1) Compound rate can be applied in the context of the increase of population to understand and predict population growth over time. In this scenario, the population acts as the principal amount, and the annual growth rate represents the compound interest.
The formula for calculating the future population using compound rate is:
A = \(P * (1 + r)^t\)
Where: A = the future population after time t
P = the current population (principal amount)
r = the annual growth rate (expressed as a decimal)
t = the number of years
For example, if a city's current population is 1,000,000 and it has an annual population growth rate of 2%, we can calculate the population after 5 years as follows:
A = \(1,000,000 * (1 + 0.02)^5\)
A = \(1,000,000 * (1.02)^5\)
A = \(1,000,000 * 1.10408\)
A = 1,104,080
So, the population of the city is estimated to be approximately 1,104,080 after 5 years, assuming a 2% annual population growth rate.
This application of compound rate in population growth helps governments, urban planners, and policymakers in making informed decisions about infrastructure, resources, and services required to accommodate the increasing population. It also aids in understanding population trends and projecting future demographics, which is crucial for effective long-term planning and development.
2) Compound rate can also be applied in the context of the decrease of population to understand and predict population decline over time. In this scenario, the population acts as the principal amount, and the negative annual growth rate represents the compound interest.
The formula for calculating the future population with a compound rate of decrease is:
A = \(P * (1 - r)^t\)
Where: A = the future population after time t
P = the current population (principal amount)
r = the annual decrease rate (expressed as a decimal)
t = the number of years
For example, if a town's current population is 50,000 and it has an annual population decrease rate of 1.5%, we can calculate the population after 10 years as follows:
A = \(50,000 * (1 - 0.015)^10\)
A = \(50,000 * (0.985)^10\)
A = \(50,000 * 0.8534\)
A = 42,670
So, the population of the town is estimated to be approximately 42,670 after 10 years, assuming a 1.5% annual decrease in population.
This application of compound rate in population decline is essential for understanding demographic changes and planning for the social and economic impacts of a shrinking population. It helps local authorities and policymakers identify potential challenges related to services, infrastructure, and resource allocation, and provides insights for designing strategies to address the consequences of population decrease.
3) Compound rate can be applied in the context of the depreciation of machines, cars, or other assets to understand how their value decreases over time. In this scenario, the initial value of the asset acts as the principal amount, and the annual depreciation rate represents the compound interest in the context of decreasing value.
The formula for calculating the future value (or remaining value) of the asset with a compound rate of depreciation is:
A = \P * (1 - r)^t\)
Where: A = the future value (or remaining value) of the asset after time t P = the initial value of the asset (principal amount) r = the annual depreciation rate (expressed as a decimal) t = the number of years
For example, if you have purchased a car for $30,000, and it has an annual depreciation rate of 10%, you can calculate the estimated value of the car after 5 years as follows:
A = \($30,000 * (1 - 0.10)^5\)
A = \$30,000 * (0.90)^5\)
A = \$30,000 * 0.59049\)
A = $17,714.7
So, the estimated value of the car after 5 years, assuming a 10% annual depreciation rate, would be approximately $17,714.7.
This application of compound rate in asset depreciation is crucial for financial planning, budgeting, and making informed decisions about the replacement or disposal of assets. It helps individuals and businesses understand the decline in the value of their investments and plan for the impact of depreciation on their overall financial picture.
Illustration: A machine depreciates in value by 5% on the first year. And by 10% in second year. If the machine was purchased by Rs 2,00,000. Find it's depreciated value at the end of second year.
Ans:
To calculate the depreciated value of the machine at the end of the second year, we need to apply the depreciation rates for each year.
Let's start with the initial value of the machine (P) which is Rs 200,000.
In the first year, the machine depreciates by 5%, so the depreciated value after the first year (A1) can be calculated as:
A1 = P * (1 - r1) = 200,000 * (1 - 0.05) = 200,000 * 0.95 = Rs 190,000
Now, for the second year, we need to take the depreciated value from the first year (A1) as the new principal amount. The machine depreciates further by 10%, so the depreciated value at the end of the second year (A2) can be calculated as:
A2 = A1 * (1 - r2) = 190,000 * (1 - 0.10) = 190,000 * 0.90 = Rs 171,000
So, the depreciated value of the machine at the end of the second year is Rs 171,000.
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