Contents

Population Growth

Depreciation


Population Growth and Deprication

Population Growth and Depreciation


To understand the formulae of this chapter more clearly, we recommend you having a look and better understanding at the derivation of Formula of Compound Interest and Compound Amount. 

Population Growth:

Population is the total number of people residing in a particular place over a particular period of time. Population is a variable factor that changes its size over time. 

In some countries, the population might be growing. While, in other countries, the population might be constant or decreasing, over time.

The size of population of a place depends on number of factors, such as: birth, death, and migration.

Population Growth is the increase in number of individuals residing in a particular place over a particular period of time. 

As we discussed in our previous chapter, Compound Interest, that calculation of compound amount is done by taking the Amount generated in the previous year as principal.

Calculation of Population Growth is also similar to calculation of Compound Amount.

When,
The population of a place at a specific period of time = P
Rate of growth of population per annum = R%
The population of the country after 'T' years = Pt

Then,

[To understand this formula, check Derivation of formula of Compound Amount]



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As we already stated above that the population is variable,
The actual population after 'T' years, Pt is given by the formula:

Pt = P(1 +R/100)^T -Deaths + In-migrants - Out-Migrants

Explanation of above formula:

The formula P(1 +R/100) predicts the population after 'T' years with the same growth rate per annum. 

But, factors like death and Out-Migration decrease the population of a place while In-Migration increases the population of the place.



Now, to find the total number of increased population over 'T' years, 

We have,
Increased Population = Pt - P
= P(1 +R/100)^T - P
= P[ (1 +R/100)^T -1]


  • If the rate of growth of population is different in every year.

Suppose,
Population Growth Rate in Year 1 = R1%
Population Growth Rate in Year 2 = R2%
Population Growth Rate in Year 3 = R3%
......
Population Growth Rate in Year T = RT%

We calculate the population after years as:

 
Pt = P (1 +R1/100) (1 +R2/100) (1 +R3/100) ....(1 +RT/100)
 

And,
Increased Population = Pt - P
=  P (1 +R1/100) (1 +R2/100) (1 +R3/100).... (1 +RT/100) - P
= P [ (1 +R1/100) (1 +R2/100) (1 +R3/100) (1 +RT/100) - 1]



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Depreciation:

When we buy products like machineries, vehicles, furniture, etc from the shop for the first time, we say those items are first hand.

When we sell those items to someone else, they become second hand items.

When they sell them again to someone else, they become third hand items.

But, you will notice that the cost of the same item will be different when it is sold in different time as compared to the cost of product when bought for the first time.

The major factor that decreases the cost of these items is the duration of their use. 

So, the decrease in the rate of an item due to its continuous use is known as depreciation.

Depreciation may be simple depreciation or compound depreciation.

In simple depreciation, the value of the item decreases constantly every year.

For example, the original cost of a machine is Rs 24000 and every year, it is depreciated by Rs 1200.

So,
The cost of item after two years is given by:

Cost of the item after 1 year = Rs 24000- Rs 1200
= Rs 22,800

Cost of the item after 2 years = Rs 22800 - Rs 1200
= Rs 21600

In simple Depreciation,

Cost of item after T years = Original Cost - T*Cost of Depreciation
 

In compound depreciation, the value of the item decreases in percentage every year. 

Calculation of the cost of item after depreciation is calculated in similar manner as the calculation of Compound amount.

However, here the amount is decreasing so, we put a '-' sign instead of a '+' sign.

When,
The original price of an item = P
Rate of Depreciation = R
Duration of time of depreciation = T
The depreciated value of the item after 'T' years = Pt


So, Pt = P (1 -R/100)^T

And, Depreciated amount = P - Pt
= P - P (1 -R/100)^T
= P [ 1 - (1 -R/100)^T]



Whereas, if the rate of depreciation is different in different years, 

Suppose,
Rate of Depreciation in Year 1 = R1%
Rate of Depreciation in Year 2 = R2%
Rate of Depreciation in Year 3 = R3%
......
Rate of Depreciation in Year T = RT%

So, the depreciated value of the item (Pt) 
= P (1 +R1/100) (1 +R2/100) (1 +R3/100) .... (1 +RT/100)

We do recommend you to check: Compound Interest Notes, to get clear about the derivation of the formulae of compound Amount and compound Interest.

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