Revision
Direct and Inverse Proportions
Question 1:
A train moves on a circular track of radius, r metres. The safe speed of the train v m/s, is
directly proportional to the square root of r. If the safe speed for a radius of 121 m is
22 m/s, calculate
a) the safe speed for a radius of 81 m,
b) the radius if the safe speed is 28 m/s.
Answers:
𝑣 = 𝑘 √𝑟
22 = 𝑘√121
22 = 11𝑘
2=𝑘
∴ 𝑣 = 2 √𝑟
a) ∴ 𝑣 = 2√81
= 18𝑚/𝑠
b) 28 = 2√𝑟
14 = √𝑟
196𝑚 = 𝑟
Revision
Direct and Inverse Proportions
Question 2:
The volume of a cylinder, V is directly proportional to the square of its radius, r.
When the radius is 3 cm, its volume is 120 cm3.
a) Find an equation connecting the V and r.
b) Find the volume of the cylinder if its radius is 8 cm, giving your answer to the
nearest whole number.
Answers:
a)
b)
𝑣 = 𝑘𝑟 2
120 = 𝑘(3)2
1
13 = 𝑘
3
1
∴ 𝑣 = 13 𝑟 2
3
1
𝑣 = 13 (8)2
3
= 853
Revision
Direct and Inverse Proportions
Question 3:
In an experiment, the speed v, of the wheel of a car is directly proportional to the
square root of the tyre pressure, p. Given that v = 40 when p = 25, find
a) the equation connecting v and p,
b) the value of v when p is 36.
Answers:
a) 𝑣 = 𝑘√𝑝
40 = 𝑘√𝑝
8−𝑘
∴ 𝑣 = 8 √𝑝
b) v =8 36 = 48
Revision
Direct and Inverse Proportions
Question 4:
y is directly proportional to x3.
When x has a certain value, y = 5. Find the value of y when x is halved.
Answers:
𝑦 = 𝑘𝑥 3
5 = 𝑘(1)3
5=𝑘
∴ 𝑦 = 5𝑥 3
1
𝑁𝑜𝑤, 𝑥 =
2
1
𝑦 = 5( )3
2
5
=
8
Revision
Direct and Inverse Proportions
Question 5:
It is given that y is directly proportional to x2. When x has a certain value, y = 8.
Find the value of y when x is halved.
Answers:
𝑦 = 𝑘𝑥 2
𝐿𝑒𝑡 𝑥 = 1
8 = 𝑘(1)2
8=𝑘
∴ 𝑦 = 8𝑥 2
1
𝑁𝑜𝑤, 𝑥 =
2
1 2
𝑦 = 8( ) = 2
2
Revision
Direct and Inverse Proportions
Question 6:
y is proportional to x2 and y = 8 and x = 4.
a) Find the value of y when x = 10.
b) What happens to y when x is doubled.
Answers:
𝑦 = 𝑘𝑥 2
8 = 𝑘(4)2
1
=𝑘
2
1
∴ 𝑦 = 𝑥2
2
1
a) 𝑦 = (10)2 = 50
2
b) 𝑁𝑜𝑤, 𝑥 = 20
1
𝑦 = (20)2
2
= 200
∴y will increase by 4 times (from 50 to 200)
Revision
Direct and Inverse Proportions
Question 7:
p is directly proportional to q3. When q has a certain value, p = 24. Find the value of p
when this value of q is doubled
Answers:
𝑝 = 𝑘𝑞 3
Let 𝑞 = 1
24 = 𝑘(1)3
24 = 𝑘
∴ 𝑝 = 24𝑞3
New 𝑞 = 2
= 24(2)3 = 192
Revision
Direct and Inverse Proportions
Question 8:
y is proportional to x 2 . x is increased by 100%. Find the percentage increase in y.
Answers:
𝑦 = 𝑘𝑥 2
𝐿𝑒𝑡 𝑥 = 1, 𝑦 = 1
1 = 𝑘(1)2
1=𝑘
𝑦 = 𝑥2
𝑁𝑒𝑤 𝑥 = 2
𝑦=4
Change in 𝑦 = 4 − 1 = 3
∴ 300% increase
Revision
Direct and Inverse Proportions
Question 9:
It is given that y is directly proportional to x2. When x is reduced by 50%, calculate the
percentage decrease in y.
Answers:
𝑦 = 𝑘𝑥 2
𝐿𝑒𝑡 𝑥 = 1, 𝑦 = 1
1 = 𝑘(1)2
1=𝑘
∴ 𝑦 = 𝑥2
𝑁𝑒𝑤 𝑥 = 0.5
𝑦 = (0.5)2 = 0.25
Change in 𝑦 = 0.25 − 1 = 0.75
∴ 75% decrease
Revision
Direct and Inverse Proportions
Question 10:
The surface area, A cm2 of a container varies directly as the square of its radius r cm.
a) Describe the effect on A when r is halved.
b) Find the percentage increase in r when A increases by 300%.
Answers:
a)
𝐴 = 𝑘𝑟 2
𝐿𝑒𝑡 𝐴 = 1, 𝑟 = 1
1 = 𝑘(1)2
1=𝑘
∴ 𝐴 = 𝑟2
So when 𝑟 = 0.5,
𝐴 = (0.5)2 = 0.25
Area is reduced by 75%.
b)
New A = 400% = 4
4 = 𝑟2
2=𝑟
Change in 𝑟 = 2 − 1 = 1
100% increase.
Revision
Direct and Inverse Proportions
Question 11:
W is directly proportional to I2. I is increased by 150%.
Find the percentage increase in W.
Answers:
𝑊 = 𝑘𝑙 2
𝐿𝑒𝑡 𝑊 = 1, 𝑙 = 1
1 = 𝑘(1)2
1=𝑘
∴ 𝑊 = 𝑙2
𝑁𝑒𝑤 𝑙 = 250% = 2.5
𝑊 = (2.5)2 = 6.25
Change in W = 6.25 − 1 = 5.25
∴ 525%
Revision
Direct and Inverse Proportions
Question 12:
It is given that y varies inversely as the square of x and y = 16 when x = 2.
Find an expression for the relationship between y and x.
Answers:
𝑘
𝑥2
𝑘
16 =
22
64 = 𝑘
64
∴𝑦= 2
𝑥
𝑦=
Revision
Direct and Inverse Proportions
Question 13:
y is inversely proportional to the square of x. It is know that y = 4 for a particular value
of x. Find the value of y when this value of x is halved.
Answers:
𝑘
𝑥2
Let 𝑥 = 1,
𝑘
4=
1
4=𝑘
4
∴𝑦= 2
𝑥
New 𝑥 = 0.5
4
𝑦=
= 16
0.52
𝑦=
Revision
Direct and Inverse Proportions
Question 14:
It is given that y is inversely proportional to x3, and that y = 640 for a particular value of
x. Find the value of y when this value of x is doubled.
Answers:
𝑘
𝑥3
𝐿𝑒𝑡 𝑥 = 1, 𝑦 = 640
𝑘
640 =
1
640 = 𝑘
New 𝑥 = 2
640
𝑦 = 3 = 80
2
𝑦=
Revision
Direct and Inverse Proportions
Question 15:
The resistance of a wire of constant length varies inversely to the square of its diameter.
The resistance is 23 ohms when the diameter is d mm. Find the resistance of the wire
when the diameter is halved.
Answers:
𝑘
𝑥2
𝑘
23 = 2
𝑑
2
23𝑑 = 𝑘
𝑅=
1
New Diameter = 𝑑
2
2
23𝑑
𝑅=
1
( 𝑑)2
2
= 92𝑜ℎ𝑚𝑠
Revision
Direct and Inverse Proportions
Question 16:
y is inversely proportional to x3. It is known that y = 32 for a particular value of x.
Find the value of y when this value of x is halved.
Answers:
𝑘
𝑥3
𝐿𝑒𝑡 𝑥 = 1, 𝑦 = 32
𝑘
32 = 3
1
32 = 𝑘
32
𝑦= 3
𝑥
𝑁𝑒𝑤 𝑥 = 0.5
32
𝑦=
0. 53
= 256
𝑦=
Revision
Direct and Inverse Proportions
Question 17:
m is inversely proportional to the square root of n.
At a certain value of n, the value of m is 20.
a) Write down an equation involving m and n.
b) Find the value of m when n is four times its original value.
Answers:
a) 𝑚 =
𝑘
√𝑛
b) 𝐿𝑒𝑡 𝑛 = 1
𝑘
20 =
√1
20 = 𝑘
20
∴𝑚=
√𝑛
𝑁𝑒𝑤 𝑛 = 4
20
𝑚=
= 10
√4
Revision
Direct and Inverse Proportions
Question 18:
The period of a pendulum, T, varies inversely as the square root of the acceleration due
to gravity, g.
a) Write down an expression for T in terms of g, and a constant k.
b) Find the percentage change in T, when g is doubled.
Answers:
a) 𝑇 =
𝑘
√𝑔
b) 𝐿𝑒𝑡 𝑇 = 1, 𝑔 = 1
𝑘
1=
√1
1
∴𝑇=
√𝑔
𝑁𝑒𝑤 𝑔 = 2
1
𝑇=
= 0.707
√2
Change in T = 0.707 − 1
= −0.2928
∴ 29.3% decrease
Revision
Direct and Inverse Proportions
Question 19:
y is inversely proportional to x2. It is known that y = 500 for a particular x. Find the
value of y when this value of x is doubled.
Answers:
𝑘
𝑥2
𝐿𝑒𝑡 𝑦 = 500, 𝑥 = 1
𝑘
500 =
1
500 = 𝑘
𝑁𝑒𝑤 𝑥 = 2
500
𝑦 = 2 = 125
2
𝑦=
Revision
Direct and Inverse Proportions
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