2.18 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICSAlgebra
2.3 VARIATION
DIRECT VARIATION :
If two variable quantities A and B be so related that as A changes B also changes in the same ratio, then A
is said to vary directly as (or simply vary) as B. This is symbolically denoted as A ∝B (read as A varies as B)
The circumference of a circle = 2 r, so circumference of a circle varies directly as the radius, for if the radius
increases (or decreases), circumference also increases or decreases.
From the above definition, it follows that :
If A varies as B, then A = KB, where K is constant ( 0)
Cor.^ :^ A^ B,^ then^ B∝^ A.^ If^ A^ ∝^ B,^ then^ A^ =^ kB.^ or,^ i.e., B ∝A.
Inverse Variation :
A is said to vary inversely as B, if A varies directly as the reciprocal of B. i.e. if A.
From A ∝ , we have A =K.^1
B or, AB = K, K is constant.
A =1
Bk. implies that, as B increases, A decreases ; or, as B decreases, A increases.
For example, for doing a piece of work, as the number of workers increases, time of completing the work
decreases and conversely. Similarly, the time of travelling a fixed distance by a train varies inversely as the
speed of the train.
Joint Variation :
A is said to vary jointly as B,C,D,....... If A varies directly as the product of B, C, D, ....... i. e., if A ∝ (B. C. D
.......). From A ∝ (B. C. D. .....) it follows A = K (B.C.D. .....), K is constant.
For example, the area of a triangle base × altitude. So it follows that area varies jointly as the base
and altitude. Similarly, the area of a rectangle varies jointly as its length and breadth (note, area = length
× breadth)
If again A varies directly as B and inversely as C, we have
A∝CB or, A K , K= CB is constant.
For example, altitude of a triangle varies directly as the area of triangle and inversely as the base (since∆ = 21
a. h. So h =^2 a∆, where ∆ indicates area)
For example, the area of a triangle (∆) varies as the base (a) when height (h) is constant and again D varies
as height when base is constant. So ∆ ∝ a. h when both a and h vary.
Some Elementary Results :
(i) If A ∝ B, then B ∝ A (ii) If A ∝ B and B ∝ C, then A ∝ C
(iii) If A ∝ B and B ∝ C, then A – B ∝ C
(iv) If A ∝ C and B ∝ C, then A – B ∝ C
(v) If A∝ C and B ∝ C, then AB C.∝
(vi) If A ∝ B, then An ∝ Bn.∝
1
B
≠
B =
A
k∝