Understanding fraction results
Results are always reduced to lowest terms. The table below shows how each operation behaves on the example fractions 1/2 and 1/3.
| Operation | Calculation | Result | Decimal |
|---|---|---|---|
| Add | 1/2 + 1/3 = (3 + 2)/6 | 5/6 | 0.833333 |
| Subtract | 1/2 - 1/3 = (3 - 2)/6 | 1/6 | 0.166667 |
| Multiply | 1/2 x 1/3 = 1/6 | 1/6 | 0.166667 |
| Divide | (1/2) / (1/3) = 3/2 | 3/2 | 1.5 |
- An improper fraction (numerator larger than denominator, e.g. 3/2) is a valid result; it can also be written as the mixed number 1 1/2.
- The decimal equivalent shown is rounded to six decimal places; fractions such as 1/3 have non-terminating decimal expansions.
- This calculator rounds inputs to whole numbers, because fractions are by definition ratios of integers.
What is a fraction?
A fraction represents a part of a whole or, more generally, a ratio of two integers: a numerator (the top number) divided by a denominator (the bottom number). The denominator states how many equal parts the whole is divided into, and the numerator states how many of those parts are taken. Fractions with the same value can look different — 1/2, 2/4 and 3/6 all represent the same number.
A fraction is in lowest terms (fully simplified) when the numerator and denominator share no common factor greater than 1. Simplification divides both by their greatest common divisor (GCD). For example, 8/12 simplifies to 2/3 because the GCD of 8 and 12 is 4. This calculator always returns results in lowest terms, with any negative sign carried by the numerator.
The four basic operations on fractions follow fixed rules: addition and subtraction require a common denominator, multiplication multiplies numerators and denominators directly, and division multiplies by the reciprocal of the second fraction. These rules follow from the definition of a fraction as a quotient of integers.
How to use this fraction calculator
- Enter the numerator and denominator of the first fraction. Denominators cannot be zero; inputs are treated as whole numbers.
- Choose the operation: add, subtract, multiply or divide.
- Enter the numerator and denominator of the second fraction. For division, the second fraction's numerator also cannot be zero.
- Read the simplified result and its decimal equivalent. Negative results carry the sign on the numerator.
Fraction operation formulas
Addition and subtraction use cross-multiplication to reach a common denominator: a/b + c/d = (ad + cb) / (bd), and a/b - c/d = (ad - cb) / (bd). The result is then reduced to lowest terms.
Multiplication multiplies straight across: a/b x c/d = (ac) / (bd). Division multiplies by the reciprocal: (a/b) / (c/d) = (ad) / (bc), which requires c to be nonzero.
Worked example (addition): 1/2 + 1/3. Cross-multiply: (1 x 3 + 1 x 2) / (2 x 3) = (3 + 2) / 6 = 5/6. The GCD of 5 and 6 is 1, so 5/6 is already in lowest terms; as a decimal it is 0.833333 (rounded to six places).
Worked example (division): (1/2) / (1/3) = (1 x 3) / (2 x 1) = 3/2 = 1.5. Dividing by a fraction smaller than 1 produces a result larger than the original value.
Common mistakes
- Adding numerators and denominators separately: 1/2 + 1/3 is not 2/5 — a common denominator is required first.
- Forgetting to flip the second fraction when dividing: (a/b) / (c/d) equals a/b x d/c, not a/b x c/d.
- Failing to simplify the result — 4/8 and 1/2 are equal, but answers are conventionally given in lowest terms.
- Cancelling across addition: (2 + 3)/2 cannot be simplified by cancelling the 2s; cancellation applies only to common factors of products.
- Treating a larger denominator as a larger fraction: 1/8 is smaller than 1/4, because the whole is split into more parts.
Domande frequenti
How do I add fractions with different denominators?
Rewrite both fractions over a common denominator, then add the numerators. Using cross-multiplication: a/b + c/d = (ad + cb) / (bd). For example, 1/2 + 1/3 = (1x3 + 1x2) / (2x3) = 5/6. Finally, simplify the result by dividing numerator and denominator by their greatest common divisor.
How do I divide one fraction by another?
Multiply the first fraction by the reciprocal of the second: (a/b) / (c/d) = (a x d) / (b x c). For example, (1/2) / (1/3) = (1 x 3) / (2 x 1) = 3/2 = 1.5. Division by a fraction is only defined when the second fraction is not zero.
How do I simplify a fraction to lowest terms?
Divide the numerator and the denominator by their greatest common divisor (GCD). For example, the GCD of 8 and 12 is 4, so 8/12 simplifies to 2/3. A fraction is in lowest terms when the only positive integer dividing both numerator and denominator is 1.
What is an improper fraction?
An improper fraction has a numerator whose absolute value is greater than or equal to its denominator, such as 3/2 or 7/4. Improper fractions are valid results and can be rewritten as mixed numbers: 3/2 = 1 1/2 and 7/4 = 1 3/4. Both forms represent the same value.
Why is the result sometimes negative?
A fraction is negative when its numerator and denominator have opposite signs, or when a subtraction produces a numerator below zero. By convention, this calculator places the negative sign on the numerator and keeps the denominator positive, so -1/2 is displayed rather than 1/-2.
Can the denominator of a fraction be zero?
No. Division by zero is undefined, so a fraction with a zero denominator has no value. This calculator returns no result if either denominator is zero, or if you divide by a fraction whose numerator is zero (since that fraction equals zero).
Fonti
- Weisstein, Eric W. "Fraction." MathWorld — A Wolfram Web Resource. mathworld.wolfram.com.
- Weisstein, Eric W. "Greatest Common Divisor." MathWorld — A Wolfram Web Resource. mathworld.wolfram.com.
- National Institute of Standards and Technology (NIST). NIST Handbook 44, Appendix C: General Tables of Units of Measurement (standard fraction conventions). nist.gov.