (1×2 + 1)/2 = 3/2.
2. Find a common denominator: the denominators are 2 and 3, so use 6. Rewrite as 9/6 + 4/6.
3. Add the numerators: 9 + 4 = 13, giving 13/6.
4. Simplify: the GCD of 13 and 6 is 1, so 13/6 is already in lowest terms.
5. Write it other ways: 13/6 = 2 1/6 as a mixed number and about 2.166667 as a decimal.

How to use this fraction calculator

1. Enter Fraction A: type the numerator and denominator in the stacked boxes. For a mixed number such as 1 1/2, put the 1 in the left whole-number box.
2. Pick the operation: tap +, −, ×, or ÷ between the fractions.
3. Enter Fraction B the same way.
4. Optional - add a third fraction: click "Add a third fraction" to chain a second operation (the operations are applied left to right).
5. Read the result: the big number is the simplified fraction; below it you also get the mixed number and the decimal, and a step-by-step card shows the work.

Who this calculator is for

• Students checking homework or learning the steps for adding, subtracting, multiplying and dividing fractions.
• Parents and tutors who want to verify an answer and explain the method clearly.
• Cooks and bakers scaling a recipe up or down (doubling 3/4 cup, halving 1 1/3 cups).
• DIY and trades adding up fractional inch measurements like 5/8 + 3/16.
• Anyone who needs a fast, exact answer with the mixed-number and decimal forms side by side.

Key fraction terms explained

• Numerator: the top number - how many parts you have.
• Denominator: the bottom number - how many equal parts make a whole. It can never be zero.
• Proper fraction: numerator smaller than denominator (e.g. 3/4); its value is less than 1.
• Improper fraction: numerator equal to or larger than denominator (e.g. 7/4); its value is 1 or more.
• Mixed number: a whole number plus a proper fraction (e.g. 1 3/4).
• Common denominator: a shared bottom number that lets you add or subtract fractions.
• Reciprocal: a fraction flipped upside down (the reciprocal of 3/4 is 4/3); dividing by a fraction means multiplying by its reciprocal.
• Greatest common divisor (GCD): the largest number that divides both the numerator and denominator; used to simplify. It is the same idea as the greatest common factor (GCF).
• Least common denominator (LCD): the smallest number that all the denominators divide into evenly - it is the least common multiple (LCM) of the denominators, and using it keeps the numbers smaller than the simple product.

More worked examples

Subtraction with unlike denominators: 3/4 − 1/6. The common denominator is 12, so this becomes 9/12 − 2/12 = 7/12. The GCD of 7 and 12 is 1, so the answer is 7/12 (about 0.583333).

Multiplication: 2/3 × 3/5. Multiply across the top (2×3 = 6) and the bottom (3×5 = 15) to get 6/15, then divide both by the GCD 3 to get 2/5 (0.4). Notice you never needed a common denominator.

Division of mixed numbers: 2 1/2 ÷ 1 1/4. Convert to improper fractions (5/2 ÷ 5/4), flip the second and multiply (5/2 × 4/5 = 20/10), then simplify to 2. The answer is a whole number, which the calculator shows as 2 with no fraction part.

Quick reference: common fractions, decimals and percents

These conversions come up constantly. The decimal is just the numerator divided by the denominator; the percent is the decimal times 100.

Tips for working with fractions

• Simplify early when multiplying: you can cancel a common factor between any top and any bottom before multiplying, which keeps the numbers small.
• The product of the denominators always works as a common denominator for adding and subtracting - it may not be the smallest, but simplifying at the end fixes that.
• "Keep, change, flip" is the memory trick for division: keep the first fraction, change ÷ to ×, and flip the second.
• Turn mixed numbers improper first. Trying to add the whole parts and fraction parts separately is where most mistakes creep in.
• Check with the decimal. If the decimal looks wrong (for example, larger than 1 when you expected a small answer), recheck your inputs.

Why two or three fractions at once

Real problems rarely stop at two fractions. A recipe might call for 1/2 cup, 1/3 cup, and 1/4 cup of three ingredients and you want the total; a board might be cut into 3/8, 1/4, and 1/8 sections. With three fractions enabled, this calculator applies the operations left to right - so 1/2 + 1/3 + 1/4 is computed as (1/2 + 1/3) + 1/4 = 5/6 + 1/4 = 13/12, which is 1 1/12. If you mix operations, remember the tool does not apply "multiply before add" precedence; it works strictly from left to right, so plan your order or use two steps if you need different grouping.

Fractions in everyday life

Outside the classroom, fractions are everywhere, and the same four operations cover almost all of it:

• Cooking: halving 3/4 cup means 3/4 × 1/2 = 3/8 cup. Doubling 2/3 cup means 2/3 × 2 = 4/3 = 1 1/3 cups.
• Construction and woodworking: a tape measure is marked in sixteenths of an inch, so adding 5/8 + 3/16 (= 10/16 + 3/16 = 13/16 in) is a daily task.
• Money and time: a quarter hour is 1/4 of 60 minutes; three quarters of an hour is 3/4 × 60 = 45 minutes.
• Sharing: splitting 3/4 of a pizza among 3 people is 3/4 ÷ 3 = 1/4 each.

Inch fractions and their decimal equivalents

For measuring work, here are the common tape-measure fractions of an inch and their decimal values - handy when a tool or plan is marked in decimals instead of fractions:

Finding the least common denominator (LCD)

The fastest common denominator to use is just the product of the two denominators, and because this calculator simplifies at the end you always reach the same reduced answer. But when you are working by hand, the least common denominator keeps the numbers small and the arithmetic easier. The LCD is the least common multiple of the denominators - the smallest number they all divide into. For example, to add 1/4 + 1/6, the product gives 24, but the LCD is only 12, so 3/12 + 2/12 = 5/12 is tidier than 6/24 + 4/24 = 10/24. A quick way to find the LCD of two denominators is to multiply them and then divide by their GCD: for 4 and 6, that is 24 ÷ 2 = 12. With three fractions, find the LCM of all three denominators in one step. Using the LCD is optional here - the calculator handles either path and reduces the result regardless - but it is a good habit for mental math and for showing your work in class.

Equivalent fractions: the same value, different numbers

Two fractions are equivalent when they describe the identical part of a whole, even though the numbers look different. You create an equivalent fraction by multiplying (or dividing) the numerator and denominator by the same non-zero number. So 1/2 = 2/4 = 3/6 = 50/100, because each is exactly half. This single rule underpins everything else on this page: finding a common denominator is just rewriting each fraction as an equivalent one with a matching bottom, and simplifying is the reverse - dividing top and bottom by their GCD to reach the smallest equivalent form. Recognising equivalents also speeds up cancellation: in 6/9 you can spot the shared factor of 3 and write 2/3 instantly. When you compare two fractions, the safest method is to convert both to a common denominator (or to decimals) and then compare the numerators - 5/8 versus 7/12 becomes 15/24 versus 14/24, so 5/8 is the larger of the two.

Converting between fractions, decimals, and percents

Because a fraction is just an unfinished division, you can move between all three forms with two simple operations. To go from a fraction to a decimal, divide the numerator by the denominator (3/8 = 3 ÷ 8 = 0.375); long division by hand is the manual version of what the long division calculator does automatically. To go from a decimal to a fraction, write the digits over the matching power of ten and simplify (0.375 = 375/1000 = 3/8), or use the dedicated decimal to fraction calculator. To reach a percent, take the decimal and multiply by 100 (0.375 × 100 = 37.5%); for the reverse, divide the percent by 100 and simplify. The percentage calculator and percentage change calculator pick up where this tool leaves off when the question is framed in percents rather than fractions. Note that not every fraction gives a clean decimal: 1/3 produces a repeating 0.3333..., and this calculator rounds such non-terminating values to six places while the fraction itself stays exact.

Comparing and ordering fractions

Deciding whether 3/5 is bigger than 4/7 is a common task, and there are three reliable ways to do it. The first is the common-denominator method: rewrite both with the same bottom (3/5 = 21/35 and 4/7 = 20/35), then compare numerators, so 3/5 wins. The second is cross-multiplication: multiply the numerator of each by the other's denominator (3 × 7 = 21 against 4 × 5 = 20), and the larger product marks the larger fraction - the same answer with less writing. The third is simply to convert each to a decimal (0.6 versus about 0.571) and compare, which is what most people do when a calculator is handy. To order a whole list of fractions, the decimal route is usually fastest; to prove it on paper, the common-denominator route is clearest. All three agree, so pick whichever fits the situation.

Fractions of a number and "fraction word problems"

Many real questions are phrased as "what is 3/4 of 60?" or "if 2/5 of the class are boys, how many is that?" The rule is the same in every case: "of" means multiply. So 3/4 of 60 is 3/4 × 60 = 180/4 = 45, and 2/5 of a 30-student class is 2/5 × 30 = 12. Going the other way - "12 is 2/5 of what number?" - you divide by the fraction: 12 ÷ 2/5 = 12 × 5/2 = 30. These two moves, multiplying by a fraction and dividing by a fraction, cover the great majority of fraction word problems, from discounts and tips to recipe scaling and probability. When the answer needs to be a percentage instead, finish by turning the fraction into a decimal and multiplying by 100, or hand the final step to the percentage calculator.

A short history and why fractions still matter

Fractions are one of the oldest pieces of mathematics in continuous use. The ancient Egyptians worked almost entirely in unit fractions - fractions with a numerator of 1, like 1/2, 1/3, and 1/4 - and wrote every other quantity as a sum of distinct unit fractions, a system preserved in the Rhind Mathematical Papyrus. The Babylonians, by contrast, used a base-60 system whose legacy survives in the 60 minutes of an hour and the 360 degrees of a circle. The horizontal fraction bar we use today was popularised by medieval Arab mathematicians and reached Europe through Latin translations. Despite the rise of decimals and calculators, fractions remain indispensable wherever an exact value matters more than a rounded one: 1/3 is precise, while 0.333 is not. That exactness is why engineers, machinists, chemists, and musicians still reason in fractions, and why a tool that keeps the exact fraction alongside the decimal is more useful than one that quietly rounds everything away.

Accuracy, rounding, and what "exact" means here

This calculator keeps every intermediate result as an exact integer fraction, so there is no accumulated rounding error in the arithmetic itself - the simplified fraction it returns is mathematically exact. The only place rounding appears is the decimal display, which is capped at six places for readability. For terminating decimals (like 0.625) those six places are exact; for repeating decimals (like 0.333333 for 1/3, or 0.142857 for 1/7) the shown decimal is a rounded approximation, and the fraction form is the true value. If you need more decimal places than the display shows, treat the fraction as the source of truth and divide it out yourself, or feed it into a scientific calculator. The tool also guards the two situations that make fractions undefined: a denominator of zero is rejected, and dividing by a fraction that equals zero is blocked, because both would imply splitting a whole into "no parts."

Related concepts and calculators

Fractions, decimals, and percents are three ways of writing the same idea, so several tools overlap with this one:

• To turn a fraction into a percentage or solve "what is X% of Y," use the Percentage Calculator.
• To simplify, scale, or solve a proportion like 2:3 = x:12, use the Ratio Calculator.
• For very large or very small numbers in scientific notation, use the Scientific Calculator.
• To average a set of numbers (which often produces a fraction), use the Average Calculator.
• To find a square root, which may be irrational rather than a neat fraction, use the Square Root Calculator.
• To convert a decimal like 0.625 back into an exact fraction, use the Decimal to Fraction Calculator.