1. UNIT
Variation
F.Y.B.Com
Prof.P.A.Navale
Dept. of Commerce

2. Variation Direct Variation:
lf two variables x and y are so related that when x increases (or decreases) in a certain ratio. y also increases (or decreases) in the same ratio. Then x is said to vary directly as y, and expressed as x α y. In such a situation we sometimes say that x varies as y or x is proportional to y.
If x1, x2,x3…be the values of x and y1, y2, y3… Be the corresponding values of y, then x α y means,
So, x α y means,
And we can write x/y = k, where k is constant.

3. Variation
Example:
a) A varies directly as b, and when b = 2, a = 7. What is the value of a when b = 10?
Solution:
The value of b has gone from 2 to 10. That is, it has increased five times. Therefore, the value of a will increase five times. It will go from 7 to 35.
Formally,
a2 : a1 = b2 : b1.
a2 : 7 = 10 : 2.
10 is five times 2. Therefore, a2 will be five times 7.
b) A varies directly as b, and when b = 8, a = 4. What is the value of a when b = 13?
Proportionally,
That is,
a2 : 4 = 13 : 8.
Alternately, 4 is half of 8. Therefore, a2 will be half of 13, which is 6.5.

4. Variation 2) Inverse variation:
If a variable x varies as the reciprocal of another variable y (i.e., x α 1/y) then we say that x varies inversely as y. In case of inverse variation when x increases (or, decreases) in a certain ratio, y decreases (or, increases) in the same ratio.
Example:
Now x α 1/y, implies x = k. 1/y or, x y = k (here k is the constant of variation)

5. Variation 3) Joint Variation:
If a variable x varies as the product of a finite number of variables u, v, w, etc., (i.e., x α uvw ...), then x is said to vary jointly with u, v, w, etc.
Here we state an important theorem, known as theorem on joint variation.  If x α y, when z is constantan x α z, when y is constant. then x α y z, when both y and z vary.
B) Some Important Results:
a) If x α y then y α x.
b) If x α y and y α z, then x α z.
c) If x α y, then x n α y n, where n is a constant
d) lf x α z and y α z, then x+y α z, x - y α z and x y α z2
e) If x α y and w α z, then x w α y z and x/w α y/z.

6. THANK YOU