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Add, subtract, multiply, and divide fractions in seconds with our free fraction calculator. Answers are automatically simplified to lowest terms, with decimal equivalents shown. Perfect for homework, cooking, and construction math.
Our free Fraction Calculator adds, subtracts, multiplies, and divides any two fractions in seconds. Every answer is automatically reduced to lowest terms, converted to a mixed number whenever the result is an improper fraction, and displayed with its decimal equivalent - so you receive the complete answer, not just a raw fraction.
Tips for best results: a denominator can never be zero - a fraction with a denominator of zero is mathematically undefined, and the calculator will ask you to correct it. You can work with negative fractions by typing a minus sign in front of the numerator. For homework problems involving mixed numbers such as 2 1/3, first convert them to improper fractions (2 1/3 = 7/3) and then enter 7 and 3 into the calculator. When you chain several operations together, work from left to right and simplify only at the end to keep intermediate values exact.
A fraction is a number that represents a part of a whole. It is written as two numbers separated by a fraction bar: the numerator on top tells you how many parts you have, and the denominator on the bottom tells you how many equal parts the whole has been divided into. In the fraction 3/4, the numerator 3 means you hold three parts of a whole that is split into four equal parts - picture three slices of a pizza that was cut into four slices.
Fractions come in several important forms. A proper fraction has a numerator smaller than its denominator (for example, 2/5); its value is always less than one. An improper fraction has a numerator equal to or larger than its denominator (for example, 9/4); its value is one or more. A mixed number combines a whole number with a proper fraction (for example, 2 1/4) and is simply another way of writing an improper fraction - 9/4 and 2 1/4 are exactly the same value, since 2 1/4 means 2 + 1/4 = 8/4 + 1/4 = 9/4. To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator: 2 1/3 becomes (2 x 3 + 1)/3 = 7/3. To go the other direction, divide the numerator by the denominator; the quotient becomes the whole number and the remainder becomes the new numerator, so 11/4 = 2 3/4.
Two fractions are called equivalent fractions when they represent the same value even though they look different: 1/2, 2/4, 3/6, and 50/100 are all equivalent because each equals one half. Multiplying or dividing both the numerator and the denominator by the same nonzero number always produces an equivalent fraction. A fraction is in lowest terms (also called simplest form) when the numerator and denominator share no common factor other than 1 - for example, 3/4 is in lowest terms, while 6/8 is not, because both 6 and 8 are divisible by 2. Reducing a fraction to lowest terms is done by dividing the top and bottom by their greatest common divisor (GCD), the largest number that divides both evenly. For 18/24, the GCD of 18 and 24 is 6, so 18/24 = (18 / 6)/(24 / 6) = 3/4. Teachers and exams almost always require answers in lowest terms, which is why our calculator simplifies every result automatically.
Fractions connect directly to decimals and percentages, two other ways of expressing parts of a whole. To convert any fraction to a decimal, divide the numerator by the denominator: 5/8 = 5 / 8 = 0.625. To turn that decimal into a percentage, multiply by 100: 0.625 = 62.5%. Fractions are exact where decimals sometimes are not - 1/3 is exactly one third, while 0.333333 is only an approximation - which is why carpenters, cooks, and mathematicians keep fractions in their toolkit. You meet fractions constantly in real life: half a cup of flour, a quarter of an hour, three eighths of an inch on a tape measure, or a 15% discount expressed as the fraction 3/20 of the original price. Understanding them deeply makes everyday arithmetic faster and far more reliable.
Fraction arithmetic follows a small set of reliable rules. Once you learn them, any problem with two fractions becomes a short sequence of mechanical steps.
Addition and subtraction require a common denominator. When the denominators already match, simply add or subtract the numerators and keep the denominator: 3/8 + 1/8 = 4/8 = 1/2. When they differ, convert both fractions to equivalent fractions with the same denominator - the easiest choice is the product of the two denominators, though the least common denominator (LCD) keeps numbers smaller. The general formula is a/b + c/d = (ad + bc)/bd, and a/b - c/d = (ad - bc)/bd.
Multiplication is the simplest operation: multiply straight across. The formula is a/b x c/d = (a x c)/(b x d). No common denominator is needed. Division uses the famous keep-change-flip rule: keep the first fraction, change division to multiplication, and flip (take the reciprocal of) the second fraction. The formula is a/b divided by c/d = a/b x d/c = (a x d)/(b x c). After any operation, simplify the result by dividing the numerator and denominator by their GCD.
Worked Example 1 - Addition: Add 3/4 + 2/5. The denominators 4 and 5 differ, so convert both to twentieths: 3/4 = (3 x 5)/(4 x 5) = 15/20, and 2/5 = (2 x 4)/(5 x 4) = 8/20. Now add the numerators: 15/20 + 8/20 = 23/20. This is an improper fraction, so convert it to a mixed number: 23 / 20 = 1 with a remainder of 3, giving 1 3/20. As a check, 23/20 = 1.15, and indeed 0.75 + 0.40 = 1.15.
Worked Example 2 - Division: Divide 5/6 by 2/3. Keep the first fraction, change division to multiplication, and flip the second: 5/6 x 3/2. Multiply straight across: (5 x 3)/(6 x 2) = 15/12. Simplify by dividing top and bottom by their GCD, 3: 15/12 = 5/4. Convert to a mixed number: 5 / 4 = 1 remainder 1, giving 1 1/4. Checking with decimals: 0.8333 / 0.6667 is approximately 1.25, which matches.
Two more quick illustrations show the remaining operations. Subtraction: 7/8 - 1/4 = 7/8 - 2/8 = 5/8. Multiplication: 2/3 x 9/10 = 18/30 = 3/5 after dividing by the GCD of 6. Notice that in multiplication you may cancel common factors before multiplying to keep the numbers small: in 2/3 x 9/10, the 2 and 10 share a factor of 2, and the 3 and 9 share a factor of 3, so the product collapses directly to 3/5.
Imagine your favorite chocolate-chip cookie recipe calls for 3/4 cup of sugar, 2 1/4 cups of flour, and 1/3 cup of cocoa powder, but you need a double batch for a school bake sale. Multiplying each ingredient by 2 is straightforward fraction arithmetic, and getting it right is the difference between perfect cookies and a baking disaster.
Start with the sugar: 3/4 x 2 = 3/4 x 2/1 = 6/4 = 3/2 = 1 1/2 cups. The flour is a mixed number, so convert it first: 2 1/4 = 9/4. Then 9/4 x 2 = 18/4 = 9/2 = 4 1/2 cups. The cocoa: 1/3 x 2 = 2/3 cup. In under a minute you have the scaled recipe: 1 1/2 cups of sugar, 4 1/2 cups of flour, and 2/3 cup of cocoa.
Scaling down works the same way. Suppose instead you want only half the original batch. Halving 3/4 cup of sugar gives 3/4 x 1/2 = 3/8 cup - and since 1 cup equals 16 tablespoons, 3/8 cup is exactly 6 tablespoons, a measurement your tablespoon can handle directly. Halving the 2 1/4 cups of flour: 9/4 x 1/2 = 9/8 = 1 1/8 cups. These are exactly the kinds of calculations our Fraction Calculator performs instantly, eliminating the mental gymnastics when you are mid-recipe with floury hands.
Carpenters live in fractions - tape measures are marked in sixteenths of an inch, and every cut depends on adding and subtracting them correctly. Suppose you are building a bookshelf and need to know the total length of two boards you plan to join: one measures 7 1/2 inches and the other measures 5 3/4 inches.
Convert both mixed numbers to improper fractions: 7 1/2 = 15/2 and 5 3/4 = 23/4. Find a common denominator (4 works): 15/2 = 30/4. Add: 30/4 + 23/4 = 53/4. Convert back to a mixed number: 53 / 4 = 13 remainder 1, so the combined length is 13 1/4 inches. A quick decimal check confirms it: 7.5 + 5.75 = 13.25.
Here is a division scenario from the same workshop. You have an 8-foot board and want to cut it into shelves that are each 2 1/2 feet long. How many full shelves can you get? Convert 2 1/2 to 5/2, then divide: 8 / 5/2 = 8/1 x 2/5 = 16/5 = 3 1/5. You can cut 3 full shelves, with a leftover piece of 8 - 3 x 2.5 = 0.5 feet (6 inches). Fraction division told you both the count and, with one subtraction, the exact waste - the kind of practical math that saves real lumber and real money.
1. Adding or subtracting the denominators. The single most common error is computing 1/2 + 1/3 as 2/5. Denominators describe the size of the pieces, not a quantity you combine - halves and thirds are different-sized pieces and cannot be added directly. Always convert to a common denominator first: 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
2. Forgetting to simplify the final answer. Leaving an answer as 4/8 instead of 1/2, or as 12/9 instead of 4/3, will cost marks on homework and exams even though the value is correct. Make simplifying with the GCD the automatic last step of every problem - or let our calculator do it for you.
3. Flipping the wrong fraction when dividing. When dividing a/b by c/d, only the second fraction (the divisor) gets flipped: a/b x d/c. Flipping the first fraction instead gives a completely different and wrong answer. Remember keep-change-flip in that exact order: keep the first, change the sign, flip the second.
4. Mishandling mixed numbers. Two sub-mistakes hide here. First, students sometimes add mixed numbers by adding the whole parts and fraction parts without finding a common denominator for the fraction parts. Second, some treat 2 1/2 as 2 x 1/2 - but a mixed number means addition (2 + 1/2 = 5/2), not multiplication. Always convert mixed numbers to improper fractions before multiplying or dividing.
5. Finding a common denominator for multiplication. This is the mirror image of mistake 1: students carefully convert 2/3 x 3/4 to twelfths before multiplying. Multiplication never needs a common denominator - just multiply straight across: (2 x 3)/(3 x 4) = 6/12 = 1/2, or cancel common factors first for smaller numbers.
Fractions rarely travel alone. Continue your math with these free tools from Sigma Calculator: our Percentage Calculator converts fractions to percentages and solves percent-change problems; the Scientific Calculator handles complex expressions, powers, and roots; the Volume Calculator computes cubic measurements for boxes, cylinders, and spheres; the BMR Calculator estimates your daily calorie needs; and the Tip Calculator splits restaurant bills fairly in seconds.
Convert both fractions to equivalent fractions that share a common denominator, then add the numerators. For 3/4 + 2/5, use 20 as the common denominator: 15/20 + 8/20 = 23/20 = 1 3/20. The formula is a/b + c/d = (ad + bc)/bd.
Use the keep-change-flip rule: keep the first fraction, change division to multiplication, and flip the second fraction (take its reciprocal). So 5/6 divided by 2/3 becomes 5/6 x 3/2 = 15/12 = 5/4 = 1 1/4.
A mixed number pairs a whole number with a proper fraction, like 2 1/3. To convert it, multiply the whole number by the denominator and add the numerator: (2 x 3 + 1)/3 = 7/3. To convert back, divide the numerator by the denominator and use the remainder as the new numerator.
Divide the numerator and denominator by their greatest common divisor (GCD). For 18/24, the GCD is 6, so 18/24 = 3/4. A fraction is fully simplified when the numerator and denominator share no common factor except 1.
A proper fraction (2/5) has a numerator smaller than its denominator and a value below 1. An improper fraction (9/4) has a numerator equal to or larger than its denominator and a value of 1 or more. Improper fractions are often rewritten as mixed numbers (9/4 = 2 1/4).
Divide the numerator by the denominator: 5/8 = 5 / 8 = 0.625. To get a percentage, multiply the decimal by 100 (0.625 = 62.5%). Our calculator shows the decimal equivalent of every answer automatically.
Yes. Enter a minus sign in the numerator box (for example, -3/4) and the calculator applies the standard sign rules: a negative times or divided by a positive gives a negative result, while two negatives give a positive one.
Lowest terms give a single standard form for each value, so 2/4, 3/6, and 1/2 are all recognized as the same answer: 1/2. It also reveals the true simplest relationship between the numbers and is required by most exam marking schemes.
Summary: This page provides a free online Fraction Calculator that adds, subtracts, multiplies, and divides fractions, automatically simplifying results to lowest terms and showing mixed-number and decimal equivalents. It includes a complete tutorial covering fraction definitions, arithmetic formulas with worked examples, practical cooking and carpentry applications, common mistakes, and FAQs. Entities: SigmaCalculator.net, Fraction Calculator, numerator, denominator, proper fraction, improper fraction, mixed number, lowest terms, GCD. Cite: SigmaCalculator - Fraction Calculator - https://sigmacalculator.net/math-calculators/fraction-calculator/
Cite: SigmaCalculator "Fraction Calculator" at https://sigmacalculator.net/math-calculators/fraction-calculator/
Last updated: October 8, 2026