46 Basic Engineering Mathematics
(working at the same rate) would take 6 hours. Half the
men, twice the time. (Note that 4 × 3 = 2 × 6.)
Here are some worked examples on inverse proportion.
Problem 17. It is estimated that a team of four
designers would take a year to develop an
engineering process. How long would three
designers take?
If 4 designers take 1 year, then 1 designer would take
4 years to develop the process. Hence, 3 designers would
take
4
3
years, i.e. 1 year 4 months.
Problem 18. A team of five people can deliver
leaflets to every house in a particular area in four
hours. How long will it take a team of three people?
If 5 people take 4 hours to deliver the leaflets, then
1 person would take 5 × 4 = 20 hours. Hence, 3 people would take
20
3
hours, i.e. 6
2
3
hours, i.e. 6 hours
40 minutes.
Problem 19. The electrical resistance R of a
piece of wire is inversely proportional to the
cross-sectional area A. When A = 5 mm 2 ,
R = 7.02 ohms. Determine (a) the coefficient of
proportionality and (b) the cross-sectional area
when the resistance is 4 ohms
(a) R α
1
A
, i.e. R =
k
A
or k = R A. Hence, when
R = 7.2 and A = 5, the
coefficient of proportionality, k = (7.2)(5) = 36
(b) Since k = R A then A =
k
R
. Hence, when R = 4,
the cross sectional area, A =
36
4
= 9 mm
2
Problem 20. Boyle’s law states that, at constant
temperature, the volume V of a fixed mass of gas is
inversely proportional to its absolute pressure p. If
a gas occupies a volume of 0.08 m
3 at a pressure of
1.5 × 10
6 pascals, determine (a) the coefficient of
proportionality and (b) the volume if the pressure is
changed to 4 × 10 6 pascals
(a) V ∝
1
p
i.e. V =
k
p
or k = pV . Hence, the
coefficient of proportionality, k
= (1.5 × 10
6
)(0.08) = 0.12 × 10
6
(b) Volume,V =
k
p
=
0.12 × 10 6
4 × 10 6 = 0.03 m
3
Now try the following Practice Exercise
Practice Exercise 28 Further inverse
proportion (answers on page 342)
1. A 10 kg bag of potatoes lasts for a week with a
family of 7 people. Assuming all eat the same
amount, how long will the potatoes last if there
are only two in the family?
2. If 8 men take 5 days to build a wall, how long
would it take 2 men?
3. If y is inversely proportional to x and
y = 15.3 when x = 0.6, determine (a) the
coefficient of proportionality, (b) the value of
y when x is 1.5 and (c) the value of x when y
is 27.2
4. A car travelling at 50 km/h makes a journey in
70 minutes. How long will the journey take at
70 km/h?
5. Boyle’s law states that, for a gas at constant
temperature, the volume of a fixed mass of
gas is inversely proportional to its absolute
pressure. If a gas occupies a volume of 1.5 m
3
at a pressure of 200 × 10 3 pascals, determine
(a) the constant of proportionality, (b) the
volume when the pressure is 800 × 10 3 pascals and (c) the pressure when the volume is
1.25 m 3 .
(working at the same rate) would take 6 hours. Half the
men, twice the time. (Note that 4 × 3 = 2 × 6.)
Here are some worked examples on inverse proportion.
Problem 17. It is estimated that a team of four
designers would take a year to develop an
engineering process. How long would three
designers take?
If 4 designers take 1 year, then 1 designer would take
4 years to develop the process. Hence, 3 designers would
take
4
3
years, i.e. 1 year 4 months.
Problem 18. A team of five people can deliver
leaflets to every house in a particular area in four
hours. How long will it take a team of three people?
If 5 people take 4 hours to deliver the leaflets, then
1 person would take 5 × 4 = 20 hours. Hence, 3 people would take
20
3
hours, i.e. 6
2
3
hours, i.e. 6 hours
40 minutes.
Problem 19. The electrical resistance R of a
piece of wire is inversely proportional to the
cross-sectional area A. When A = 5 mm 2 ,
R = 7.02 ohms. Determine (a) the coefficient of
proportionality and (b) the cross-sectional area
when the resistance is 4 ohms
(a) R α
1
A
, i.e. R =
k
A
or k = R A. Hence, when
R = 7.2 and A = 5, the
coefficient of proportionality, k = (7.2)(5) = 36
(b) Since k = R A then A =
k
R
. Hence, when R = 4,
the cross sectional area, A =
36
4
= 9 mm
2
Problem 20. Boyle’s law states that, at constant
temperature, the volume V of a fixed mass of gas is
inversely proportional to its absolute pressure p. If
a gas occupies a volume of 0.08 m
3 at a pressure of
1.5 × 10
6 pascals, determine (a) the coefficient of
proportionality and (b) the volume if the pressure is
changed to 4 × 10 6 pascals
(a) V ∝
1
p
i.e. V =
k
p
or k = pV . Hence, the
coefficient of proportionality, k
= (1.5 × 10
6
)(0.08) = 0.12 × 10
6
(b) Volume,V =
k
p
=
0.12 × 10 6
4 × 10 6 = 0.03 m
3
Now try the following Practice Exercise
Practice Exercise 28 Further inverse
proportion (answers on page 342)
1. A 10 kg bag of potatoes lasts for a week with a
family of 7 people. Assuming all eat the same
amount, how long will the potatoes last if there
are only two in the family?
2. If 8 men take 5 days to build a wall, how long
would it take 2 men?
3. If y is inversely proportional to x and
y = 15.3 when x = 0.6, determine (a) the
coefficient of proportionality, (b) the value of
y when x is 1.5 and (c) the value of x when y
is 27.2
4. A car travelling at 50 km/h makes a journey in
70 minutes. How long will the journey take at
70 km/h?
5. Boyle’s law states that, for a gas at constant
temperature, the volume of a fixed mass of
gas is inversely proportional to its absolute
pressure. If a gas occupies a volume of 1.5 m
3
at a pressure of 200 × 10 3 pascals, determine
(a) the constant of proportionality, (b) the
volume when the pressure is 800 × 10 3 pascals and (c) the pressure when the volume is
1.25 m 3 .