1) The document contains examples of direct, inverse, and joint variations. It provides the definitions and formulas for each type of variation. 2) Examples are given for expressing variables in terms of other variables for different situations involving direct, inverse, and joint variations. The values of constants are calculated. 3) Tables are included that require calculating missing values based on the given variations and values.

1. PPR Maths nbk
VARIATIONS
Guided Practice:
A Direct Variation
1. Given that E varies directly as J. 3. Given that p varies directly as square
Express E in terms of J when E = 6 and root of q. Express p in terms of q when p
J = 12. = 10 and q = 25.
Solution:
EαJ Solution:
p α q
E = kJ k is a constant
p =k q
Substitute the given values of E and J to
find the value of k.
6 = k (12)
6
=k
12
1
k=
2
1
Hence, E= J
2
2. Given that R varies directly as the 4. x 32 m
square of Q and R = 48 when Q = 4, y 4 2
express R in terms of Q.
The table shows the values of x and y.
Solution: Given that x varies directly as y3,
R α Q2 calculate the value of m.
Solution:
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2. PPR Maths nbk
B Inverse Variation
1. Given that W varies inversely as X. 2. Given that g varies inversely as h.
Express W in terms of X when W = 8 Express g in terms of h when g = 25
and X = 2. and h = 0.6.
Solution:
1 Solution:
W α
X 1
g α
h
1 k is a constant
W=k
X k
g =
k h
W =
X
k
8 =
2
8(2) = k
k = 16
16
Hence, W=
X
3. Given that M varies inversely as the 4. F 2 e
square of T and M = 8 when T = 2, y 4 8
express M in terms of T.
The table shows the values of F and y.
Given that F varies inversely as y2,
calculate the value of e.
5. 6.
p 5 a 1
t 16 4 X a
2
t 4 2
The table shows the values of p and t.
Given that p varies inversely as the The table shows the values of X and t.
square root t, calculate the value of a. Given that X varies inversely as the
square of t, calculate the value of a.
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3. PPR Maths nbk
C Joint Variation
1. Given that m varies directly as n2 and p. 2. Given that h varies inversely as n3 and
Express m in terms of n and p when m m and h = 2 when n = 2 and m = 121.
= 270, p = 6 and n = 3. Express h in terms of n and m.
3. Given that J varies directly as r3 and 4. 1
inversely as m2 and J = 144 when r = 4 D 10
6
and m = 2. 1
a. Express J in terms of r and m. e 2 3
b. Find the value of 3
i. J when r = 1 and m = 6, 1
f 81
ii. m when J = 4.5 and r = 2. 5
Given that D varies inversely as e2 and f.
Complete the table.
5. If p varies directly as q and p = 71 6.
when q = 25, find F 10 20
a. p when q = 9, n 40 60 90
b. q when p = 355. d 20 45
Given F varies directly as n and
inversely as d. Complete the table.
7. Given that m is directly proportional to 8. It is given that y varies directly as the
2
n and m = 64 when n = 4, express m in square root of x and y = 24 when x = 9.
terms of n. Calculate the value of x when y = 40.
(SPM 2003) (SPM 2005)
A m = n2 C m = 16n2 A 5 C 25
B m = 4n2 D m = 64n2 B 18 D 36
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4. PPR Maths nbk
VARIATIONS (ANSWERS)
Guided Practice:
A Direct Variation
1. Given that E varies directly as J. 3. Given that p varies directly as square
Express E in terms of J when E = 6 and root of q. Express p in terms of q when p
J = 12. = 10 and q = 25.
Solution:
EαJ Solution:
p α q
E = kJ k is a constant
p =k q
Substitute the given values of E and J to
find the value of k.
6 = k (12) 10 = k 25
6 10
=k =k
12 5
1 k=2
k=
2
Hence, p =2 q
1
Hence, E= J
2
2. Given that R varies directly as the 4. x 32 m
square of Q and R = 48 when Q = 4, y 4 2
express R in terms of Q.
The table shows the values of x and y.
Solution: Given that x varies directly as y3,
R α Q2 calculate the value of m.
Solution:
R = kQ2 x α y3
48 = k (4)2 x = ky3
48 32 = k (4)3
=k
16
32
=k
k=3 64
Hence, R = 3Q2 1 1 3
k= , Hence, x= y
2 2
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5. PPR Maths nbk
B Inverse Variation
1. Given that W varies inversely as X. 2. Given that g varies inversely as h.
Express W in terms of X when W = 8 Express g in terms of h when g = 25
and X = 2. and h = 0.6.
Solution:
1 Solution:
W α
X 1
g α
h
1 k is a constant
W=k
X k
g =
k h
W =
X
k
25 =
k 0.6
8 =
2
k = 15
8(2) = k
15
Hence, g=
k = 16 h
16
Hence, W=
X
3. Given that M varies inversely as the 4. F 2 e
square of T and M = 8 when T = 2, y 4 8
express M in terms of T. The table shows the values of F and y.
Given that F varies inversely as y2,
calculate the value of e.
1
Answer:
2
Answer: M = 32
T2
1
5. p 5 A 6. X a
2
t 16 4
t 4 2
The table shows the values of p and t. The table shows the values of X and t.
Given that p varies inversely as the Given that X varies inversely as the
square root t, calculate the value of a. square of t, calculate the value of a.
Answer: 10 Answer: 2
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6. PPR Maths nbk
C Joint Variation
1. Given that m varies directly as n2 and p. 2. Given that h varies inversely as n3 and
Express m in terms of n and p when m m and h = 2 when n = 2 and m = 121.
= 270, p = 6 and n = 3. Express h in terms of n and m.
Answer: m = 5pn2
Answer: h = 176
n3 m
3. Given that J varies directly as r3 and 4. 1
D 10
inversely as m2 and J = 144 when r = 4 6
and m = 2. 1
a. Express J in terms of r and m. e 2 3
3
b. Find the value of 1
i. J when r = 1 and m = 6, f 81
ii. m when J = 4.5 and r = 2. 5
Given that D varies inversely as e2 and f.
9r 3
Answer: a. J = 2 Complete the table.
m 18
1 Answer: D= 2
b.i. fe
4
D = 2 and f = 27
ii. 4
5. If p varies directly as q and p = 71 6.
F 10 20
when q = 25, find n 40 60 90
c. p when q = 9, d 20 45
d. q when p = 355.
Given F varies directly as n and
Answer: p = 42.6
inversely as d. Complete the table.
q = 625
5n
Answer: F =
d
F = 10 and d = 15
7. Given that m is directly proportional to 8. It is given that y varies directly as the
2
n and m = 64 when n = 4, express m in square root of x and y = 24 when x = 9.
terms of n. Calculate the value of x when y = 40.
(SPM 2003) (SPM 2005)
A m = n2 C m = 16n2 A 5 C 25
B m = 4n2 D m = 64n2 B 18 D 36
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7. PPR Maths nbk
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