Variation - direct and inverse proportion

Direct Proportion

If two quantities exist such that the increase in one leads to an increase in the other, they are in direct proportion.

E.g. The speed of a car and the distance covered during a certain time - the greater the speed , the greater the distance.

E.g.1

The cost of five books is £20. Find the cost of 7 books. How many books can someone buy for £36?
5 books ----------> £20
1 book -----:-5----> £4
7 books ----X4------> £28
Now, reverse the calculation as follows:
£20 ----------> 5books
£1 -----:-20----> 5/20 book
£36 ----X36------> 9 books

Inverse Proportion

If two quantities exist such that the increase in one leads to the decrease in the other, they are said to be in inverse proportion.
E.g.
The speed of a car is in inverse proportion to the time taken for a certain journey.

E.g.2

If a car moves at 20 mph, it covers a distance in 6 hrs. How long will it take to cover the same distance at 30 mph? It the time taken for the same journey is 8 hrs, calculate the speed.
20 mph ----------> 6 hrs
:-20 ----------> X 20
1 mph --------->120 hrs
X 30 ----------> :- 30
30 mph ----------> 4 hrs
Now, reverse the calculation as follows:
6 hrs ----------> 20 mph
:-6 ----------> X 6
1 hour --------->120 mph
X 8 ----------> :- 8
8 hrs ----------> 15 mph

Direct Variation

When one quantity increases, if another quantity increases with that, it is called direct variation.
E.g.
The length of a rubber band (l) and the force that stretches it (F), are in direct variation.
l ∝ F
l = kF, where k is a constant.

Please move the slider and experiment with it.

l ∝ F

E.g.3

The length of a rubber band, when stretched by 20N is 6cm. Find the length, when the force is raised to 60N. Find the Force that keeps the length at 15cm.
l ∝ F
l = kF
When l = 6 and F = 20,
6 = k x 20
k = 6/20
When F = 60,
l = 6/20 x 60 = 18
F = 18 N.
When l = 15,
15 = 6/20 x F
F = 50N.

E.g.4

The power of a certain device (P) is directly proportional to the square of the current through it. The power is 60W, when the current is 2Amps. Find the power, when the current is 3Amps. Find the current when the power is 90W.
P ∝ I²
P = k I²
When I = 2 and P = 60,
60 = K X 4
k = 15
When I = 3,
P = 15 x 9
P = 135W.
When P = 90,
90 = 15 * I²
I = 2.3A.

Inverse Variation

When one quantity increases, if another quantity decreases with that, it is called inverse variation.
The volume of a fixed mass of gas is inversely proportional to the pressure.
V ∝ 1/P
V = k1/P

Please move the slider and experiment with it.

P ∝ 1 / V

E.g.5

The volume of a gas container is 3cm³ when pressure is 12Pa. Find the volume, when the pressure is 36N. At what pressure, will the volume be 9cm³?
When P = 12 and V = 3,
3 = k x 1/12
k = 36
When P = 36,
V = 36 x 1/36 = 1
V = 1cm³
When v = 9,
9 = 36 x 1/P
P = 4Pa.

E.g.6

The force between two magnets is inversely proportional to the distance between them. When the distance is 3cm, the force is 12N. Find the force, when the distance is 2cm. Calculate the distance, when the force is 432N as well.
F ∝ 1/d²
F = k 1/d²
When d = 3 and F = 12,
12 = k x 1/9
k = 108
When d = 2,
F = 108 x 1/4
F = 27N.
When F = 432,
432 = 108 x 1/d²
d² = 1/4
d = 1/2cm.

Now, in order to complement what you have just learnt, work out the following questions:

1. The distance d through which a feather falls from rest is proportional to the square of the time taken t. if the feather falls 15m in 3 seconds, how far will it fall in 9 seconds? How long will it take to fall 45m?
2. The energy E stored in a square-plated capacitor varies as the square of the length of the side x. When the length is 3cm, the energy stored is 81 Joules. What is the energy stored when the length is 5cm? What is the length when the stored energy is 243Joules?
3. The length L of an embryo of a certain mammal is proportional to the square root of the number of hours left before its birth. If the embryo is 12 cm long before 2 hours, how long will it be after 8 hours? How long will it take to grow to a length of 18cm?
4. p is directly proportional to q. Fill in the banks.
   

   p	q
   x	60
   4	12
   1/3	y

5. L is directly proportional to (M - 2). L is 36 when M is 4. Find L when M = 10. Find M when L = 108.
6. The kinetic energy of an object is proportional to the square of its speed. The kinetic energy is 64J when the speed is 4m/s. Find the kinetic energy when the speed is 20m/s. At which speed will the Kinetic energy be 900J?
7. The frictional force against the motion of a car is proportional to the square of the speed of the car. If this force is 16000N at 10 m/s, what is the force at 30 m/s? At what speed is the force equal to 900N?
8. The force of attraction between any two stars is inversely proportional to the square of the distance between them. The force between two, when they are 4 light years apart is 24x10⁶⁴N.Find the force when the distance is 3 light years. If the force is 48x10⁶⁴N, find the distance between them too.
9. In the following table, p is inversely proportional to q. Fill in the blanks.
   

   p	q
   x	60
   4	3
   1/3	y

10. y is inversely proportional to (x - 3). y = 4 when x = 5. Find y when x = 18. Find x when y = 144.