Fractions, division and ratios

A fraction n/d represents numerator n divided by non-zero denominator d. The same rational value can have many written forms: 3/4, 6/8 and 75/100 are equal because numerator and denominator have been multiplied by the same non-zero factor. Reduction removes common factors until the numerator and denominator are coprime. The reduced exact result is the strongest answer for further fraction work because it has not been rounded.

A ratio such as 3:4 compares two quantities and is closely related to the fraction 3/4, but context matters. A part-to-part ratio of boys to girls is not the same as the fraction of all learners who are boys. The ratio output on this calculator simply rewrites the final reduced numerator and denominator with a colon; it does not reinterpret what the two numbers represent.

Addition and subtraction

Fractions can be added or subtracted only after their parts are expressed over a common denominator. For a/b + c/d, cross-multiplication gives (ad + cb)/(bd). Subtraction changes the plus sign to a minus. The product bd is always a common denominator, although it may not be the least common denominator. Final reduction removes any shared factor, so the exact answer is still simplified.

For the default 3/4 + 5/6, the numerator becomes 18 + 20 and the denominator becomes 24. That gives 38/24, which reduces to 19/12. The mixed form is 1 7/12. A useful estimate is that 3/4 is 0.75 and 5/6 is a little more than 0.8, so a result near 1.58 is sensible. Estimation catches misplaced signs and inverted terms before they spread into later working.

Multiplication and division

Multiplication is direct: multiply the numerators and multiply the denominators. Cancelling a common factor across a numerator and the opposite denominator before multiplying can keep the working numbers small. The calculator multiplies first and then reduces, which is mathematically equivalent for inputs within the supported numeric range.

Dividing by a fraction means multiplying by its reciprocal. Thus (a/b) ÷ (c/d) becomes ad/bc. If c is zero, the second fraction equals zero and division is undefined, so the page stops with an error. A negative sign may be written in the numerator or denominator; the final formatting moves it to the numerator so the denominator remains positive.

Improper and mixed-number forms

An improper fraction has a numerator whose absolute value is at least as large as its denominator. It is not mathematically wrong. In fact, improper fractions are usually easier for calculations because one numerator and one denominator carry the whole value. A mixed number separates a whole part and a proper fractional remainder, which can be easier to read in recipes, lengths and everyday descriptions.

For negative results, the sign applies to the whole mixed value. The output −1 1/2 means negative one-and-a-half, equivalent to −3/2. When converting a typed mixed number into this calculator, first multiply the whole magnitude by the denominator, add the numerator, then apply the original sign to the combined numerator.

Exact values versus decimal checks

Some fractions terminate in base-ten notation, such as 1/4 = 0.25. Others repeat indefinitely, such as 1/3 = 0.333…. The decimal panel is limited to a practical number of places and is therefore an approximation for repeating expansions. Converting that rounded decimal back into a fraction can produce a different rational number.

Keep the reduced fraction for algebra, probability and exact proportions. Use a decimal where measurement, currency or a system interface requires it, and state an appropriate rounding rule. In South African financial work, a fraction of a rand may eventually be rounded to cents, but applying rounding at every intermediate line can produce a different total from rounding once at the end.

Units and proportional reasoning

Fractions are dimensionless only when numerator and denominator use the same unit. A rate such as kilometres per litre is written like a fraction but carries compound units. Before adding two quantities, convert them to compatible units. Three-quarters of a metre and five-sixths of a centimetre cannot be added by entering 3/4 and 5/6 without first expressing both in metres or both in centimetres.

When scaling a recipe, drawing or mixture, distinguish a ratio from a total. A cement-to-sand ratio of 1:3 has four total parts, so cement is one-quarter of the combined dry parts. The page can simplify arithmetic, but it cannot infer whether the specification is part-to-part, part-to-whole, by mass or by volume. Preserve the original instruction beside the answer.

Manual checks before use

Check that no denominator is zero, estimate the sign and size, and substitute a decimal only as a secondary verification. For addition of two positive proper fractions, the answer must be positive and larger than either addend, although it can exceed one. Multiplying two positive proper fractions produces a smaller positive value. Dividing by a positive fraction below one produces a larger value.

Large whole-number inputs can exceed the exact-integer range of browser arithmetic even before the displayed result looks extreme. This calculator is intended for ordinary educational, household and business checks. Use arbitrary-precision rational software for cryptography, very large combinatorial values or formal exact computation.

Percentages from fractions

To express a fraction as a percentage, divide the numerator by the denominator and multiply by 100. The fraction 3/4 becomes 0.75 and then 75%. Moving in the other direction, place a whole-number percentage over 100 and reduce: 15% is 15/100, or 3/20. A percentage above 100% is valid when a quantity exceeds the chosen base, but the base must be stated.

Percentage change uses the original value as its denominator, not simply whichever number is smaller. Keep that modelling choice separate from the fraction operation so the result answers the intended comparison.

Sign checks with zero

A zero numerator makes the fraction zero as long as its denominator is non-zero. Adding zero leaves the other fraction unchanged, and multiplying by zero gives zero. A zero result should normally reduce to 0/1, even if the intermediate denominator is much larger. Division is different: zero divided by a non-zero fraction is zero, but dividing by a zero fraction is undefined. Separating those cases prevents the phrase “zero is not allowed” from being applied too broadly.