Answer:
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. We have direct variation if the two variables
change in the same sense, i.e. if one increases, so does the other.
We have indirect variation if one going up causes the other to go
down. 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. 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.
(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. 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. 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: 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. 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.
