The **simple rule of three** is a mathematical tool used to quickly solve problems that involve a direct proportional relationship between two variables. For example:* A motorcycle travels 320 kilometers in 150 minutes, how many kilometers per hour did it travel?*.

For** correctly pose a simple rule of three **Three data must be known, and only one is the one that operates as an unknown: if A (known value) maintains a certain relationship with B (known value), and it is known that C (known value) with D (unknown value and called by such “unknown” ratio) have the same relationship, it is possible to calculate the unknown value D using the values A, B and C.

### Examples of application of the simple rule of three

- With forty hours of work a week, a worker earned $ 12,000. How much will he earn if the following week he can work fifty hours?
- A motorcycle travels 320 kilometers in 150 minutes, how many kilometers per hour did it travel?
- This year there were 42 days with rain, what percentage of the year does that mean?
- In 50 liters of seawater there are 1300 grams of salt, in how many liters will 11600 grams be contained?
- A machine makes 1,200 screws in six hours. How long will it take the machine to make 10,000 screws?
- If a person can live in New York for 10 days with $ 650. How many days can afford if you only have $ 500?
- With 5 liters of paint, 90 m of fence have been painted. Calculate how many meters of fence can be painted with 30 liters.
- Three taps take 10 hours to fill a water tank. How many hours will it take 5 bobbins to do it?
- If I have to sow 30 corn seeds per row, how many seeds will I need to plant a 20-row batch?
- If in two and a half hours a motorcyclist has covered a distance of 320 kilometers. Have you exceeded the speed limit, which is 80 km / h?

### Simple rule of three features

The way to solve the unknown is very** simple and easy to memorize**In fact, it is one of the first reasonings that children are taught during primary school, where they begin to handle basic operations (addition, subtraction, multiplication and division).

If the data whose positive relationship is known are noted above, and below and in column, the known data of the other series is noted on one side (generally by convention the left).

The unknown will result from **multiply the two values** known diagonally, C x B, and divide that product by the remaining known value, that is, A; thus the unknown value D.

### The linear function in the simple rule of three

The mathematical explanation of the simple rule of three presumes the existence of a **lineal funtion** which links two variables.

It happens that the linear function is one of the easiest to understand and visualize, because to determine all its behavior it is enough to know two points through which that line or line passes: the linear character makes the trajectory always the same, persisting towards negative and positive infinity.

Therefore, the deduction after the simple rule of three allows **fully know the function** to which it is referring: the quotient between the subtractions of both variables (in the case we have seen, the result of (DB) divided (CA) is the slope, that is, how far the variable that contains D and B advances when it advanced by one containing the C and A.

Note that in some cases the **domain is restricted**, since things like negative time (-10 hours) or a non-integral quantity of screws or cars cannot exist.

### Direct and inverse proportionality

Within the simple rule of three, it is important to differentiate between direct proportionality and inverse proportionality: the latter occurs when **the relationship instead of being positive** (as explained) **is negative**, with a line in the opposite direction, and then when one variable goes in a certain sense the other goes in the opposite direction.

If it is stated, for example, that 2 workers (known value, A) take 6 hours to make a wall (known value, B), and the proportional character is trusted, 4 workers (known value, C) will not take 12 hours to build that same wall, but on the contrary, 3 hours (unknown value, D).

This figure arises from doing in this case of inverse proportionality **A x B / C** (instead of B x C / A), which is what was raised earlier for direct proportionality.

Something important is that proportionality, whether direct or inverse, does not apply to all cases, since not all mathematical relationships follow this linear pattern.

The vast majority of natural and social relationships deviate from this pattern, making them much more difficult to address and predict.