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