Direct & Inverse Variation Problems.

Example 1.

If W varies directly with F and when W = 24 , F = 6 . Find the value W when F = 10.

Solution.

FW

kFW When W = 24, F = 6

624 k

246 k4k

FW 4

When F = 10 W = ?

104W

40W

Example 2.

If g varies directly with the square of h and when g = 100 , h = 5 . Find the value h when g = 64.

Solution.2hg

2khg When g = 100 , h = 5

25100 k10025 k

4k

24hg

When g = 64 , h = ?

2464 h

644 2 h

162 h16h

h = 4 or h = - 4

Example 3.

If d varies inversely with w and when d = 3 , w = 9 . Find the value d when w = 3.

Solution.

wd 1

wkd

When d = 3 , w = 9

93 k

k = 27

wd 27

When w = 3 d = ?

327

d

d = 9

Example 4 .

If r varies inversely with the square root of f and when r = 32 , f = 16. Find f when r = 32.

Solution.

fr 1

fkr

When r = 32 , f= 16.

1632 k

432 k

k = 128

fr 128

When r = 32 , f = ?

f12832

12832 f

32128

f

4f

16f

Example 5 .

If t varies jointly with m and b and t = 80 when m = 2 and b = 5. Find t when m = 5 and b = 8 .

Solution.

mbt kmbt

When t = 80 , m = 2 and b = 5

5280 k

8010 k8k

mbt 8

When m = 5 , b = 8 , t = ?

858 t320t

Example 6 .

c varies directly with the square of m and inversely with w. c = 9 when m = 6 and w = 2 . Find c when m = 10 and w = 4 .

Solution.

wmc

2

wkmc

2

When c = 9 , m = 6 and w = 2

269

2k

19

236

k

K = ½

wmc2

2

When m = 10 , w = 4 and c = ?

42102

c =12.5

Examination Questions. Example 1.

The time,T minutes ,taken for a stadium to empty varies directly as the number of spectators , S, and inversely as the number of open exits, E.

(a) Write down a relationship connecting T,S and E.

It takes 12 minutes for a stadium to empty when there are 20 000 spectators and 20 open exits.

(b) How long does it take the stadium to empty when there are 36 000 spectators and 24 open exits ?

Solution.

(a)EST

ESKT

K is the constant of variation.

(b) T = 12 , S = 20 000 and E = 20

Substitute to find the value of K.

2020000

112 K

Cross multiply.

20 000 K = 20 x 12

5006

20000240

K

EST

5006

Now S = 36 000 and E = 24 .

Substitute.

24500360006

T

T = 18 minutes

Example 2.

The number of letters, N , which can be typed on a sheet of paper varies inversely as the square of the size, S , of the letters used.(a) Write down a relationship connecting N and S .

(b) The size of the letters used is doubled.

What effect does this have on the number of letters which

can be typed on the sheet of paper ?Solution.

(a)2

1S

N

2SKN

(b) Letter size = 2S

2)2( SKN

24SKN

By doubling the size of letters the number of letters is quartered.

Example 3.

A frictional force is necessary for a car to round a bend. The frictional force , F kilonewtons , varies directly as the square of the car’s speed , V metres per second, and inversely as the radius of the bend, R metres.

(a) Write down a relationship between F, V and R.

A frictional force of 20 kilonewtons is necessary for a car , travelling at a given speed , to round a bend.

(b) Find the frictional force necessary for the same car , travelling at twice the given speed , to round the same bend.

Solution.

RVF

2

(a)

RKVF

2

(b) Let the speed = 2V

RVKF

2)2(

RVKF

24

RKVF

24

By doubling the speed the frictional force F required to round the bend becomes 4 times greater.

F = 4 x 20 = 80