MultiplyDivide

Multiplydivide is a descriptive term used to refer to the combined arithmetic operation that applies a multiplication by a factor and a division by another factor to a number. In practice, it is most commonly written as x * m / d, where d is nonzero. Because division by d is equivalent to multiplying by 1/d, the expression is algebraically the same as x * (m/d).

If all factors are real numbers and d ≠ 0, the operation acts linearly on x, producing a result that scales with x. The order of multiplication and division does not change the value for real numbers, since (x * m) / d = x * (m/d). In integer arithmetic, however, the intermediate steps can affect precision or overflow, so simplification (when possible) or choosing a safe evaluation order is important. If d divides m and one works with integers, the expression can be simplified to x * (m/d) to reduce intermediate magnitude.

Examples illustrate common uses. To scale a quantity x by a ratio 3:4, compute x * 3 / 4. For unit conversion, converting inches to centimeters uses x * 2.54, since 1 inch equals 2.54 centimeters. In financial or data-processing contexts, multiplydivide patterns are used to apply a rate or factor to a value in a single expression.

Computational considerations include rounding errors and overflow in finite-precision arithmetic. Some languages allow alternative evaluation orders (for example, x / d * m) to mitigate precision loss when x and d are integers. In more robust implementations, rational arithmetic or careful floating-point handling is preferred.

See also: ratio, fraction, unit conversion, scaling, and fused multiply-divide techniques in numerical computing.