Proportion (Copy)
Question 1
y is directly proportional to x.
(a) Write the relation between y and x.
(b) If y = 15 when x = 5, find y when x = 12.
Question 2
y is inversely proportional to x.
(a) Write the relation between y and x.
(b) If y = 8 when x = 3, find y when x = 12.
Question 3
y is directly proportional to x².
(a) Write the relation between y and x.
(b) If y = 50 when x = 5, find y when x = 8.
Question 4
y is directly proportional to √x.
If y = 20 when x = 25, find y when x = 81.
Question 5
y is inversely proportional to √x.
If y = 15 when x = 9, find y when x = 36.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Question 6
y is directly proportional to x³.
If y = 64 when x = 2, find y when x = 5.
Question 7
y is inversely proportional to x³.
If y = 40 when x = 2, find y when x = 4.
Question 8
The time (t) taken to travel a fixed distance is inversely proportional to the speed (v).
If t = 2 hours when v = 60 km/h, find t when v = 80 km/h.
Question 9
The volume (V) of a cube is directly proportional to the cube of its side length (s).
If V = 343 cm³ when s = 7 cm, find V when s = 10 cm.
Question 10
The period (T) of a pendulum is directly proportional to the square root of its length (L).
If T = 2 seconds when L = 1 m, find T when L = 4 m.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Question 11
The force (F) is directly proportional to the acceleration (a).
If F = 40 N when a = 8 m/s², find F when a = 15 m/s².
Question 12
The resistance (R) of a wire is directly proportional to its length (L).
If R = 10 Ω when L = 2 m, find R when L = 7.5 m.
Question 13
The brightness (B) of a lamp is inversely proportional to the square of the distance (d) from the lamp.
If B = 20 units when d = 2 m, find B when d = 5 m.
Question 14
The gravitational force (F) between two masses is inversely proportional to the square of the distance (r) between them.
If F = 100 N when r = 2 m, find F when r = 6 m.
Question 15
The energy (E) stored in a spring is directly proportional to the square of its extension (x).
If E = 72 J when x = 6 cm, find E when x = 10 cm.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Question 16
y is directly proportional to x.
If y = 25 when x = 5, find the constant of proportionality and write the equation.
Question 17
y is inversely proportional to x².
If y = 4 when x = 3, find y when x = 6.
Question 18
y is directly proportional to the cube root of x.
If y = 18 when x = 27, find y when x = 64.
Question 19
The current (I) through a conductor is directly proportional to the voltage (V).
If I = 2 A when V = 12 V, find I when V = 30 V.
Question 20
The mass (M) of a solid sphere is directly proportional to the cube of its radius (r).
If M = 36 g when r = 2 cm, find M when r = 5 cm.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Question 21
A car’s stopping distance (d) is directly proportional to the square of its speed (v).
If d = 20 m when v = 20 km/h, find d when v = 60 km/h.
Question 22
The pressure (P) of a gas is inversely proportional to its volume (V).
If P = 100 kPa when V = 2 litres, find P when V = 5 litres.
Question 23
The area (A) of a circle is directly proportional to the square of its radius (r).
If A = 28.26 cm² when r = 3 cm, find A when r = 7 cm.
Question 24
y is inversely proportional to the cube root of x.
If y = 12 when x = 8, find y when x = 27.
Question 25
The frequency (f) of a vibrating string is inversely proportional to its length (L).
If f = 300 Hz when L = 0.5 m, find f when L = 0.25 m.
Question 26
y is directly proportional to x and inversely proportional to z.
If y = 12 when x = 6 and z = 2, find y when x = 9 and z = 3.
Question 27
The gravitational potential energy (E) is directly proportional to the mass (m) and the height (h).
If E = 200 J when m = 5 kg and h = 4 m, find E when m = 8 kg and h = 10 m.
Question 28
The centripetal force (F) is directly proportional to the square of the speed (v).
If F = 180 N when v = 6 m/s, find F when v = 10 m/s.
Question 29
y is directly proportional to x² and inversely proportional to √z.
If y = 20 when x = 4 and z = 9, find y when x = 6 and z = 16.
Question 30
The time period (T) of a wave is inversely proportional to its frequency (f).
If T = 0.05 s when f = 20 Hz, find T when f = 50 Hz.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Answer key and explanations — 2.8 Proportion
1.
y ∝ x → y = kx.
Given y = 15 when x = 5 → 15 = 5k → k = 3.
So y = 3x.
When x = 12 → y = 36.
2.
y ∝ 1/x → y = k/x.
Given y = 8 when x = 3 → 8 = k/3 → k = 24.
So y = 24/x.
When x = 12 → y = 24/12 = 2.
3.
y ∝ x² → y = kx².
50 = k(25) → k = 2.
So y = 2x².
When x = 8 → y = 2(64) = 128.
4.
y ∝ √x → y = k√x.
20 = k√25 → 20 = k(5) → k = 4.
So y = 4√x.
When x = 81 → y = 4(9) = 36.
5.
y ∝ 1/√x → y = k/√x.
15 = k/3 → k = 45.
So y = 45/√x.
When x = 36 → y = 45/6 = 7.5.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
6.
y ∝ x³ → y = kx³.
64 = k(8) → k = 8.
So y = 8x³.
When x = 5 → y = 8(125) = 1000.
7.
y ∝ 1/x³ → y = k/x³.
40 = k/8 → k = 320.
So y = 320/x³.
When x = 4 → y = 320/64 = 5.
8.
t ∝ 1/v → t = k/v.
2 = k/60 → k = 120.
So t = 120/v.
When v = 80 → t = 120/80 = 1.5 hours.
9.
V ∝ s³ → V = ks³.
343 = k(343) → k = 1.
So V = s³.
When s = 10 → V = 1000 cm³.
10.
T ∝ √L → T = k√L.
2 = k√1 → k = 2.
So T = 2√L.
When L = 4 → T = 2(2) = 4.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
11.
F ∝ a → F = ka.
40 = k(8) → k = 5.
So F = 5a.
When a = 15 → F = 75 N.
12.
R ∝ L → R = kL.
10 = k(2) → k = 5.
So R = 5L.
When L = 7.5 → R = 37.5 Ω.
13.
B ∝ 1/d² → B = k/d².
20 = k/4 → k = 80.
So B = 80/d².
When d = 5 → B = 80/25 = 3.2.
14.
F ∝ 1/r² → F = k/r².
100 = k/4 → k = 400.
So F = 400/r².
When r = 6 → F = 400/36 ≈ 11.11 N.
15.
E ∝ x² → E = kx².
72 = k(36) → k = 2.
So E = 2x².
When x = 10 → E = 200 J.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
16.
y ∝ x → y = kx.
25 = k(5) → k = 5.
Equation: y = 5x.
17.
y ∝ 1/x² → y = k/x².
4 = k/9 → k = 36.
So y = 36/x².
When x = 6 → y = 36/36 = 1.
18.
y ∝ ³√x → y = k³√x.
18 = k(3) → k = 6.
So y = 6³√x.
When x = 64 → y = 6(4) = 24.
19.
I ∝ V → I = kV.
2 = k(12) → k = 1/6.
So I = V/6.
When V = 30 → I = 5 A.
20.
M ∝ r³ → M = kr³.
36 = k(8) → k = 4.5.
So M = 4.5r³.
When r = 5 → M = 4.5(125) = 562.5 g.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
21.
d ∝ v² → d = kv².
20 = k(400) → k = 0.05.
So d = 0.05v².
When v = 60 → d = 0.05(3600) = 180 m.
22.
P ∝ 1/V → P = k/V.
100 = k/2 → k = 200.
So P = 200/V.
When V = 5 → P = 40 kPa.
23.
A ∝ r² → A = kr².
28.26 = k(9) → k ≈ 3.14 (π).
So A = 3.14r².
When r = 7 → A ≈ 153.86 cm².
24.
y ∝ 1/³√x → y = k/³√x.
12 = k/2 → k = 24.
So y = 24/³√x.
When x = 27 → y = 24/3 = 8.
25.
f ∝ 1/L → f = k/L.
300 = k/0.5 → k = 150.
So f = 150/L.
When L = 0.25 → f = 600 Hz.
26.
y ∝ x/z → y = kx/z.
12 = k(6/2) = 3k → k = 4.
So y = 4x/z.
When x = 9, z = 3 → y = 12.
27.
E ∝ mh → E = kmh.
200 = k(5×4) = 20k → k = 10.
So E = 10mh.
When m = 8, h = 10 → E = 800 J.
28.
F ∝ v² → F = kv².
180 = k(36) → k = 5.
So F = 5v².
When v = 10 → F = 500 N.
29.
y ∝ x²/√z → y = kx²/√z.
20 = k(16/3) → k = 60/16 = 3.75.
So y = (3.75x²)/√z.
When x = 6, z = 16 → y = (3.75×36)/4 = 33.75.
30.
T ∝ 1/f → T = k/f.
0.05 = k/20 → k = 1.
So T = 1/f.
When f = 50 → T = 0.02 s.
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course