Add, subtract, multiply, and divide fractions and mixed numbers, simplify to lowest terms, and convert between fractions and decimals. Results show as a mixed number, an improper fraction, and a decimal.
How to use the fraction calculator
Enter the whole-number, numerator, and denominator parts of each value, choose an operation, and calculate. Leave the whole-number box at zero for an ordinary fraction. The result includes the reduced improper fraction, a mixed number when applicable, and a decimal approximation.
A negative sign may appear on the whole number or numerator. The denominator
must be nonzero. Before doing arithmetic, each mixed number is converted to an
improper fraction: w a/b = (wb + a)/b for a positive mixed number.
Common denominators for addition and subtraction
The product bd in the general formula always works, but the least common
denominator usually keeps the intermediate numbers smaller. For example, the
least common multiple of 6 and 8 is 24, so 1/6 + 3/8 becomes 4/24 + 9/24 = 13/24. The LCM Calculator finds that common
denominator, and the GCF Calculator helps
reduce the final numerator and denominator.
Multiplication, division, and simplification
For multiplication, cancel common factors before multiplying when possible.
In 6/35 × 14/15, cancel 6 with 15 and 14 with 35 to get 2/5 × 2/5 = 4/25.
For division, invert only the second fraction and then multiply. Dividing by a
zero fraction is undefined because its reciprocal does not exist.
A fraction is in lowest terms when its numerator and denominator share no factor greater than 1. Dividing both by their greatest common factor preserves the value. The sign is conventionally kept in the numerator so the denominator stays positive.
Fractions, decimals, and percentages
Divide the numerator by the denominator to get a decimal. Some denominators
terminate, such as 3/8 = 0.375; others repeat, such as 1/3 = 0.333…. The
displayed decimal is therefore an approximation when the expansion does not
terminate. Multiply the decimal by 100 to express the same value as a percentage,
or use the Percentage Calculator for
percentage-change and reverse-percentage questions.
Examples
3/4 + 1/6
Common denominator 12: 9/12 + 2/12 = 11/12.
3/4 ÷ 1/6
3/4 × 6/1 = 18/4 = 9/2, or 4 1/2 as a mixed number.
Formula
Addition and subtraction need a common denominator:
a/b + c/d = (ad + bc) / bd
a/b − c/d = (ad − bc) / bd
Multiplication and division do not:
a/b × c/d = ac / bd
a/b ÷ c/d = a/b × d/c = ad / bc
Always reduce by dividing numerator and denominator by their greatest common factor.
A denominator of 0 is undefined. The calculator rejects a zero denominator and division by a zero fraction.
Sources
- OpenStax Prealgebra 2e: Add and Subtract Fractions with Different Denominators
- OpenStax Prealgebra 2e: Multiply and Divide Fractions
Fraction limits
Denominators must be nonzero. Mixed numbers are converted to improper fractions before arithmetic, and final fractions are reduced by their greatest common factor. Decimal output is an approximation when the reduced denominator contains prime factors other than 2 and 5.
Exact decimal to fraction, including repeating digits
Use Exact decimal when you want the fraction represented by the digits you type. Enter 0.125 to get 1/8, or -1.375 to get -11/8 and the mixed number -1 3/8. Parentheses identify a repeating block: 0.(3) is 1/3, while 0.1(6) is 1/6. Do not type an ellipsis or assume a finite string repeats automatically.
A terminating decimal with n decimal places is an integer over 10ⁿ, reduced by the greatest common divisor. For a repeating decimal, the denominator also contains one 9 for each repeating digit. For 0.1(6), subtracting 1 from 16 gives a numerator of 15 over 90, which reduces to 1/6. The steps table shows the unreduced fraction, divisor and reduced result.
Exact digits versus an approximation
The original approximate conversion remains available. It finds a nearby simple rational number using continued fractions. Exact decimal instead preserves the meaning of the entered text with integer arithmetic, up to 200 digits. Thus 0.333 is exactly 333/1000, whereas 0.(3) is exactly 1/3. The secondary decimal display is rounded; the fraction is the exact output.
Do not use the exact result to imply that a measured quantity is infinitely precise. A measured 0.333 meter still has the uncertainty of that measurement, even when expressed as 333/1000 meter.