Cross-multiplication

The rule of three (formerly also: the rule de tri [ from French Règle de tri, from Latin regula de tribus ], also ratio equation, proportionality, final accounts or short -circuits called ) is a mathematical method of three given values ​​of a ratio of the unknown fourth to calculate value. A ( simplified ) variant is the two- sentence. The rule of three is not a mathematical theorem, but a solution method for proportional tasks. He is especially taught in school mathematics. One can use the rule of three problems based on simple insights or even solve quite schematically, without completely see through the underlying mathematical laws. Anyone familiar with proportionalities, the three-set no longer needed, because then he can get the results by simple mathematical operations.

  • 2.1 Inverse rule of three
  • 2.2 Generalized rule of three
  • 3.1 Example 1
  • 3.2 Example 2 (simple and reverse rule of three )
  • 3.3 Example 3 ( generalized rule of three )
  • 3.4 Example of incorrect use

Simple rule of three

  • It is a law of the type "the more A, the more as" before ( direct proportionality ): When doubling ( tripling, ...) of A is also B doubles ( triples, ...).
  • Given a ratio of units of a size A to a size B. Units
  • Is asked for the number of units of size B, which stand in the same relation to units of A.

In a table, the "like " values ​​are to be written to each other:

Content- solving

The three-set task is very easy to solve in three steps of thinking:

In the table, a new row is inserted. In both columns of the table is divided or multiplied by the same value.

In computing, resulting fractures are reduced in each step (see Example 1).

Background

Relations among the elementary mathematical knowledge and appear already in Euclid's Elements. The three- sentence rule is specified (without justification) as regularized de tri in the arithmetic books of Adam Ries. The term three-set stems from the three given, inserted in the bill ( in old German, " gesatzten " ) sizes. Today's German textbooks suggest the designation often called the " dissolution in three sets ." In algebraic notation, it is in the three- set task to a rate equation:

By rearranging the equation one gets the solution (Example 2a).

Extensions

Inverse rule of three

  • It is a law of the kind " The less A, all the more as" before (indirect proportionality, Example 2b): When halving ( thirds, ...) of A is B doubles ( triples, ...).
  • Here, units found a size A units with a size B, a constant product.
  • Is asked for the number of units of size B, which give the same product with units of A: .

In two columns of the table opposite arithmetic operations are carried out:

Generalized rule of three

When generalized rule of three products go several sizes in the ratio of one (see Example 3).

Based on, one can determine the solution of the problem in two ways. The simple rule of three is repeatedly applied ( first one goes from to, then from to and finally to ). Alternatively, all steps may be performed at the same time:

Examples

Example 1

In 3 hours, a vehicle shall at constant speed 240 km back, how far it comes in 7 hours? The following applies:

Statement in tabular form:

Solution: In seven hours the vehicle is 560 km far.

Example 2 (simple and reverse rule of three )

The following examples have the same numbers but different ratios. In the first example, the quantitative data relate to a fixed period (one working day ). In the second example, the times refer to a fixed amount specified ( a certain amount of overburden).

A) 21 trucks to transport 35 tons of overburden on a working day. How many tons of overburden create in the same time 15 truck?

  • 21 trucks 35 tons
  • 15 trucks x tons
  • X = 15 * 35/21 = 25, so 25 tons.

B ) 21 trucks require 35 days for the removal of a certain amount of overburden. How long does this 15 truck?

  • 21 Trucks 35 days
  • 15 trucks x days
  • X = 35 * 21/15 = 49, so 49 days.

Example 3 ( generalized rule of three )

2 cows eat grass 48 kg in one day. How much kg of grass cows eat 5 to 6 hours?

Under the assumption that the cows on all the time even eat a lot of grass.

Example of incorrect use

See: Potato paradox

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