How to Solve Proportions
This calculator solves proportions in the form A : B = C : D, finding the unknown term when the other three are known. It uses the fundamental property of proportions.
The Fundamental Property
In a proportion A : B = C : D, it always holds that A x D = B x C (the product of the extremes equals the product of the means). This property, proven by Euclid in the Elements (circa 300 BC), allows solving any proportion.
To find the unknown term:
- A = (B x C) / D
- B = (A x D) / C
- C = (A x D) / B
- D = (B x C) / A
The Rule of Three
The "rule of three" is the traditional name for solving proportions. If 3 kg of apples cost 6 dollars, how much do 5 kg cost? Set up the proportion 3 : 6 = 5 : x, giving x = (6 x 5) / 3 = 10 dollars.
Types of Proportionality
There are two main types:
- Direct proportionality: as one quantity increases, the other increases proportionally (price/quantity, distance/time at constant speed)
- Inverse proportionality: as one quantity increases, the other decreases (speed/time at constant distance, workers/days to complete a job)
Applications of Proportions
Proportions are among the most versatile mathematical tools: scaling recipes, maps and scales, percentages, unit conversions, and pharmaceutical dosages based on body weight.