We will learn about what is variation, direct variation, indirect variation and joint variation.
In Mathematics, we usually deal with two types of quantities-Variable quantities ( or variables ) and Constant quantities ( or constants). If the value of a quantity remains unaltered under different situations, it is called a constant. On the contrary, if the value of a quantity changes under different situations, it is called a variable.
For example: 4, 2.718, 22/7 etc. are constants
while speed of a train, demand of a commodity, population of a town etc. are variables.
In problems relating to two or more variables, it is seen that the value of a variable changes with the change in the value ( or values ) of the related variable (or variables). Suppose a train running at a uniform speed of v km./h. travels a distance of d km. in t hours. Obviously, if t remains unchanged then v increases or decreases according as d increases or decreases. But if d remains unchanged, then v decreases or increases according as t increases or decreases. This shows that the change in the value of a variable may be accompanied differently with the change in the values of related variables. Such relationship with regards to the change in the value of a variable when the values of the related variables change, is termed as variation.
We will discuss about such variations, which are classified into three types:
(1) Direct Variation
(2) Inverse Variation and
(3) Joint Variation.
If two variables A and B are so related that when A increases ( or decreases ) in a given ratio, B also increases ( or, decreases ) in the same ratio, then A is said to vary directly as B ( or, A is said to vary as B ). This is symbolically written as, A ∝ B (read as, ‘A varies as B’ ). Suppose a train moving at a uniform speed travels d km. in t minutes. Now, consider the following table:
A variable quantity A is said to vary inversely as another variable quantity B, when A varies as the reciprocal of B i.e., when A varies as 1/B
Thus, if A varies inversely as B, we write A ∝ 1/B or, A = m ∙ (1/B) or, AB = m where’m (≠ 0) is the constant of variation. Hence, if one variable varies inversely as another, then the product of the corresponding values of the variables is constant.
Conversely, if AB = k where A and B are variables and k is a constant, then A ∝ 1/B Hence, if the product of the corresponding values of two variables is constant, then one quantity varies inversely as another.
Again, if A varies inversely as & then AB = constant ; but AB = constant implies that when A increases in a given ratio, B decreases in the same ratio and vice -- versa. Thus, if two variables are so related that an increase (or decrease ) in the value of one variable in a certain ratio corresponds to a decrease (or increase) in the same ratio in the value of the other variable then one variable varies inversely as another.
Let a m. and b m. be the length and breadth r area 160 sq. m. Now, consider the following table:
One variable quantity is said to vary jointly as a number of other variable quantities, when it varies directly as their product. If the variable A varies directly as the product of the variables B, C and D, i.e., if.A ∝ BCD or A = kBCD (k = constant ), then A varies jointly as B, C and D.
We know, area of a triangle = ½ × base × altitude. Since ½ is a constant, hence area of a triangle varies oint1y as its base and altitude.
A is said to vary directly as B and inversely as C if A ∝ B ∙ 1/C or A = m ∙ B ∙ 1/C (m = constant of variation) i.e., if A varies jointly as B and 1/C.
If x men take y days to plough z acres of land, then x varies directly as z and inversely as y.