Formula
For addition/subtraction: a/b ± c/d = (ad ± bc) ÷ bd. For multiplication: a/b × c/d = ac ÷ bd. For division: a/b ÷ c/d = ad ÷ bc. Simplify by dividing numerator and denominator by their greatest common divisor.
Fraction Calculator
Add, subtract, multiply or divide two fractions, reduce the answer to simplest form, show the mixed number and decimal check, and print a classroom or measurement worksheet record.
Worked example
For 3/4 + 2/5, use a common denominator: (3 × 5 + 2 × 4) ÷ (4 × 5) = (15 + 8) ÷ 20 = 23/20. As a mixed number, 23/20 is 1 and 3/20. As a decimal, it is 1.15.
Professional note
Master’s Tip: keep the exact fraction beside the decimal when teaching, quoting recipe changes or checking construction marks. A rounded decimal can hide whether the result is exact or only an approximation.
Regional and unit assumptions
Standard or basis: integer fraction arithmetic using common denominators and greatest-common-divisor simplification. No measurement, tax, legal or curriculum standard is claimed.
Operation proof
Same fractions, different operations
The page keeps the exact fraction result visible first, then shows the decimal check. Addition and subtraction use a common denominator, multiplication uses numerator times numerator and denominator times denominator, and division uses the reciprocal of the second fraction.
Visible checks
What the page now proves
- Common denominator trail
- Greatest-common-divisor simplification
- Mixed-number and decimal cross-check
Assumptions and limitations
- Denominators are rounded to whole numbers and must not be zero.
- The operation selector uses add, subtract, multiply and divide buttons; internally these map to deterministic arithmetic modes.
- The calculator reduces the final fraction after the operation, not each intermediate display only.
- Division by a zero second fraction is undefined.
- Decimal output is a checking aid; the exact simplified fraction is the primary result.
Methodology & Accuracy
How this calculator is checked
CalculationTime pages are built around visible arithmetic: the formula, assumptions, worked example and practical limitations are shown so the result can be checked rather than simply trusted.
Formula used
For addition/subtraction: a/b ± c/d = (ad ± bc) ÷ bd. For multiplication: a/b × c/d = ac ÷ bd. For division: a/b ÷ c/d = ad ÷ bc. Simplify by dividing numerator and denominator by their greatest common divisor.
Standard or basis
Standard or basis: integer fraction arithmetic using common denominators and greatest-common-divisor simplification. No measurement, tax, legal or curriculum standard is claimed.
Where a calculator follows a named legal, trade or industry standard, that standard is cited visibly. Otherwise the page uses transparent general arithmetic and states its limits.Master's Tip
Master’s Tip: keep the exact fraction beside the decimal when teaching, quoting recipe changes or checking construction marks. A rounded decimal can hide whether the result is exact or only an approximation.
Related calculators
Questions
How do you add fractions?
Use a common denominator, add the adjusted numerators, then reduce the result by the greatest common divisor.
How do you subtract fractions?
Use the same common-denominator method as addition, but subtract the adjusted second numerator from the adjusted first numerator.
How do you multiply fractions?
Multiply the numerators together, multiply the denominators together, then simplify the final fraction.
How do you divide fractions?
Multiply the first fraction by the reciprocal of the second fraction, then simplify the final fraction.
Why does the calculator show a decimal too?
The decimal helps with checking, measuring and comparison, but the simplified fraction remains the exact result when the inputs are exact whole-number fractions.
Calculation note
Fraction arithmetic is one of the oldest practical forms of calculation because people needed to divide food, land, materials, money and time before decimal notation was common. A useful fraction calculator should show the exact fraction, the simplification step and the decimal check instead of hiding the working.
Fractions record parts of a whole
A fraction writes a relationship between a numerator and a denominator: how many parts are counted and how many equal parts make the whole. That makes fractions useful anywhere a whole object, length, quantity or batch is split into equal parts.
Common denominators make addition and subtraction visible
Adding unlike fractions requires both fractions to describe the same size parts. Multiplying each numerator by the other denominator creates equivalent fractions over a common denominator, which is why the calculator shows the cross-multiplication method.
Simplifying protects readability
Fractions such as 46/40 and 23/20 have the same value, but the reduced version is easier to read and compare. The greatest common divisor gives a deterministic way to reduce the answer without changing its value.
Decimal checks are useful but not always exact
Some fractions have terminating decimals, while others repeat forever in base ten. Keeping the simplified fraction beside the decimal helps students, cooks, makers and tradespeople avoid treating a rounded decimal as exact.
Sources and further readingEncyclopaedia Britannica: FractionNIST: Guide for the Use of the International System of Units