# Rule of three

The rule of three (in Austria instead: final calculation ; earlier also: Regeldetri , Regel Detri , Regel de Tri or Regula de Tri from the Latin regula de tribus [terminis] 'rule of three [members]' or French Règle de trois ; also golden rule , Ratio equation , proportionality , final calculation or inferences ) is a mathematical procedure to calculate the unknown fourth value from three given values ​​of a ratio . A (simpler) variant is the double clause . The rule of three is not a mathematical theorem , but a solution procedure for proportional problems. It is particularly taught in school mathematics. With the rule of three, problems can be solved on the basis of simple insights or even very schematically, without fully understanding the underlying mathematical laws. Those who are familiar with proportionalities no longer need the rule of three, because they can then get the results through simple mathematical operations.

## Simple rule of three

• There is a law of the type “The more A, the more B.” (direct proportionality ): When A is doubled (tripled, ...), B is also doubled (tripled, ...).
• There is a ratio of units of size A to units of size B.${\ displaystyle a}$${\ displaystyle b}$
• The question is asked about the number of units of size B that are in the same ratio to units of A.${\ displaystyle x}$${\ displaystyle c}$

The "similar" values ​​are to be written below one another in a table:

 Size a Size B. ${\ displaystyle a}$ ${\ displaystyle b}$ ${\ displaystyle c}$ ${\ displaystyle x}$

### Content solving

The rule of three problem can be solved very easily in three thinking steps:

1. ${\ displaystyle a}$Units of A correspond to units of B.${\ displaystyle b}$
2. One unit of A corresponds to units of B.${\ displaystyle b / a}$
3. ${\ displaystyle c}$Units of A correspond to units of B.${\ displaystyle x = c \ cdot b / a}$

An additional line is inserted in the table. The same value is used to divide or multiply in both columns of the table.

 Size a Size B. Calculation step ${\ displaystyle a}$ ${\ displaystyle b}$ ${\ displaystyle \ div a}$ ${\ displaystyle 1}$ ${\ displaystyle b \ div a}$ ${\ displaystyle \ cdot c}$ ${\ displaystyle c}$ ${\ displaystyle c \ cdot (b \ div a)}$

Fractions that arise during the calculation are shortened in each step (see example 1 ).

### background

Relationships belong to the elementary mathematical knowledge and appear already in Euclid's elements . The rule of three is given (without justification) as regula de tri in Adam Ries' arithmetic books . The term rule of three is due to the three given in the bill inserted (in old German: "enacted") sizes. Today's German school books often interpret the term as “solving in three sentences”. In algebraic notation, the rule of three is a ratio equation :

${\ displaystyle a: b = c: x}$

The solution is obtained by rearranging the equation ( Example 2a ). ${\ displaystyle x = c \ cdot b / a}$

## Extensions

### Reversed rule of three

• There is a law of the type “The less A, the more B.” ( indirect proportionality , example 2b ): When halving (thirds, ...) of A, B is doubled (tripled, ...).
• Thereby giving units of a size A with units of a size B is a constant product.${\ displaystyle a}$${\ displaystyle b}$
• Is asked according to the number of size B, with the units of the same units of A product yield: .${\ displaystyle x}$${\ displaystyle c}$${\ displaystyle a \ cdot b = c \ cdot x}$

Opposite arithmetic operations are carried out in both columns of the table:

 Calculate: Size a Size B. Calculate: by ${\ displaystyle a}$ ${\ displaystyle a}$ ${\ displaystyle b}$ times ${\ displaystyle a}$ times ${\ displaystyle c}$ ${\ displaystyle 1}$ ${\ displaystyle a \ cdot b}$ by ${\ displaystyle c}$ ${\ displaystyle c}$ ${\ displaystyle a \ cdot b / c}$

### Generalized rule of three

With the generalized rule of three, products of several sizes are included in the ratio (see example 3 ).

Starting from there are two ways to determine the solution to the problem . The simple rule of three is to be applied several times (one goes first from to , then from to and finally from to ). Alternatively, all steps can be carried out at the same time: ${\ displaystyle a_ {0} \ cdot b_ {0} \ cdot c_ {0} \ {\ mathrel {\ widehat {=}}} \ d_ {0}}$${\ displaystyle a_ {1} \ cdot b_ {1} \ cdot c_ {1} \ {\ mathrel {\ widehat {=}}} \ x}$${\ displaystyle a_ {0}}$${\ displaystyle a_ {1}}$${\ displaystyle b_ {0}}$${\ displaystyle b_ {1}}$${\ displaystyle c_ {0}}$${\ displaystyle c_ {1}}$

1. ${\ displaystyle a_ {0} \ cdot b_ {0} \ cdot c_ {0} \ {\ mathrel {\ widehat {=}}} \ d_ {0}}$
2. ${\ displaystyle 1 \ {\ mathrel {\ widehat {=}}} \ {\ frac {d_ {0}} {a_ {0} \ cdot b_ {0} \ cdot c_ {0}}}}$
3. ${\ displaystyle x \ {\ mathrel {\ widehat {=}}} \ {\ frac {d_ {0} \ cdot a_ {1} \ cdot b_ {1} \ cdot c_ {1}} {a_ {0} \ cdot b_ {0} \ cdot c_ {0}}}}$

## Examples

### example 1

A vehicle covers 240 km in 3 hours at constant speed, how far does it go in 7 hours? The following applies:

3 to 240 is like 7 to "x"

Invoice in tabular form:

 Time in h Distance in km Calculate: 1. 3 240 : 3 2. 1 80 · 7 3. 7th 560

Solution: The vehicle can travel 560 km in 7 hours.

### Example 2 (simple and reverse rule of three)

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

a) 21 trucks transport 35 tons of spoil in one working day. How many tons of overburden can 15 trucks manage in the same time?

• 21 trucks 35 tons${\ displaystyle {\ mathrel {\ widehat {=}}}}$
• 15 trucks x tons${\ displaystyle {\ mathrel {\ widehat {=}}}}$
• x = 15 · 35/21 = 25, i.e. 25 tons.

b) 21 trucks need 35 days to remove a certain amount of overburden. How much time do 15 trucks need for this?

• 21 trucks 35 days${\ displaystyle {\ mathrel {\ widehat {=}}}}$
• 15 trucks x days${\ displaystyle {\ mathrel {\ widehat {=}}}}$
• x = 35 21/15 = 49, i.e. 49 days.

### Example 3 (generalized rule of three)

Two cows eat 48 kg of grass in one day. How many kg of grass do 5 cows eat in 6 hours?

1. 2 cows eat 48 kg of grass in 24 hours
2. 1 cow eats 1 kg of grass in 1 hour
3. 5 cows eat 30 kg of grass in 6 hours

assuming that the cows eat an even amount of grass over the whole time.