Sec 3 Mathematics Worksheet (1) : Variation
Name : ______________________ (
) Class (
)
Date : ___________
The table below shows a relationship between the number of rooms cleaned and the cost
of the total job for 1 through 5 rooms.
Number of Rooms
1
2
3
4
5
Cost of the Job
$20
$40
$60
$80
$100
(a) As the number of rooms increases, the total cost of the job
________________.
As the number of rooms doubled, the total cost is also
________________.
As the number of rooms halved, the total cost is also
________________.
(b) Compute the ratio for each pair of corresponding values.
e.g
20

1
,
40

2
,
What do you notice?
The ratio of the two corresponding variables is a __________.
6
(c) Plot the ordered pairs on the
5
grid provided.
Is it possible to draw a line
passing through all the
4
3
points?
2
What do you notice?
The graph which represents
1
the data is a _____________
passing through
-40the ________.
-20
20
40
60
80
-1
The _____________ of the line is equal to the ratio between the variables.
-2
The above relationship between the number of rooms and the cost is an
example of -3
________________ variation.
-4
100
1
The table below shows the time taken by a car to travel a distance of 120 km at various
speeds.
Speed, x (km/h)
20
30
40
60
120
Time taken, y (in hours)
6
4
3
2
1
(a) As the speed increases, the time taken is
______________.
Pg 24 New Syllabus D Mathematics 3
As the speed is doubled, the time taken is ______________.
As the speed is halved, the time taken is
______________.
(b) Compute the product for each pair of corresponding values. What do you notice?
140
The product of the two corresponding variables is a __________.
(c) Plot the ordered pairs on the grid
provided.
120
What do you notice?
The graph which represents the
100
data is a ___________________
1
Now compute the values of
for
y
each of the values in our table.
80
60
1
.
y
40
1
is180a
y
__________________but not passing
160
20
Plot a graph of speed x against
The graph of x against
through the ____________.
-2
2
4
6
140
-20
120
This relationship between
the speed and time taken
is an example of
100
80
60
________________
40
variation.
20
-0.2
0.2
-20
0.4
0.6
0.8
1
Sec 3 Mathematics Worksheet (2) : Variation
Name : ______________________ (
) Class (
)
Date : ___________
(A) Direct Variation
For any two variables x and y, where y varies directly as x or y is directly proportional to x
the following properties exist :
(a) as one variable increases, the other variable also increases.
(b) the ratio between any pair of corresponding values is constant and equals to k, the
constant of proportionality.
The notation used to express y varies directly as x is given by y  x or y  kx .
(c) the graph which represents the variables x and y, is a straight line passing through the
origin with gradient equal to the constant of proportionality.
1
For the given data establish whether direct variation exists between x and y using :
(a) a numerical method,
(b) a graphical method.
x
y
2
3
4
6
(a) Numerical Method
8
12
10
15
(b) Graphical Method
12
10
8
6
4
2
-10
-5
5
-2
2
Pg 149 Math Quest 11 General Mathematics
If n varies directly as the square root of
-4m, find the missing values in the table below.
m
n
1
4
1
2
-63
4
-8
3
10
25
1
Pg 160 Math Quest 11 General Mathematics
Which of the following represents direct variation between x and y?
-10
-12
A
B
C
D
E
Pg 153 Math Quest 11 General Mathematics
4
The area of a circle, A, varies directly as the square of its radius r. Sketch graph,
(a) to show the relationship between the area and its radius
(b) to show the relationship between the area and the square of its radius.
5
Kepler’s law states that the square of the time taken for a plant to revolve around the
sun once varies as the cube of the distance of the planet from the sun. It is known
that the distances from the Earth and Venus to the sun are 150 and 108 million km
respectively. Find the time taken for Venus to rotate once around the sun if the time
taken for the Earth to revolve once around the sun is 365 days. Give your answer to
the nearest whole number.
Pg 22 New Syllabus D Mathematics 3 [223 days]
6
Given the table of values for S and T, write down a formula expressing S in terms of T
for these values :
T
0
1
2
3
S
0
7
28
63
Pass GCE “O” Level Examination Elementary Mathematics
(B) Inverse Variation
For any two variables x and y, where y varies inversely as x or y is inversely proportional to x
the following properties exist :
(a) as one variable increases, the other variable decreases.
(b) the product of any pair of corresponding values is constant and equal to k, the constant
of proportionality.
k
The notation used to express y varies inversely as x is given by y  1 or y  .
x
x
(c) the graph which represents the variables x and y, is a hyperbola.
(d) neither variable is equal to 0.
7
For the data represented in the table below, establish whether an inverse variation
exists between x and y using :
(a) a numerical method,
(b) a graphical method.
x
y
2
12
(a) Numerical Method
3
8
4
6
6
4
(b) Graphical Method
12
10
8
6
4
2
8
-2inversely as the square of its diameter, d.
The electrical resistance, R, of a wire varies
(a) What happens to the resistance when the diameter of the wire is doubled?
(b) If a wire with diameter of 4 mm has the-4
resistance of 4 ohms. Find
(i) the resistance of a wire with diameter 1.2 cm,
(ii) the diameter of the wire when the resistance
is 8 ohms.
-6
-8
-10
Pg 172 Math Quest 11 General Mathematics []
-12
9
In an experiment, a drug is added to identical flasks each containing the same
amount of a bacteria. The drug is allowed to react with the bacteria for various times,
t hours. It is found that the amount of bacteria left, s units, varies inversely as
 t  2 hours. In one flask, there were 6 units of bacteria left after 5 hour, calculate
the amount of units of bacteria left in another flask after 11 hours.
Pg 26 New Syllabus D Mathematics 3 [2 units]
10 The expression y 
A
1
could be represented by :
x
B
C
D
E
Pg 169 Math Quest 11 General Mathematics
11 The force of attraction between two magnets is F Newtons. This force is inversely
proportional to the square of the distance, d centimeters, between the magnets.
(a) Sketch a graph to show the relation between the force and the distance.
(b) When the magnets are a certain distance apart, the force is 10 Newtons.
What is the force when this distance is doubled?
(c) (i) Write down a formula connecting F, d and a constant k.
(ii) When the magnets are 3 cm apart, the force is 2 Newtons.
Find the force when they are 5 cm apart.
Q19 P1 NOV02 UCLE [ (b) 2.5 N, (c) 0.72 N ]
12 The weight of an object above the earth varies inversely as the square of its distance
from the centre of the earth. A space vehicle in an elliptical orbit has a maximum
distance from the centre of the earth (apogee) of 10800 km. Its minimum distance
from the centre of the earth (perigee) is 6600 km. If an astronaut in the vehicle
weighs 26 kg at its apogee, what does the astronaut weigh at the perigee?
Pg 351 Mathematical Ideas [70kg]
13 A tank is filled up with water by 7 taps. If an additional tap is used, the tank can be
filled up 5 minutes faster. Given that the time required to fill up the tank is inversely
proportional to the number of taps used, how many additional taps must be used in
order for the tank to be filled up 12 minutes faster.
Pass GCE “O” Level Examination Elementary Mathematics
(C) Joint Variation
In real life there are many situations that involve more than two variables.
In such situations, where one variable is directly proportional to the product or quotient of other
variables, we say that joint variation takes place.
For example, if one quantity y, varies directly as the product of two other quantities, x and z, it is
said that y varies jointly as x and z and is written as
y  xz or y  kxz .
14 The variable m varies directly as the square root of n and inversely as the square of
p. When n = 16 and p = 3, then m = 2.4.
Write the equation which describe the relationship between m, n and p.
Pg 181 Math Quest 11 General Mathematics
15 Write the equation of variation for each of the following .
(a) Power, P, varies directly as the square of the voltage, V , and inversely as the
resistance, R.
(b) Power, P, varies jointly as the resistance, R, and the square of the current, I.
(c) Kinenetic energy, E, is directly proportional to the square of the velocity, v and
the mass m.
(d) Force of a circular motion, F, varies directly as the mass, m, and inversely as the
radius, R.
(e) Frequency of a sound, f, varies directly as the square root of the tension, t and
inversely as the square root of the mass per unit length, m.
Pg 180 Math Quest 11 General Mathematics
16 Coulomb’s Law states that the force between two charges at rest, F, is directly
proportional to the product of the charges q and q , and inversely proportional to
2
1
the square of the distance between the charges, d .
(i) Write the equation which represents this relationship.
(ii) What effect will the following changes have on the size of the force, F?
(a) The distance between the charges is doubled.
(b) The distance between the charges is halved.
(c) One of the charges is doubled.
(d) Both charges are doubled.
Pg 183 Math Quest 11 General Mathematics
17 The electrical resistance, R, of a wire varies jointly as its length, l and the reciprocal
of the square of its diameter, d. Find the percentage change in the resistance if the
length of the wire is increased by 30% and its diameter is decreased by 15%.
Give your answer to 2 decimal places.
Pg 183 Math Quest 11 General Mathematics
[79.93%]
(D) Part Variation
Relationships may also consist of two or more parts added together. In this situation,
we say that part variation takes place. When part variation occurs, each of the parts will
have its own constant of variation.
If the relationship between two variables x and y is such that y varies partly as x and is
partly constant, it is written as y  ax  b and is called part linear variation.
18 The total cost of the electricity supplied to a house is found by adding two charges.
These are – a fixed standing charge and a charge for each unit of electricity used.
The total cost of 100 units is $22 and other costs are given in the table below.
Number of units
100
200
500
1000
used
Total cost ($)
22
27
42
67
(i)
Find the fixed standing charges.
(ii)
The total cost when n units are used is C dollars.
Write down the formula for C in terms of n.
Pass GCE “O” Level Examination Elementary Mathematics
19 The relationship between the velocity of the body, v and the time, t, is described by
part linear variation. The velocity of the body moving in a straight line with uniform
acceleration is 20 m/s after 5 seconds and is 26 m/s 3 seconds later.
Find the equation of linear variation.
Pg 187 Math Quest 11 General Mathematics
20 The variable y varies as the sum of two quantities, one of which varies directly as x
and the other inversely as x 2 . When x  1, y  17 and when x  2, y  1 . Find the
equation for y in terms of x .
Pg 185 Math Quest 11 General Mathematics
Miscellaneous
1
If m is inversely proportional to n and m  6 when n  1 , then which of the following
2
belongs to the same relationship?
A
3, 14 
B
12, 1
C
 3, 1
D
 3, 6 
E
121 , 1
Pg 169 Math Quest 11 General Mathematics
2
The frequency of a string of a musical instrument, F, varies jointly as the square root
of the tension, t, in the string, the reciprocal of the length of the string, l, and the
reciprocal of the square root of the mass per unit length, m. The equation that
describe the relationship between the 4 variables could not be written as
F
t
k t
t
Fk
t
Fl
t
A F
B Fl  k
C
D
E



k l m
l m
m
l
m
k
m
Pg 182 Math Quest 11 General Mathematics
3
From Physics, we know that the work done, W, when a certain object is pushed can
be calculated by multiplying the force, F, applied to it by the distance, d, it was
pushed. Which of the three variables should be fixed as a constant so that the other
two would vary inversely?
Pg 171 Math Quest 11 General Mathematics
4
Find the relationship between x and y for each table of data.
x
0
1
2
3
4
y
2
5
8
11
14
x
0
1
2
3
4
y
5
1
3
7
11
x
0
1
2
3
4
y
0
2
16
54
128