Variation, in general, will concern two variables, say height and
weight of a person, and how when one of these changes, the other might
be expected to change. Â We have direct variation if the two variables
change in the same sense, i.e. if one increases, so does the other. Â
We have indirect variation if one going up causes the other to go
down. Â An example of this might be speed and time to do a particular
journey, so the higher the speed the shorter the time.
Normally we let x be the independent variable and y the dependent
variable, so that a change in x produces a change in y. Â For example,
if x is number of motor cars on the road and y the number of
accidents, then we expect an increase in x to cause an increase in y. Â
(This obviously ceases to apply if number of cars is so large that
they are all stationary in a traffic jam.)
When x and y are directly proportional then doubling x will double the
value of y, and if we divide these variables we get a constant result.
Since if y/x = k  then (2y/2x) = k  where k is called the constant of
proportionality
We could also write this  y = kx
Thus if I am given the value of x, I multiply this number by k to find
the value of y.
Example: Given that y and x are directly proportional and y = 2 when
x = 5, find the value of y when x = 15.
We first find value of k, using y/x = k
                2/5 = k
Now use this constant value in the equation y = kx for situation when
x = 15
   y = (2/5)*15
    = 30/5 = 6
If you want to do this quickly in your head, you could say, x has been
multiplied by a factor 3 (going from 5 to 15), so y must also go up by
a factor of 3. Â That means y goes from 2 to 6.
Indirect Variation.
We gave an example of inverse proportion above, namely speed and time
for a particular journey. Â In this case if you double the speed you
halve the time. So the product speed x time = constant
In general, if x and y are inversely proportional then the product xy
will be constant.
       xy = k
 or      y = k/x
Example: Â If it takes 4 hours at an average speed of 90 km/hr to do a
certain journey, how long would it take at 120 km/hr
  k = speed*time = 90*4 = 360 (k in this case is the distance)
Then time = k/speed
     = 360/120
     = 3 hours.
To do this in your head, you could say that speed has changed by a
factor 4/3, so time must change by a factor 3/4. Â However, for the
usual type of problem, go through the steps I outlined above.
Step-by-step explanation:
I hope these examples have made the idea of variation (both direct and
inverse) reasonably clear.