Function of inverse proportionality y = m/x

Next to the problems of proportionality, you have those of inverse proportionality, in which not the quotient but the product of corresponding values is constant.

Task: If you spend daily 12 Baht, your money lasts 20 days. How long would it last, if you spend daily half that amount (6 Baht) or twice that much (24 Baht)?

12 Baht (x₁) spent daily, your money lasts 20 days (y₁):
x₁·y₁=12·20=240=m.

6 Baht (x₂) spent daily, your money lasts 40 days (y₂) - twice as long: x₂·y₂=12·20=240=m,

24 Baht (x₃) spent daily, your money lasts 10 days (y₃) - half as long: x₃·y₃=24·10=240=m, etc.

The functional equation is xy = m or y = m/x.

Given the function xy = m, the product of two corresponding values is constant, and inversely, if the product of two corresponding values is constant, one has the functional equation xy = m. Once m has been determined from given data (in the above example 240), the function yields all corresponding number pairs.

Frequently, you will not seek the value of m, but look for a second pair of numbers; you proceed then as follows:

x₁·y₁ = x₂·y₂ = m yields: y₁/y₂ = x₂/x₁

_{You write down this proportion and compute from three given values the fourth proportional.}

_{A comparison of} y₁/y₂ = x₂/x_{1 with} y₁/y₂ = x₁/x_{2 shows that the sequence of the terms on one side is inverted. Hence you say that the two y values behave inversely to the corresponding x values or y is inversely proportional to x.}

In direct proportionality, the quotient of corresponding values is constant. The functional equation is y = mx.

In inverse proportionality, the product of corresponding values is constant. The functional equation is y = m/x.

In both cases, the factor of proportionality is m.

Just as for tasks with direct proportionality, there arise different modes of computation. You can solve tasks of inverse proportionality by

a) computation, b) proportions, c) the function method, d) graphics.

We will treat the graphical method later on and confine ourselves here to the solution of a problem by the methods a) to c).

Example: If 400 litres are used daily, the freshwater supply of a ship lasts 90 days. How much water can be used daily, if the supply need only last for 80 days?

Obviously, we are concerned with inverse proportionality.

a) Solution by computation: During 90 days, 400 litres can be used daily, in 1 day, 90 times that much, that is 36000 litres, during 80 days 36000/80 = 450 litres.

b) Solution by proportion: The number of 100 litres (y) is inversely proportional to the number of days (x),

that is, y/4 = 90/80, that is y = 90·400/80 = 450 litres.

c) Solution by functional method: Since you are concerned with inverse proportionality, the equation is y = m/x. You must determine m from a pair of corresponding values. For x = 400 and y = 20, 90 = m/400 or m = 36000, whence the equation is y = 36000/x. For x = 80, y = 36000/80 = 450.

Graphical representation of the function y = m/x.

Set m = 1 and consider the function y=1/x. You have inverse proportionality. In order to obtain an image of the function, set up a table. The curve you obtain by graphing corresponding values is called a hyperbola. It has two parts - branches.

Since you must use tables for each value of m to draw these hyperbolae, as a rule, you not do so graphically because it is too laborious.

Whenever two variables are related by a proportion, you must always make sure whether it is direct or inverse - whether the function is y = mx or y = m/x. You must also take into account its range of validity.

Frequently, proportionality is assumed to be valid when it is not! 20 workers build a villa during three months, 60 workers during a third of this time = one month, 1800 workers during a thirtieth of a month, - one day. If you were to employ even more workers you could erect the villa in one second. In this case, more complicated functional relations hold instead of simple inverse proportionality.

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