THE rule of three is a technique used to find a measure when we know three others, as long as these four measures form a proportion. This method, known as the rule of three, makes use of some important knowledge: fundamental property of proportions, greatnesses and measurements, reasons and proportions. It can be said that the union of all this knowledge results, among other things, in what we know as the rule of three.
Rule of three
Let's say a toy factory can produce 500 pieces a day with just 12 employees. How many employees does it take to produce 750 pieces daily?
To solve this type of problem, we use ruleinthree. Note that there are two greatnessesproportional in the problem, one is the number of employees and the other is the number of daily items. Also note that three measures of these quantities are known and the other we want to find out. That's why this technique is known as the rule of three.
building the proportion regarding this problem, we have:
12 = x
500 750
To find the value of x, just use the knowledge from the equations or use the
propertyfundamentalof theproportions: the product of the extremes is equal to the product of the means. This property is also known as “cross multiplication”. To apply it, just multiply 500 by x and 12 by 750:500x = 12·750
Solving this equation, we have:
500x = 9000
x = 9000
500
x = 18
It will take 18 employees to produce 750 toys a day.
Inversely proportional quantities
In the previous example, notice that by increasing the number of employees, we also increase the number of toys produced per day. When two quantities have this property, they are called directly proportional quantities. Whenever two quantities are directly proportional, the calculation of the rule of three can be done as in the previous example.
On the other hand, when we increase the measure relative to one quantity and the other decrease as a result, the quantities are said inversely proportional.
Example: a car travels at 50 km/h and takes 2 hours to reach its destination. How long would that same car take if it were at 100 km/h?
Note that by increasing the speed, the time spent on the course decreases, so these greatnessesthey areinverselyproportional. In this case, we will build the ratio by putting speed in one fraction and time in the other:
50 = 2
100x
This construction is necessary because, with inversely proportional quantities, before applying the fundamental property of proportions, we will invert one of the fractions.
50 = x
100 2
Applying the property, we have:
100x = 2·50
100x = 100
x = 100
100
x = 1
Therefore, the car will only spend 1 hour on the route.
Fundamentals of the Rule of Three: Ratio and Proportion
One reason is a division usually expressed as a fraction. The reasons are used to represent divisions in between measuresingreatnesses. The result obtained in a ratio can be evaluated in several ways, for example, when we divide the number of males from the population of a city by the total number of people living in that city, we will find a decimal called rate, which is the result of dividing two measurements between greatnesses.
On the other hand, when we divide the measure of the distance traveled by an automobile by the time spent by that automobile, we obtain another quantity, known as average speed.
an equality between two reasons is known as proportion. Note that, for a proportion to exist, there must be four measures, two relating to one magnitude and two relating to another.
Example: for a test, a car was placed on a 100 km route and it took 2 hours to cover it. In a second moment, he was placed on a 200 km course and took 4 hours to cover it. THE proportion relating to this experiment is:
100 = 200 = 50
2 4
Note that the two reasons between distance covered and speed are the same, as both result in 50 (kilometers per hour). So the two reasons form a proportion and distance and time quantities are called proportional.
THE ruleinthree is used when one of the four measures present in the reasons above is not known and we need to discover it.
By Luiz Paulo Moreira
Graduated in Mathematics
Source: Brazil School - https://brasilescola.uol.com.br/o-que-e/matematica/o-que-e-regra-tres.htm