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Monday, July 22, 2013

Direct and Inversely Proportional


Proportionality:

A quantity is said to be proportional to another quantity if change of one of the quantities is always accompanied by the change of the other. This property is known as proportionality.

Proportionality is of two types:

(i)                  Direct proportionality

(ii)                Inverse proportionality

Direct Proportionality:

A quantity is said to be directly proportional to another quantity if the change in both of them is in the same direction. This means that if one of the quantities increases then the other also increases. If one of the quantities decreases then the other also decreases.

It is denotes by the symbol a. If ‘a’ is directly proportional to ‘b’ then:

We write as a a b ==> a = k * b where k is the proportionality constant.

i.e.  a / b = k = constant à a1 / b1 = a2 / b2

Graph:

Let us consider that x a y. If we plot the values of x and y on a graph sheet we obtain the graph showing the relation between these two quantities. Generally, the graph of directly proportional quantities is a straight line. Thus by seeing the graph we can conclude the relation and proportionality between two quantities.

Inversely proportionality:

A quantity is said to be inversely proportional to another quantity if the change in both of them is in the opposite direction. This means that if one of the quantities increases then the other quantity decreases. If one decreases then the other increases.

Inverse proportionality is also uses the symbol a but the reciprocal of the second quantity is written.

If ‘a’ is inversely proportional to b then:

We write as a a 1 / b

You can see that the reciprocal of b is written to indicate inverse proportionality. We can also say that ‘a’ is directly proportional to the reciprocal of ‘b’.

If a a 1 / b à a = k / b where k is the proportionality constant.

i.e. a * b = k = constant à a1 * b1 = a2 * b2

Graph:

Let us consider x is inversely proportional to y i.e. x a 1 / y

Now plot the values of x and y on the on a graph sheet we obtain the graph showing the relation between the two quantities. Generally the graph is not a straight line but a curve. On seeing the graph we can analyze the relation and proportionality between the two quantities.

Problem:

If the volume of a gas at a given temperature is 2 liters when its pressure is 1 bar, then what will its volume when the pressure increases to 3 bars? (Volume in inversely proportional to pressure)

Sol: Given, initial volume v1 = 2 l

Initial pressure p1 = 1 bar

Final pressure p2 = 3 bars

As volume is inversely proportional to pressure we have, p1 v1 = p2 v2

Now v2 = p1 v1 / p2 = (2 * 1) / 3 = 0.66 liters (approx.)

Thus final volume the gas is 0.66l

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