Greatness is what can be measured. THE greatness it is not the object that can be measured, but the measure that it is possible to be observed in it, such as: distance, Weight, velocity etc. The quantities can also be checked in reasons, as is the case with velocity, which is a quantity resulting from the division between distance and time, which, in turn, are two other quantities.
What is proportionality between quantities?
THE reason between two greatnesses it is a common thing that can be done to evaluate them and to get other quantities and properties as a result. When there is an equality between two distinct ratios, obtained by dividing two quantities at different times, it is called proportion, and the quantities, in this case, are said proportional. This is the form used for calculations involving rule of three, for example.
Let's say that a car travels at 50 km/h and, in a given period of time, travels 100 km. If this car were at 100 km/h, within that same time interval, the space covered by it would be 200 km. THE
reason in between velocity and space covered by this car can be evaluated at two different times and has the same results: 0.5. 50 = 100 = 0,5
100 200
This means that the greatnesses they are proportional, that is, the variation of one of the quantities causes the other to also undergo variation at the same rate as the first. Thus, when we double the speed of the car, we also double the space traveled by it in the same time interval.
Directly proportional quantities
by the fact of two greatnesses be proportional, when the values of one are changed, the values of the other are also changed, consequently, in the same proportion than the first. We say that the quantities A and B are directly proportional when, increasing the measure of greatness A, the measure of quantity B increases, as a result, in the same proportion.
if two greatnesses go directlyproportional, decreasing the measure of quantity A will make the measure of quantity B also decrease in the same proportiontherefore the word directly is used to represent this type of proportionality between quantities.
In the situation presented above, the car doubled its speed, and this made the space covered to double. The consequence of the increase in speed was an increase in the space traveled. proportion of speed. For this reason, the magnitudes velocity and space traveled they are directlyproportional in the assessed situation.
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Inversely proportional quantities
two quantities that are inverselyproportional they still vary as a consequence of the other and in the same proportion, however, the increase of the measure related to the first causes the measure related to the second to decrease. If we decrease the measure relative to the first greatness, this will cause the measure relative to the second to increase. That's why this proportionality is called inverse.
Example: In a shoe factory with 25 employees, a certain amount of shoes is produced in 10 hours. If the number of employees is 50, that same amount of shoes will be produced in 5 hours.
Clearly twice as many employees will get the job done in half the time. This is because the greatnessesworked hours and number of employees they are inverselyproportional.
Rule of three
THE ruleinthree is the tool used to find one of the measurements of a proportion. It is also valid for when this proportion is obtained through quantities.
when the greatnesses go directlyproportional, assemble the proportion between the observed measurements and use the fundamental property of proportions to find the desired measurement.
Example: A car at 50 km/h travels 100 km. If this car were at 75 km/h, how many kilometers would it have covered in the same period of time?
50 = 75
100x
50x = 75·100
50x = 7500
x = 7500
50
x = 150 km.
Also, when the greatnesses go inverselyproportional, it will be necessary to invert one of the fractions of the proportion formed by them before applying the fundamental property of proportions.
Example: A car travels at a speed of 50 km/h and takes two hours to reach its destination. How many hours would this same car take if it were at 75 km/h?
assembling the proportion, we will have:
50 = 2
75 x
By increasing the speed, the time spent on the route should decrease, therefore, the greatnesses they are inverselyproportional. Inverting one of the fractions, we will have:
50 = x
75 2
Applying the fundamental property of proportions, we will have:
75x = 50·2
75x = 100
x = 100
75
x = 1.33
This means that the time taken will be one hour and 20 minutes. (1.33 h is in decimal base, so it needs to be converted to hours, which can also be done by rule of three).
By Luiz Paulo Moreira
Graduated in Mathematics
