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Variations
VARIATIONS
IMPORTANT CONCEPTS
Direct Variations
Statement Using symbol `∝ ’, Equation with k as constant of
variation
y varies directly as x y ∝ x
y = kx
P varies directly as Q2
P ∝ Q2 P = kQ
2
M is directly proportional to
N
M ∝ N M = k N
Inverse Variations
Statement Using symbol `∝ ’, Equation with k as constant of
variation
y varies inversely as x y ∝
x
1 y =
x
k
P varies inversely as Q2
P ∝ 2
1
Q P =
2Q
k
M is inversely proportional to
N M ∝
N
1 M =
N
k
Joint Variations
Statement Using symbol `∝ ’, Equation with k as constant of
variation
w varies directly as x and y w ∝ xy
w = kxy
P varies directly as q and r2
P ∝ q r2
P = kq r2
s varies directly t and
inversely as u s ∝ u
t s =
u
kt
d varies directly as e2 and
inversely as f d ∝
f
e2 d =
f
ke2
R varies inversely as M and
N R ∝
NM
1 R =
NM
k
4 steps to solve problems involving variations:
• STEP 1 : Change the statement using the symbol ∝ .
• STEP 2 : Write down the equation connecting the variables using k as the constant of
variation.
• STEP 3 : Find the value of k.
• STEP 4 : Find the value of variable required.
Variations
A) DIRECT VARIATION
EXAMPLE 1:
Given that P varies directly as Q and P = 3
1 when Q = 2.
Express P in terms of Q and find the value of P when Q = 18.
• STEP 1 : Symbol - P ∝ Q
• STEP 2 : Equation - P = kQ , k = constant
• STEP 3 : Find the value of k, 3
1= k(2)
3
1= 2k
k = 6
1
THEN, substitute k = 6
1 in the equation P = kQ.
HENCE, P = 6
1 Q
• STEP 4:Find the value of P when Q = 18, P = 6
1×18
P = 3
Exercises:
1. Given that y varies directly as x and y = 15 when x = 5.
a) Express y in terms of x.
b) Find the value of y when x = 4.
2. Given that P ∝ Q2 and P = 10 when Q = 2.
a) Express P in terms of Q.
b) Find the value of P when Q = 4.
Variations
3. Given that y varies directly as x and y = 6 when x = 9.
c) Express y in terms of x.
d) Find the value of y when x = 25.
4. It is given that m varies directly as n3 and m = 32 when n = 2. Express m in terms of
n and find the value of m when n = 2
1 .
EXAMPLE 2 :
P varies directly as x2 where x = 5 + y. Given that P = 72 when y = 1.
a) Express P in terms of x.
b) Find the values of y when P = 200.
• STEP 1 : Symbol - P ∝ x2
• STEP 2 : Equation - P = k x2 , k = constant
• STEP 3 : Find the value of k, substitute and x = 5 + y in the above equation:
P = k(5 + y)2
Then substitute P=72 and y = 1, 72 = k(5 + 1)2
72 = 36k
k = 6
72
k = 2
THEN, substitute k = 2 in the equation P = k x2
HENCE, P = 2 x2
• STEP 4:Find the value of y when P = 200 by substitute x = 5 + y
200 = 2(5 + y)2
100 = (5 + y)2
± 100 = 5 + y
± 10 = 5 + y
y = 10 – 5 and y = -10 – 5
y = 5 and y = -15
Variations
Exercises:
1. M varies directly as x2 where x = 3 + y. Given that M = 64 when y = 1.
a) Express M in terms of x.
b) Find the values of y when M = 400.
2. Given that T ∝ S2 and S = 2w – 3, and T = 12 when W = 1.
a)Express T in terms of S
b)Find the value of T when W = 5
B) INVERSE VARIATION
EXAMPLE 1:
Given that y varies inversely as x and y = 6 when x = 4.
Express y in terms of x and find the value of y when x = 3.
• STEP 1 : Symbol - y ∝ x
1
• STEP 2 : Equation - y = x
k , k = constant
• STEP 3 : Find the value of k, 6 = 4
k
k = 24
THEN, substitute k = 24 in the equation y = x
k.
HENCE, y = x
24
• STEP 4:Find the value of y when x = 3, y = 3
24
y = 8
Variations
Exercises:
1. Given that y varies inversely as x and x = 4 when y = 3.
a. Express y in terms of x.
b. Find the value of y when x = 6.
2. Given that P varies inversely as x and P = 6 when x = 3
1. Express P in terms of x and
find the value of P when x = 3
13 .
3. Given that y varies inversely as x2 and y = 2 when x = 4.
a) Express y in terms of x
b) Find the value of y when x = 2.
4. Given that S varies inversely as square root of r and S = 5 when r = 16.
a) Express S in terms of r.
b) Find the value of S when r = 25.
Variations
5. Given that P varies inversely as Q . Complete the following table.
6. The table below shows some values of the variables M and N such that N varies inversely
as the square root of M.
Find the relation between M and N.
C) JOINT VARIATION
EXAMPLE 1: (DIRECT VARIATION & DIRECT VARIATION) y varies directly as x and z. Given that y = 12 when x = 2 and z = 3.
a) Express y in terms of x and y,
b) Find the value of y when x = 5 and z = 2.
• STEP 1 : Symbol - y ∝ xz
• STEP 2 : Equation - y = kxz , k = constant
• STEP 3 : Find the value of k, 12 = k (2)(3)
12 = 6k
k = 2
THEN, substitute k = 2 in the equation y = kxz.
HENCE, y = 2xz
• STEP 4:Find the value of y when x = 5 and z = 2, y = 2(5)(2)
y = 20
P 6 2
Q 9 4
M 4 36
N 6 2
Variations
Exercises:
1. p varies directly as q and r. Given that p = 36 when q = 4 and r = 3.
a) Express p in terms of q and r,
b) Find the value of p when q = 3 and r = 3.
2. Given that y ∝ mn and y = 20 when m = 2 and n = 5.
a) Express y in terms of m and n
b) Find the value of y when m = 3 and n = 4.
EXAMPLE 2: ( INVERSE VARIATION & INVERSE VARIATION)
Given that y varies inversely as x and z. y = 10 when x = 2 and z = 4.
Express y in terms of x and z, then find the value of y when x = 4 and z = 5.
• STEP 1 : Symbol - y ∝ xz
1
• STEP 2 : Equation - y = xz
k , k = constant
• STEP 3 : Find the value of k, 10 = )4(2
k
k = 80
THEN, substitute k = 80 in the equation y = xz
k.
HENCE, y = xz
80
• STEP 4:Find the value of y when x = 4 and z = 5, y = )5(4
80
y = 20
80, y = 4.
Variations
Exercises:
1. Given that m ∝ yx
1 and m = 3 when x = 3 and y = 16. Express m in terms of x and y. Find
the value of y when m = 9 and x = 12.
2. Given that y ∝ ed 2
1 and y = 2 when d = 3 and e = 4, calculate the value of e when y = 3 and
d = 4.
EXAMPLE 3 (DIRECT VARIATION + INVERSE VARIATION)
1. Given that y varies directly as x and varies inversely as v and y = 10 when x = 4 and v=5.
a) Express y in terms of x and v
b) Find the value of y when x = 2 and v = 15.
• STEP 1 : Symbol - y ∝ v
x
• STEP 2 : Equation - y = v
kx , k = constant
• STEP 3 : Find the value of k, 10 = 5
4k
50 = 4k
k = 2
25
THEN, substitute k = 2
25 in the equation y =.
v
kx
HENCE, y = v
x
2
25
• STEP 4:Find the value of y when x = 2 and v = 15, y = )15(2
)2(25
y = 30
50, y =
3
5.
Variations
Exercises:
1. M varies directly as N and varies inversely as P and m = 6 when N = 3 and P = 4.
a) Express M in terms of N and P.
b) Calculate the value of M when N = 5 and P = 2.
2. Given that w ∝ y
x and w = 10 when x = 8 and y = 16. Calculate the value of w when
x= 18 and y = 36.
3. F varies directly as G to the power of two and varies inversely as H. Given that F = 6
when G = 3 and H = 2, express F in terms of G and H. Find the value of G when F = 27
and H = 4.
4. Given that S ∝ 3M
P and S = 6 when P = 3 and M = 2. Calculate the value of P when S =
2 and M = -4.
Variations
5. Given that p varies directly as x and inversely as the square root of y. If p = 8 when x = 6
and y = 9.
a. express p in terms of x and y,
b. calculate the value of y when p = 6 and x = 9.
EXAMPLE 4:
The table shows some values of the variables d, e and f.
d 12 10
e 9 25
f 4 m
Given that d ∝ f
e
a) express d in terms of e and f
b) calculate the value of m.
• STEP 1 : Symbol - d ∝ f
e
• STEP 2 : Equation - d = f
ek , k = constant
• STEP 3 : Find the value of k, substitute d = 12, e = 9, and f = 4
12 = 4
9k
48 = 3k
k = 16
THEN, substitute k = 16 in the equation d =. f
ek
HENCE, d = f
e16
• STEP 4:Find the value of m (f) when d = 10 and e = 25, 10 = m
2516
10m = 16 × 5,
10m = 80
m = 10
Variations
Exercises:
1. The table shows the relation between three variables D, E and F.
D E F
2 3 4
2
1
6 m
Given that D ∝ EF
1
a) express D in terms of E and F
b) calculate the value of m.
2. The table shows some values of the variables w, x and y, such that w varies directly as
the square root of x and inversely as y.
w x y
18 9 2
3 K 36
a) find the equation connecting w x and y.
b) calculate the value of K.
Variations
Objective Questions
1. The table shows relation between the variables, x and y. If y∝ x , find the value of m.
A. 7
10 B.
5
14 C. 8 D.
2
35
2.
m 2 X
n 9 25
The table shows the relation between the variables, m and n. If m varies inversely as the
square root of n, then x =
A. 25
18 B.
5
6 C.
3
28 D.
9
50
3. It is given that y ∝ w
v 2, and y = 6 when v = 4 and w = 8, calculate the positive value of v
when y = 25 and w = 3.
A. 5 B. 6 C. 12.5 D. 25
4. Given that y varies inversely as x and y = 5 when x = 3
1, express y in terms of x.
A. y = x
15 B. y = 15x C. y =
x3
5 D. y =
x5
3
5. Given that y varies directly as x, and y = 6 when x = 2, express y in terms of x.
A. y = x3
1 B. y = 3x C. y = 12x D. y =
x
12
6. Given that P varies inversely as Q , and P = 2 when Q = 9, express P in terms of Q.
A. P = 3
2Q B. P = 6 Q C. P =
Q
6 D. P =
Q
18
X 2 7
Y 5 M
Variations
7. The relation between the variables, P, x and y is represented by P ∝ xmyn. If P varies
directly as the square of x and inversely as the cube of y, then m + n =
A. -1 B. 1 C. 5 D. -6
8. Given that y varies directly as xn. If x is the radius and y is the height of the cyclinder whose
volume is a constant, then the value of n is
A. -2 B. 1 C. 2 D. 3
9. The table shows the corresponding values of d and e. The relation between the variables d and
e is represented by
A.e d∝ B. e ∝ d
1 C. e ∝
d
1 D. e ∝
2d
1
10. T varies directly as the square root of p and inversely as the square root of g. This joint
variation can be written as
A. T ∝ pg B. T g
p 2 C. T ∝
g
p D. T ∝
2g
p
11. It is given that m varies inversely as s and t. If m = 3 when s = 2 and t = 4, find the value of
m when s = 2 and t = 6.
A. 2 B. 3 C. 4 D. 5
12. It is given that G ∝ 2H
1 H = 5M -1. If G =3 when M = 2, express G in terms of H.
A. G = 2H
9 B. G =
2H
27 C. G =
2H
243 D. G =
H
243
d 1 4 9 16
e 30 15 10 7.5
Variations
13.
d 6 8
e 3 X
f 2 5
The table shows the relation between the three variables d, e and f. If d ∝ f
e, calculate the
value of x
A. 6 B. 10 C. 12 D. 13
14. It is given that y ∝ xn. If y varies inversely as the square root of x, then the value of n is
A. 2
1 B.
2
1− C. -1 D. – 2
15.
F G H
6 9 2
12 m 3
The table shows the relation between the three variables, F,G and H. If F varies directly as the
square root of G and inversely as H, then the value of m is
A. 3 B. 9 C. 81 D. 144
16.
S 2 3
P 3 1
M 4 X
The table shows the relation between the variables, S, P and M. If S ∝ MP
1, calculate
the value of x
A. 4 B. 8 C. 16 D. 64
17. Given that p ∝ y
x 2 and p = 6 when x = 2 and y = 3, express p in terms of x and y
A. p = y2
x9 2
B. p = y9
x2 2
C. p = y9
x2 D. p =
y
x9 2
Variations
18. Given that y varies directly as x2 and that y = 80 when x = 4, express y in terms of x
A. y = x2 B. y = 5x C. y = 5x
2 D. y =
2x
1280
19.
w 2 3
x 12 24
y 9 m
The table shows the relation between the three variables, w, x and y. If w ∝ y
x, calculate
the value of m
A. 2 B. 4 C. 8 D. 16
20. If M varies directly as the square root of N, the relation between M and N is
A. M ∝ N B. M ∝ N2
C. M ∝ N 2
1
D. M ∝
2
1
N
1
PAST YEAR QUESTIONS
1. SPM 2003(Nov)
Given that p is directly proportional to n2 and p = 36, express p in terms of n
A. p = n2 B. p = 4n
2 C. p = 9n
2 D. p = 12n
2
2. SPM 2003(Nov)
w 2 3
x 8 18
y 4 n
The table shows some values of the variables, w, x and y which satisfy the relationship w ∝
y
x, calculate the value of n
A. 6 B. 9 C. 12 D. 36
Variations
3. SPM 2004(Nov) P varies directly as the square root of Q. The relation between P and Q is
A. P ∝ 2
1
Q B. P ∝ 2Q C. P ∝
2
1
1
Q
D. P ∝ 2
1
Q
4. SPM 2004(Nov)
P M r
3 8 4
6 w 9
The table shows the relation between the three variables p, m and r. Given that p ∝
r
m, calculate the value of w
A. 16 B. 24 C. 36 D. 81
5. SPM 2004(Jun)
The relation between p, n and r is p ∝r
n3
. It is given that p = 4 when n = 8 and r = 6.
Calculate the value of p when n = 64 and r = 3
A. 16 B. 24 C. 32 D. 48
6. SPM 2004(Jun)
It is given that p varies inversely with w and p = 6 when w = 2. Express p in terms of w.
A. p = w
3 B.
w
12 C. p = 3w D. p = 12w
7. SPM 2005(Nov)
The table shows some values of the variables x and y such that y varies inversely as the
square root of x.
x 4 16
y 6 3
Find the relation between y and x.
A. y = 3 x B. x
12 C.
2
8
3x D.
2
96
x
8. SPM 2005(Nov)
It is given that y varies directly as the square root of x and y = 15 when x = 9. Calculate the
value of x when y = 30.
A. 5 B. 18 C. 25 D. 36
Variations
9. SPM 2005(Nov) The table shows some values of the variables w, x and y such that w varies directly as the
square of x and inversely as y.
W x y
40 4 2
M 6 4
Calculate the value of m.
A. 90 B. 45 C. 30 D. 15
10. SPM 2005(Jun) Table shows values of the variables x and y.
x 3 m
y 5 15
It is given x varies directly with y. Calculate the value of m.
A. 6 B. 9 C. 12 D. 15
11. SPM 2005(Jun)
P varies directly with the square of R and inversely with Q. It is given that P = 2 when Q = 3
and R = 4. Express P in terms of R and Q.
A. P =Q
R
8
3 2
B. 23
32
R
Q C.
Q
R3 D. P =
R
Q
3
4
12. SPM 2006(Jun)
It is given that y varies inversely with x and y = 21 when x = 3. Express y in terms of x.
A. y = 7x B. y = 7
x C. y =
x63
1 D. y =
x
63
13. SPM 2006(Jun)
Table 2 shows two sets of values of Y, V and W.
Y V W
5
3
3 12
m 5 18
It is given that Y varies directly with the square of V and inversely with W. Find the value of
m.
A. 3
5 B.
9
4 C.
9
10 D.
25
6
Variations
14. SPM 2007(Nov) Table 1 shows some values of the variables x and y.
x 2 n
y 4 32
It is given that y varies directly as the cube of x. Calculate the value of n.
A. 4 B. 8 C. 16 D. 30
15. SPM 2007(Nov) P varies inversely as the square root of M. Given that the constant is k, find the relation
between P and M.
A. 2
1
kMP = B.
2
1
M
kP = C.
2kMP = D. 2M
kP =
16. SPM 2007(Nov)
The relation between the variables x, y and z is z
yx∞ . It is given that x =
4
5 when y = 2
and z = 8. Calculate the value of z when x = 3
5 and y = 6.
A. 2 B. 18 C. 32 D. 72
17. SPM 2007(Jun)
It is given that P varies inversely with Q and P = 5
2 when Q =
2
1. Find the relation between
P and Q.
A. QP5
4= B. QP
5
1= C.
QP
5
4= D.
QP
5
1=
18. SPM 2007(Jun)
Table shows some values of the variables F, G and H that satisfy F H
G 2
α .
F G H
20 2 3
108 6 p
Calculate the value of p.
A. 5 B. 9 C. 10 D. 18
Variations
19. SPM 2008(Nov)
Table shows some values of the variables R and T. It is given that R varies directly as T .
R 54 72
T 36 y
Find the value of y.
A. 24 B. 27 C. 48 D. 64
20. SPM 2008(Nov)
Given y varies inversely as x3, and that y = 4 when x = ½ . Calculate the value of x when y =
16
1.
A. 8
1 B. ½ C. 2 D. 8
21. SPM 2008(Nov)
It is given that P varies directly as the square root of Q and inversely as the square of R. Find the
relation between P, Q and R.
A. R
QP
2
α B. 2R
QPα C.
Q
RP
2
α D. 2Q
RPα
22. SPM 2008 (Jun)
It is given that p varies directly as the square root of w and that p = 5 when w = 4 . Express p in
terms of w.
A. 2
16
5wp = B.
2
80
wp = C. wp
2
5= D.
wp
10=
23. SPM 2008(Jun)
Table shows some values of the variables m and n, such that m varies inversely as the cube of n
m
2
1
x
n 2 3
Calculate the value of x.
A. 27
4 B.
9
4 C.
16
9 D.
16
27
Variations
ANSWERS
Chapter 21 Variations
A) DIRECT VARIATION
Example 1: No. 1. a) y = 3x No. 3. a) y = 2x
b) y = 12 b) y = 50
No. 2. a) P = 2
2
5Q No. 4. m =
3
2
1n , m =
16
1
b) P = 40
Example 2: No. 1. a) M = 4x2
b) y = 7 and y = -13
B) INVERSE VARIATION
Example 1: No. 1. a) y = x
12 No. 4. a) S =
r
20
b) y = 2 b) S = 4
No. 2. a) P = x
2 No. 5. P = 9 , Q = 81
b) P = 5
3 N0. 6. k = 12, N =
M
12
No. 3. a) y = 2
16
x
b) y = 4
C) JOINT VARIATION
Example 1: No. 1. a) k = 3, p = 3qr
No. 2. a) k = 2, y = mn
b) y = 24
Example 2: No. 1. k = 36, m = yx
36, y = 9
No. 2. k = 72, y = 2
72
de, e =
2
3
Example 3: No. 1. a) k = 8, M = P
N8
b) M = 20
No. 2. k = 20, w = y
x20, w = 2
Variations
No. 3. k = 3
4, F =
H
G
3
4 2
, G = 9±
No. 4. k = 16, S = 3
16
M
P, P = -8
No. 5. a) k = 4, p = y
x4
b) y = 36
Example 4: No. 1. a) k = 24, D = EF
24
b) m = 16
No. 2. a) k = 12, w = y
x12
b) K = 81
Objective Questions.
1.D 6. C 11.A 16. C
2.B 7. A 12.C 17. A
3.A 8. A 13.B 18. C
4.C 9. C 14.B 19. D
5.B 10. C 15.C 20. C
PAST YEAR QUESTIONS
1.B 8.D 15.B 22.C
2.B 9.B 16.B 23. A
3.A 10.B 17.D
4.B 11.A 18. A
5.A 12.A 19. C
6.B 13.C 20. C
7.B 14. A 21.B