COMPOUND PROPORTIONALITY: EXPLANATION, COMPOUND RULE OF THREE, EXERCISES - MATHS - 2025

The composite or multiple proportionality is the ratio of over two magnitudes, which can be observed directly and inverse proportionality between the data and the unknown. This is a more advanced version of simple proportionality, although the techniques used in both procedures are similar.

For example, if 7 people are needed to unload 10 tons of merchandise in 3 hours, the compound proportionality can be used to calculate how many people it will take to unload 15 tons in 4 hours.

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To answer this question, it is convenient to make a table of values ​​to study and relate the magnitudes and unknowns.

We proceed to analyze the types of relationships between each magnitude and the present unknown, which for this case corresponds to the number of people who will work.

As the weight of the merchandise increases, so does the number of people required to unload it. Because of this, the relationship between weight and workers is direct.

On the other hand, as the number of workers increases, working hours decrease. Due to this, the relationship between people and working hours is of the inverse type.

How to calculate compound proportionalities

To solve examples like the one above, the compound rule of three method is mostly used. This consists of establishing the types of relationships between quantities and unknowns and then representing a product between fractions.

With respect to the initial example, the fractions corresponding to the table of values ​​are organized as follows:

But before solving and solving the unknown, the fractions corresponding to the inverse relationship must be inverted. Which for this case correspond to the variable time. In this way, the operation to solve will be:

Whose only difference is the inversion of the fraction corresponding to the time variable 4/3. We proceed to operate and clear the value of x.

Thus, more than eleven people are needed to be able to unload 15 tons of merchandise in 4 hours or less.

Explanation

Proportionality is the constant relationship between quantities that are subject to change, which will be symmetrical for each of the quantities involved. There are directly and inversely proportional relationships, thus defining the parameters of simple or compound proportionality.

Direct rule of three

It consists of a proportion relationship between variables, which show the same behavior when modified. It is very frequent in the calculation of percentages referring to magnitudes other than one hundred, where its fundamental structure is appreciated.

As an example, 15% of 63 can be calculated. At first glance, this percentage cannot be easily appreciated. But implementing the rule of three, the following relationship can be made: if 100% is 63, then 15%, how much will it be?

100% ---- 63

15% ---– X

And the corresponding operation is:

(15%. 63) / 100% = 9.45

Where the percentage signs are simplified and the figure 9.45 is obtained, which represents 15% of 63.

Inverse rule of three

As its name indicates, in this case the relationship between the variables is the opposite. The inverse relationship must be established before proceeding to the calculation. Its procedure is homologous to that of the direct rule of three, except for the investment in the fraction to be calculated.

For example, 3 painters need 5 hours to finish a wall. In how many hours would 4 painters finish it?

In this case, the relationship is inverse, since as the number of painters increases, the work time should decrease. The relationship is established;

3 painters - 5 hours

4 painters - X hours

As the relationship is reversed, the order of operation is reversed. This being the correct way;

(3 painters). (5 hours) / 4 painters = 3.75 hours

The term painters is simplified, and the result is 3.75 hours.

Condition

To be in the presence of a compound or multiple proportionality, it is necessary to find both types of relationship between magnitudes and variables.

- Direct: The variable has the same behavior as the unknown. That is, when one increases or decreases, the other is altered equally.

- Inverse: The variable has an antonym behavior to that of the unknown. The fraction that defines said variable in the table of values ​​must be inverted, in order to represent the inversely proportional relationship between variable and unknown.

Verification of results

It is very common to confuse the order of quantities when working with compound proportionalities, unlike what happens in the usual proportion calculations, whose nature is mostly direct and solvable by means of a simple rule of three.

For this reason, it is important to examine the logical order of the results, verifying the coherence of the figures produced by the compound rule of three.

In the initial example, making such a mistake would result in 20 as the result. That is, 20 people to unload 15 tons of merchandise in 4 hours.

At first glance it does not seem like a crazy result, but it is curious an increase of almost 200% in staff (from 7 to 20 people) when the increase in merchandise is 50%, and even with a greater margin of time to carry out the work.

Thus, the logical verification of the results represents an important step in implementing the compound rule of three.

Clearance

Although more basic in nature with respect to mathematical training, the clearance represents an important step in cases of proportionality. A wrong clearance is enough to invalidate any result obtained in the simple or compound rule of three.

History

The rule of three became known in the West through the Arabs, with publications by various authors. Among them Al-Jwarizmi and Al-Biruni.

Al-Biruni, thanks to his multicultural knowledge, had access to vast information regarding this practice in his trips to India, being responsible for the most extensive documentation on the rule of three.

He states in his research that India was the first place where the use of the rule of three became common. The writer assures that it was carried out in a fluid way in its direct, inverse and even composed versions.

The exact date when the rule of three became part of the mathematical knowledge of India is still unknown. However, the oldest document addressing this practice, the Bakhshali manuscript, was discovered in 1881. It is currently in Oxford.

Many historians of mathematics claim that this manuscript dates from the beginning of the present era.

Solved exercises

Exercise 1

An airline must carry 1,535 people. It is known that with 3 planes it would take 12 days to get the last passenger to the destination. 450 more people have arrived at the airline and 2 planes are ordered to be repaired to help with this task. How many days will it take the airline to transfer every last passenger to their destination?

The relationship between the number of people and days of work is direct, because the greater the number of people, the more days it will take to carry out this work.

On the other hand, the relationship between airplanes and days is inversely proportional. As the number of airplanes increases, the days needed to transport all passengers decrease.

The table of values ​​referring to this case is made.

As detailed in the initial example, the numerator and denominator must be inverted in the fraction corresponding to the inverse variable with respect to the unknown. The operation is as follows:

X = 71460/7675 = 9.31 days

To transfer 1985 people using 5 planes, it takes more than 9 days.

Exercise 2

A 25-ton corn crop is taken to the cargo trucks. It is known that the previous year it took them 8 hours with a payroll of 150 workers. If for this year the payroll increased by 35%, how long will it take them to fill the cargo trucks with a 40-ton crop?

Before representing the table of values, the number of workers for this year must be defined. This increased by 35% from the initial figure of 150 workers. A direct rule of three is used for this.

100% ---- 150

35% ---– X

X = (35,100) / 100 = 52.5. This is the number of additional workers with respect to the previous year, obtaining a total number of workers of 203, after rounding the amount obtained.

We proceed to define the corresponding data table

For this case, the weight represents a variable directly related to the unknown time. On the other hand, the workers variable has an inverse relationship with time. The greater the number of workers, the shorter the working day.

Taking these considerations into account and inverting the fraction corresponding to the workers variable, we proceed to calculate.

X = 40600/6000 = 6.76 hours

The journey will take just under 7 hours.

Proposed exercises

- Define 73% of 2875.

- Calculate the number of hours Teresa sleeps, if it is known that she only sleeps 7% of the total for the day. Define how many hours you sleep a week.

- A newspaper publishes 2000 copies every 5 hours, using only 2 printing machines. How many copies will he produce in 1 hour, if he uses 7 machines? How long will it take to produce 10,000 copies using 4 machines?

References

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3. Rational Approximation of Real Functions. PP Petrushev, Vasil Atanasov Popov. Cambridge University Press, Mar 3. 2011.
4. Elementary arithmetic for teaching in schools and colleges in Central America. Darío González. Tip. Arenales, 1926.
5. The Study of Mathematics: On the study and difficulties of mathematics. Augustus De Morgan. Baldwin and Cradock, 1830.