Rule of three (mathematics)

From Wikipedia, the free encyclopedia

It has been suggested that this article or section be merged with Cross-multiplication. (Discuss)

In mathematics, the rule of three is the method of finding the fourth term of a mathematical proportion when the first three terms are known, that is, where the first term is in proportion to the second as the third is to the unknown fourth term. To find the fourth term, multiply the second and third terms (this is often called "taking the cross-product" or "cross-multiplying"), then divide their product by the first term.^[1] It was also known as the Golden Rule.^[2]

Using mathematical notation, with a, b and c as the known three terms of the proportion, and x as the unknown fourth term to be found, the problem can be stated as

${a \over b}={c \over x} .$

According to the rule of three,

$x={b \cdot c \over a} .$

To give an example, say that a car, driving at a constant speed, in 3 hours travels 90 miles. How far can the car drive in 7 hours if it maintains the same speed? Substituting the numbers for the letters using the rule of three,

$x={90 \cdot 7 \over 3} ={630 \over 3} = 210 \, \textrm{ miles}.$

The rule of three is based on the principle that, in a proportion, the product of the first and fourth terms (called the extremes) is equal to the product of the second and third terms (called the means). Or, where

${a \over b}={c \over d} \ , \textrm { then } \ a \cdot d=b \cdot c .$

In the expressions above, a and d are the extremes, and b and c are the means, of the proportion. It should be noted that since you can't divide by zero in mathematics, this rule won't apply when $b$ is equal to zero.