Rule of three

The rule of three makes it possible to solve many problems relating to of the phenomena proportional.

Explanation

The principle of the rule of three consists with to be reduced to the unit .

Let us take an example:

The question that we wish to solve is:

So to manufacture 5 objects it takes 7 work hours, how much hours is necessary it to manufacture 8 objects?

• Let us determine the time necessary with the production of an object:

In 7 hours, 5 objects are manufactured. Donc the manufacture of 1 object lasts $\backslash \; frac\; 75$ work hours (5 times less time).

• We can thus deduce the time necessary with the production from it from 8 objects:

So for 1 object one needs $\backslash \; frac\; 75$ hours, then for 8 objects one needs 8 times more time is $\backslash \; frac\; 75\; \backslash \; times\; 8$ work hours.

The term of Rule of three comes owing to the fact that it utilizes 3 numbers (here 5,7,8). The installation of a rule of three requires a rigorous drafting to place these three numbers in the final fraction. This drafting can be advantageously replaced by a table of proportionality. Moreover, the use of such a table makes it possible to use the equality of the product in cross (equality of the product of the diagonals).

That is to say $x$ the time of manufacture of 8 objects: :

Here, one passes from the first column to the second column while dividing by 5, then second column with the third column while multiplying by 8.

the missing number is thus $x\; =\; 11,2\; H\; =\; 11:12\; min$.

The table of proportionality makes it possible to still shorten the reasoning by mechanizing calculation. One can directly find the result in this way:

The end result is obtained by carrying out the product of the two terms of a diagonal and while dividing by the remaining term.

$x\; =\; \backslash \; frac\; \{7\; \backslash \; times\; 8\}\; \{5\}$.

It is in this form that it is now presented in France.

Examples

Example 1

The price of apples of 5 € kg, I 1,5 kg of them, how much have will I buy am to pay?

$x\; =\; \backslash \; frac\; \{1,5\; \backslash \; times\; 5\}\; \{1\}\; =\; 7,5$

Therefore, I will have to pay 7,5 €.

Example 2

One has a plan whose scale indicates that: 2 cm $\backslash \; hat\; =$ 15 km.
One wants to know the distance between 2 as the crow flies villes.
For that, one measures on the level the distance between the points indicating the geographical place of these cities. 12,2 cm.
is found

According to the rule of three, one carries out this small calculation:

$x\; =\; \backslash \; frac\; \{12,2\; \backslash \; times\; 15\}\; \{2\}\; =\; 91,5$

Thus the distance on the ground is: 91,5 km.

Extensions

Rule of three opposite

There are sizes which decrease with proportion of an increase in data. For example, if one how long asks in 10 workmen build a certain wall which 15 workmen could brick up in 12 days, one considers that it is necessary, to build such a wall, a work equal to

$W\; =\; 15\; \backslash \; times\; 12\; =\; 180$ hommes×jours;

work which is, on the whole, independent of the number of men or the serviceable time, but depends only on the size of the wall. Thus, required time T must be such as: $W\; =\; 10\; \backslash \; times\; T\; =\; 180$ thus $t\; =\; 18$ days. In short, the rule of three is written in this case: $T\; =\; \backslash \; frac\; \{15\; \backslash \; times\; 12\}\; \{10\}\; =\; 18$

Rule of three made up

One encounters sometimes problems of proportion utilizing two connected “rules of three”, or even more. Here is an example:

18 workmen working at a rate of 8 hours per day paved in 10 days a long street of 150 Mr. One asks how much workmen are needed working 6 hours per day to pave in 15 days a 75 m long street, of the same street width than the preceding one.

Lagrange proposes the following rule: If a quantity increases at the same time, in the proportion that one or more other quantities increase, and that other quantities decrease, it is the same thing as if it were said that the quantity suggested increases like the product of the quantities which increase at the same time, divided by the product of those which decrease at the same time.

In the example which one has just given,

• it is necessary more workmen if it took more workmen or a greater day laborer duration of work, or more days to carry out a certain length of road, or if the length of street to be paved increases;
• it is necessary less workman if the day laborer duration of work increases, if the number of days granted to do the work increases, or if a certain work made it possible to make a bigger length of road.

Thus the number NR of workmen sought is given by: $N\; =\; \backslash \; frac\; \{18\; \backslash \; times\; 8\; \backslash \; times\; 10\; \backslash \; times\; 75\}\; \{6\; \backslash \; times\; 15\; \backslash \; times\; 150\}$

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