Direct rule of three calculator

Enter a first quantity, its corresponding value, and a second quantity. The calculator returns the value that corresponds to the second quantity, along with the scale multiplier and the arithmetic used. It is useful for prices, recipes, distances, production estimates, and any situation where doubling one amount should double the other.

Recognize a direct proportion before calculating

A direct rule of three applies when two quantities change in the same proportion. If one quantity is multiplied by a factor, its corresponding value must be multiplied by that same factor. For example, if four notebooks cost ten currency units, seven notebooks cost more because the number of notebooks increased. This relationship is direct: buying twice as many at a constant unit price costs twice as much. The method is not appropriate when one quantity rises while the other falls, such as the time needed for a fixed job when more equally productive workers participate. It also does not model bulk discounts, fixed fees, stepped rates, or other relationships where the rate changes. Before entering numbers, identify one complete known pair: the first quantity and the value belonging to it. Then identify the second quantity whose matching value you want. Keeping corresponding measurements in the same positions is essential. If the pairs are reversed or mixed, the arithmetic may be valid but the answer will describe the wrong relationship.

Enter the three known values in corresponding positions

Use first_quantity for the known amount in the complete pair and first_value for the value associated with that amount. Use second_quantity for the new amount whose value is unknown. Suppose four kilograms of material cost ten units and you need the cost of seven kilograms. Enter 4, 10, and 7 in that order. The calculator first finds how the quantity changed by dividing seven by four. That multiplier is then applied to ten. Equivalently, it cross-multiplies seven by ten and divides the product by four. Both views express the same direct proportion: first quantity is to first value as second quantity is to the result. Decimal and negative finite numbers are accepted for algebraic work, although real-world quantities may impose their own restrictions. The first quantity cannot be zero because it forms the divisor and cannot define a usable rate. Use consistent units within each side of the relationship; convert grams to kilograms or minutes to hours beforehand when necessary, so the resulting number has a clear meaning.

Read and verify the fourth proportional

The result is the fourth proportional, the missing number that makes both ratios equal. The output also includes the multiplier, a proportion equation, and the exact calculation layout. These details make the answer easy to audit instead of presenting an unexplained number. A quick reasonableness check is valuable: when the second quantity is larger than a positive first quantity and the known value is positive, the result should also be larger. If the second quantity is half the first, the result should be half the first value. You can verify the calculation by dividing the first value by the first quantity and comparing that unit rate with the result divided by the second quantity; subject to ordinary floating-point rounding, the rates should agree. The calculator uses deterministic arithmetic and returns stable numeric precision, so identical inputs produce identical output without network access or random behavior. For automated workflows, each request costs $0.002. Treat the result as a mathematical proportional estimate, and apply domain-specific rounding only afterward, such as rounding money to the required currency precision or materials to purchasable package sizes.