Fraction Calculator
Add, subtract, multiply, and divide fractions. Simplify fractions, convert between mixed numbers and improper fractions.
A fraction represents a part of a whole. It consists of a numerator (the top number) and a denominator (the bottom number). The denominator tells you how many equal parts the whole is divided into, while the numerator tells you how many of those parts you have. For example, 3/4 means 3 out of 4 equal parts.
Fractions can also be expressed as mixed numbers, which combine a whole number with a proper fraction (e.g., 2 3/4). Any mixed number can be converted to an improper fraction and vice versa.
Addition & Subtraction: To add or subtract fractions, you need a common denominator. Find the least common multiple (LCM) of the denominators, convert each fraction, then add or subtract the numerators.
Multiplication: Multiply the numerators together and multiply the denominators together. Simplify the result by dividing both parts by their greatest common divisor (GCD).
Division: Flip the second fraction (take its reciprocal) and then multiply. For example, 2/3 ÷ 4/5 becomes 2/3 × 5/4 = 10/12 = 5/6.
Simplifying a fraction means reducing it to its lowest terms by dividing both the numerator and denominator by their greatest common divisor (GCD). The GCD can be found using the Euclidean algorithm: repeatedly divide the larger number by the smaller and take the remainder until the remainder is zero.
A fraction is fully simplified when the only common factor of the numerator and denominator is 1. For example, 24/36 simplifies to 2/3 because the GCD of 24 and 36 is 12.
Fractions are used everywhere in daily life. In cooking, recipes call for fractional measurements like 1/2 cup or 3/4 teaspoon. In construction, dimensions are often expressed in fractions of an inch. Financial calculations use fractions when splitting costs or calculating interest rates.
In science and engineering, fractions help express ratios, probabilities, and proportions. Understanding fractions is also foundational for learning algebra, calculus, and other advanced mathematics.
Fraction work gets easier when you identify the type of problem before doing any arithmetic. Adding and subtracting compare parts of the same whole, so the denominators must describe equal-sized pieces. Multiplying scales one quantity by another and does not require a common denominator. Dividing asks how many groups of one fraction fit into another, which is why multiplying by the reciprocal works. The calculator can carry out each operation, but knowing the purpose of the operation helps you spot answers that do not make sense.
Common denominators are the main step for addition and subtraction. If you add 1/3 and 1/4, the pieces are different sizes, so adding the numerators directly would mix thirds and fourths. The least common denominator is 12, so the problem becomes 4/12 + 3/12 = 7/12. A larger common denominator would still work, but the result would need more simplification afterward. The least common denominator keeps the numbers smaller.
Multiplication has a different meaning. If a recipe calls for 3/4 cup of flour and you make half the recipe, you need 1/2 x 3/4 = 3/8 cup. The result is smaller because you are taking a fraction of a fraction. If you multiply by a number greater than one, the result grows. This is a useful reasonableness check: multiplying by 2/3 should reduce a positive value, while multiplying by 5/4 should make it larger.
Division by a fraction can feel odd until you read it as grouping. If you have 3/4 of a yard of ribbon and each bow uses 1/8 of a yard, the question is how many one-eighth pieces fit into three-fourths. The calculation is 3/4 ÷ 1/8 = 3/4 x 8/1 = 6. The reciprocal method works because division asks for the missing factor that would produce the dividend.
Mixed numbers are easier to read but harder to calculate with. Convert 2 3/5 to 13/5 before multiplying or dividing, then convert back if a mixed number is clearer for the final answer. Improper fractions are not wrong. They are often the cleanest form during the calculation. At the end, the best format depends on the context: 7/4 may be useful in algebra, while 1 3/4 is easier for a tape measure or recipe.
Every fraction answer should pass a size check. Adding two positive proper fractions should produce a value larger than either addend. Subtracting should produce a smaller value when the second fraction is positive. Multiplying by a proper fraction should make the value smaller. Dividing by a proper fraction should make the value larger. These checks catch many numerator, denominator, and reciprocal mistakes before they reach a final worksheet, recipe, or estimate.
Simplification makes answers easier to compare. The fraction 18/24 is mathematically correct, but 3/4 is easier to recognize. To simplify, divide the numerator and denominator by their greatest common divisor. If both numbers are even, divide by 2 first. If the digits add to a multiple of 3, divide by 3. Larger numbers may need the Euclidean algorithm, but the idea is the same: remove common factors until none remain.
Fractions and decimals both have uses. Decimals are convenient for money, measurements in metric units, and calculator displays. Fractions are better when exact ratios matter. One third is exact as 1/3 but becomes a repeating decimal. A construction measurement of 5/16 inch should usually stay as a fraction because that is how the ruler is marked. A fuel price, interest rate, or spreadsheet value may be easier as a decimal or percentage.
Unit labels matter with fractions. Half of a cup, half of an hour, and half of a mile all use the same fraction but describe different quantities. When solving word problems, write the unit next to each value. If the units change during the work, convert them before combining fractions. Adding 1/2 foot and 3 inches is not 3 1/2 of anything until the inches are converted to feet or the feet are converted to inches.
Negative fractions follow the same arithmetic rules as negative integers. The negative sign may be written in front of the fraction, with the numerator, or with the denominator, but not with both. The forms -3/5, (-3)/5, and 3/(-5) are equivalent. The form (-3)/(-5) is positive 3/5. Keeping the sign outside the fraction often makes the final answer easier to read.
Fractions often appear inside ratios and rates. A ratio compares two quantities, such as 3 parts water to 1 part concentrate. That can be written as 3:1, 3/1, or as a total mixture where concentrate is 1/4 of the whole. A rate compares quantities with different units, such as 45 miles per 3/4 hour. Dividing by the time gives 60 miles per hour. The fraction rules are the same, but the units tell you what the answer means.
Proportions are useful when scaling. If 2/3 cup of rice serves two people, then serving six people means multiplying by 3. The amount is 2 cups. If a map uses 1 inch to represent 5 miles, then 3 1/2 inches represents 17 1/2 miles. Set up the relationship before calculating so the numerator and denominator describe matching things on both sides of the proportion.
Probability uses fractions to describe outcomes. If a bag has 5 red marbles and 7 blue marbles, the chance of drawing red is 5/12. If a red marble is not replaced before a second draw, the next fraction changes because the total and possibly the red count changed. Fractions make this visible. The denominator is the number of possible outcomes under the current condition, not the starting condition.
In algebra, keeping fractions exact can make later work cleaner. Rounding 1/3 to 0.333 may be fine for an estimate, but it introduces a small error that can grow after several steps. Exact fractions can be simplified at the end, converted to decimals when needed, or left in fractional form when they reveal the structure of the problem.
A denominator cannot be zero because division by zero has no defined value. The fraction 5/0 does not describe five parts of a whole split into zero equal pieces. In algebra, this means any value that makes a denominator zero must be excluded from the domain before simplifying. If an expression has x - 3 in the denominator, x = 3 is not allowed even if later cancellation makes the expression look simpler.
Denominators also control the size of the pieces. With the same numerator, a larger denominator means smaller pieces. Three eighths is smaller than three fifths because eighths are smaller pieces than fifths. This idea helps when estimating before calculating. If an answer says 3/8 is larger than 3/5, something went wrong in the comparison or conversion.
When comparing fractions with different numerators and denominators, cross multiplication can avoid decimals. To compare 5/7 and 7/10, compare 5 x 10 with 7 x 7. Since 50 is greater than 49, 5/7 is slightly larger. This works because both products use a common denominator without writing it out.
To add fractions with different denominators, first find the least common denominator (LCD) by determining the least common multiple of the two denominators. Then convert each fraction so that it uses the LCD as its denominator. Finally, add the numerators and keep the common denominator. Simplify the result if possible.
A proper fraction has a numerator smaller than its denominator (e.g., 3/4), so its value is less than 1. An improper fraction has a numerator equal to or greater than its denominator (e.g., 7/4), meaning its value is 1 or greater. Improper fractions can be converted to mixed numbers.
To simplify (or reduce) a fraction, find the greatest common divisor (GCD) of the numerator and denominator, then divide both by that number. For example, 12/18 simplifies to 2/3 because the GCD of 12 and 18 is 6.
Multiply the whole number by the denominator, then add the numerator. Place that result over the original denominator. For example, 2 3/4 becomes (2 × 4 + 3)/4 = 11/4.
Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a/b is b/a. So a/b ÷ c/d becomes a/b × d/c. This rule follows from the mathematical definition of division.
Yes. A fraction is negative when exactly one of the numerator or denominator is negative. Writing −3/4, 3/−4, or −(3/4) are all equivalent. If both the numerator and denominator are negative, the fraction is positive.
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