The document discusses various mathematical methods for population studies relevant to water supply projects, focusing on the arithmetic, geometric, saturation, and increasing factor methods. Each method is illustrated with examples, comparing their accuracy for population predictions. The saturation method is highlighted as the most accurate, considering factors like saturation limits, whereas the arithmetic method is deemed the least reliable due to its linear assumptions.

1. 7th
March 2017
Faculty of Engineering – Suez canal University
Methods of Population
Studies
Sanitary Engineering
By: Amira Abdallah Youssef
Level3 – Civil Eng. Department
For: Dr. Abeer El-Shahawy

2. INTRODUCTION
- One of the basic studies that need to be undertaken in water supply Projects
is Population studies.
- In this report, I shall shade the light on the mathematical methods of
population studies, apply on an example and compare between results to
find which one is most accurate.
METHODS OF POPULATION STUDIES:
- The following charts illustrate different methods for calculating population
Methods of
Calculations
Graphical
Methods
Extention
Method
Comparsion
Method
Mathimatical
Methods
Arithmetic
Method
Geometric
method
Increasing
Factor
method
Saturation
method

4. MATHEMATICAL METHODS
1.Arithmetic Method
- This Method is used in small cities which have almost constant rate of
population growth due to the stability of the rate of migration and adapting
to living conditions.
- This method depends on the linear increase of population.
Pf = Po + Ka(Tf − Ti)
Where
- Pf : Population in future
- Po : Population in present
- Ka : Arithmetic constant = average of rate (
∆P
∆t
)
- Tf : The design year
- Ti : The year of known population
- (Tf − Ti): Design Period

5. Example 1:
- Given
Time (years) 0891 0881 0111 0112
Population
(capita)
01211 00111 00111 08111
- Required Find the population at 2030
- Solution
Time (year) Population
(capita)
∆P ∆t ∆P
∆t
1980 20500 1500
4000
3000
10
10
5
150
400
600
1990 22000
2000 26000
2005 29000
- Ka =
∑
∆P
∆t
no of
∆P
∆t
=
150+400+600
3
= 383.33
- P𝑜 = 29000
𝑡 𝑜 = 2005
𝑡 𝑓 = 2030
P𝑓 = P𝑜 + K 𝑎(T𝑓 − T𝑜) = 29000 + 383.33(2030 − 2005) = 38583.25
≅ 38584 𝑐𝑎𝑝𝑖𝑡𝑎

6. 2.Geometric Method
- This method depends on linear increase of the logarithm of population
- This method is used in new cities where population increase rapidly due
to availability of jobs so we use logarithmic scale
lnP𝑓 = ln P𝑖 + K 𝑔(T𝑓 − 𝑇𝑖)
Where
- Pf : Population in future
- Po : Population in present
- Kg : Geometric constant = average of rate (
∆ln P
∆t
)
- Tf : The design year
- Ti : The year of known population
- (Tf − Ti): Design Period
Example 2 (same example with geometric method)
- Given
Time (years) 0891 0881 0111 0112
Population(capita) 01211 00111 00111 08111
- RequiredFind the populationat 2030
- Solution
Time (year) Population
(capita)
∆P LNP ∆t ∆ LN P
∆t
1980 20500
1500
4000
3000
0.07
0.162
0.11
10
10
5
0.007
0.0162
0.022
1990 22000
2000 26000
2005 29000

7. K 𝑔 =
∑
Δln P
Δ𝑡
𝑛𝑜 𝑜𝑓
Δ lnP
Δ𝑡
=
0.007+0.0162+0.022
3
= 0.015
ln P𝑓 = ln P𝑖 + K 𝑔(T𝑓 − 𝑇𝑖) = LN 29000+ 0.015(2030 − 2005) = 42194.75
≅ 42195 𝑐𝑎𝑝𝑖𝑡𝑎
3. Saturation Method
- Also called decreasing rate of increase method.
- The rate of growth decreases with the saturation limit known in the
problem.
- This method assumes that rate of population growth with time is directly
proportional to (saturation limit – current population) and the constant of
proportionality is 𝑘 𝑑.
𝑃𝑓 = S − (S − 𝑃𝑜)e−𝑘 𝑑(𝑡 𝑓−𝑡 𝑜)
Where
- Pf : Populationin future
- Po : Population in present
- Tf : The design year
- Ti : The year of known population
- (Tf − Ti): Design Period
Kd =
−ln(
S − P2
S − P1
)
tf − to

8. Example 3 (same example with Saturation Method)
- Given S=150,000 capita
Time (years) 0891 0881 0111 0112
Population(capita) 01211 00111 00111 08111
- RequiredFind the populationat 2030
- Solution
year pop
S - P
K
1980 20500 129500
1990 22000 128000 0.988 -0.012 10 0.0012
2000 26000 124000 0.969 -0.031 10 0.0031
2005 29000 121000 0.976 -0.024 5 0.0048
∑ 𝑘 = 0.0091
𝑘 𝑑 = 𝑘 𝑎𝑣 =
∑ 𝑘
𝑛
=
0.0091
3
=0.003
𝑃𝑓 = S − (S − 𝑃𝑜)e−𝑘 𝑑(𝑡 𝑓−𝑡 𝑜)
= 150000 − 12100𝑒−0.003(2030−2050)
= 37743 CAPITA
𝑆 − 𝑃2
𝑆 − 𝑃1
ln(
𝑆−𝑃2
𝑆−𝑃1
) ∆𝑡 = 𝑡1 − 𝑡2

9. 4.Increasing factor method
If
- Pf : Populationin future
- Po : Population in present
- Tf : The design year
- Ti : The year of known population
- (Tf − Ti): Design Period
- X : increasing factor (the annualrate of populationgrowth)
𝑷 𝒇 = 𝑷 𝒐(𝟏 +
𝒙
𝟏𝟎𝟎
)(𝒕 𝒇−𝒕 𝒐)
Example 4
Find the population of a city at year 2026 if the population at 1996 was 39000
capita and the increasing rate is 2.5%
- Solution
- Pf = Po(1 +
x
100
)(tf−to)
= 39000(1 +
2.5
100
)2026−1996
= 81805 capita

10. General View
- When comparingbetween the first three methods’ results :
Method Arithmetic Method Geometric Method Saturation Method
Population at 2030
(capita)
38584 42195 37743
Since Arithmetic method assumes linear increase in population, It’s the most
inaccurate way as the population may decrease due to diseases or wars.
The Geometric method is more accurate than arithmetic method as it uses
logarithmic relation which is more reliable than linear one, But it also assumes
that growth of population is in continuous increase which makes it inaccurate as
well.
The Saturation method appears to have the smallest value but the most accurate
one, because it took into consideration the saturation limitof the city or the
population of the city when it’s completely filled