It is said that two measurements are proportional when one can pass from the one to the other by multiplying by a constant called proportionality factor .

Example: If, in a store, the price of apples is of 2 euros kg, there is proportionality between the price to pay and the bought apple weight. The proportionality factor is 2.

• For 1 kg, one must pay two euros.
• For 3 kg, one must pay 6 euros.
• For 1,5 kg, one must pay 3 euros.

It is noticed that the quotient of the two quantities is constant and is equal to the proportionality factor.

$\backslash \; frac\; \{2\}\; \{1\}\; =\; \backslash \; frac\; \{6\}\; \{3\}\; =\; \backslash \; frac\; \{3\}\; \{1,5\}\; =\; 2$

The Old ones as Euclide would have written that 2 is to 1 as 6 is to 3 or as 3 is to 1,5.

Table of proportionality

It is a table where one of the lines is proportional to the different one.

One can add a column to a table of proportionality by adding two columns, or by multiplying a column by a constant.

• 3 + 1,5 = 4,5 and 6 + 3 = 9 thus

• 3 × 2 = 6 and 6 × 2 = 12 thus

If two columns are chosen, the product of the numbers located in a diagonal is equal to the product of the numbers located in the other diagonal ( produces in cross ) 3 × 3 = 6 × 1,5

Fourth proportional

The fourth proportional one is the fourth number to be put in a table of proportionality of which 3 boxes are already filled. This fourth number is obtained by making the product of the numbers located on the same diagonal and while dividing by the third number.

This technique was called a long time the Rule of three.

Example: it takes 6 hours to write a ratio of 33 pages, how much hours is necessary it for a ratio of 55 pages?

Table of proportionality: Answer: $\backslash \; frac\; \{6\; \backslash \; times\; 55\}\; \{33\}\; =\; 10$ hours

Proportionality and geometry

The proportionality in geometry is mainly used in the Théorème from Thalès and in the similar triangles. But one also finds it in the coordinates of vectors colinéaires. In dimension 2, the equality of the products in cross ab' = ba' becomes ab' then - ba'= 0 (determining no one)

Chart

Is so there proportional to X, the points of coordinates (X; there) all are located on the same line passing by the origin. The proportionality factor, real K such as there = k×x is called slope or directing coefficient of the right-hand side. It is also said that there linear function is X.

Quantities inversely proportional

Two quantities are inversely proportional, if one is proportional contrary to the other.

To traverse 100 km, time is inversely proportional to the speed.

• with 100 km.h^-1, one needs 1:00
• with 50 km.h^-1, one needs 2:00
• with 10 km.h^-1, one needs 10:00

It is noticed that, if two quantities are inversely proportional, their product is constant

• 100 × 1 = 50 × 2 = 10 × 10

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