Fraction Calculator
Add, subtract, multiply and divide fractions and mixed numbers, with the answer simplified, as a mixed number and as a decimal.
Mixed numbers: whole number, space, fraction (2 1/2). A leading minus applies to the whole mixed number.
Steps
- As improper fractions: 31/4 - 21/8
- Common denominator 8: 62/8 - 21/8 = 41/8
- Simplify by the greatest common divisor: 41/8
Updated Formula covered by automated tests
Fractions show up wherever things get divided into parts: tape measures marked in sixteenths of an inch, recipes that call for 2/3 cup, lumber cut lists, sewing patterns and math homework. Type two fractions, whole numbers or mixed numbers such as 7 3/4, pick add, subtract, multiply or divide, and the calculator returns the answer reduced to lowest terms, as a mixed number, and as a decimal.
The arithmetic is exact. Nothing is rounded along the way, so 1/3 + 1/3 + 1/3 comes out as exactly 1 rather than 0.9999. When a decimal never ends, the repeating block is shown in parentheses, for example 0.1(6) for 1/6, so you can see the pattern instead of a cut-off string of digits.
Each result also shows the steps a teacher would want: the improper-fraction form of mixed numbers, the common denominator used for adding or subtracting, and the greatest common divisor used to simplify. Finite decimals such as 0.75 can be typed in as well and are turned into exact fractions before the calculation.
The formula
- a, c
- Numerators: how many parts you have
- b, d
- Denominators: how many equal parts make one whole; can never be zero
- w
- Whole-number part of a mixed number
- GCD
- Greatest common divisor of numerator and denominator, found with Euclid’s algorithm and divided out to reach lowest terms
Fractions can only be added or subtracted when they describe parts of the same size, which is why both are first rewritten over a common denominator. Multiplying denominators always works; the least common denominator gives smaller numbers, and either way the result is simplified at the end. Multiplication needs no common denominator because it scales one fraction by the other, and dividing by a fraction is the same as multiplying by its reciprocal, which is where "flip and multiply" comes from.
To simplify, the tool divides top and bottom by their greatest common divisor using Euclid’s algorithm: repeatedly replace the larger number by the remainder of dividing it by the smaller until the remainder is zero. For 84/36 the remainders go 84 mod 36 = 12, then 36 mod 12 = 0, so the GCD is 12 and the fraction reduces to 7/3. A zero denominator is rejected because dividing a whole into zero parts has no meaning.
Worked example
Marcus has a shelf board 7 3/4 feet long. He cuts off 2 5/8 feet for a small shelf and wants to split what is left into two equal pieces.
- 1 Convert to improper fractions: 7 3/4 = (7 × 4 + 3)/4 = 31/4, and 2 5/8 = (2 × 8 + 5)/8 = 21/8.
- 2 Use the common denominator 8: 31/4 = 62/8, so 62/8 − 21/8 = 41/8.
- 3 Write the leftover as a mixed number: 41 ÷ 8 = 5 remainder 1, so 5 1/8 feet, or 5.125 feet.
- 4 Divide by 2 by multiplying by 1/2: 41/8 × 1/2 = 41/16.
- 5 Convert: 41 ÷ 16 = 2 remainder 9, so each piece is 2 9/16 feet, which is 2.5625 feet.
Result: After the first cut Marcus has 5 1/8 feet left, and each of the two equal pieces is 2 9/16 feet long (ignoring the width of the saw blade).
Before you rely on the result
- Only finite decimals can be typed as input; a repeating decimal such as 0.333 is treated as exactly 333/1000, not as 1/3.
- Measurement answers ignore real-world losses such as saw kerf, seam allowance or spillage, so leave a margin when cutting or pouring.
- Very long repeating periods are cut off after 60 decimal digits and marked with an ellipsis; the fraction result itself remains exact.
Frequently asked questions
How do I enter a mixed number?
Type the whole number, a space, then the fraction: 3 1/2. A minus sign in front applies to the whole value, so -3 1/2 means negative three and a half, which is -7/2, not -3 + 1/2. Plain fractions (5/8), whole numbers (4) and finite decimals (0.375) work too.
Why is my answer an improper fraction like 41/8?
Improper fractions are easier to keep working with, so the result is shown both ways: 41/8 and the mixed number 5 1/8. Use the mixed form for measuring and the improper form if the number feeds into another calculation.
Why does 2/3 show as 0.(6) instead of 0.67?
Two thirds never ends as a decimal; the 6 repeats forever. The parentheses mark the repeating block so the decimal stays exact. Round it yourself to the precision you need: 0.67 for money, 0.667 for most measurements.
Do I need the least common denominator, or will any common denominator do?
Any common denominator gives the right answer once the result is simplified. The least common denominator just keeps the numbers smaller. For 3/4 and 5/8 it is 8 rather than 32, which is why the worked example shows 62/8 − 21/8.