Understanding Annual Compound Growth Through Real-World Examples
As a SEO expert at Google, navigating the complex world of compound growth rates is a key skill for analyzing data and providing insights on business growth. This article explores a practical example—calculating the annual compound growth rate of a city whose population increases by 15 every 5 years. By breaking down the calculation and providing a step-by-step approach, we aim to offer clarity and context for those in similar analytical roles.
The Problem: Growth Rates and Population Increase
Imagine a city that grows by 15 every 5 years. Our goal is to determine the annual compound growth rate over this same period. This kind of problem is common in understanding population dynamics, business growth, and economic forecasting. If the city starts with a population of 100 and grows to 115 in 5 years, we need to find the annual rate at which this growth occurs.
Step-By-Step Calculation of the Annual Compound Growth Rate
The formula for calculating the compound growth rate is given as:
( 1r^5 115 )
Where:
( 1r ) is the annual growth rate that we need to find. 5 years is the total period of growth. 115 is the final value after 5 years, indicating an increase from the initial value of 100.Starting with the equation:
( 1r^5 115 )
First, divide 115 by the initial value to find the factor by which the population has grown:
( 1r^5 frac{115}{100} 1.15 )
Next, we apply logarithms to solve for ( 1r ). Logarithms help us to deal with exponents in algebraic expressions, making it easier to isolate the variable:
( log 1r^5 log 1.15 )
By using the logarithm properties, we can separate the exponent:
( 5 log 1r log 1.15 )
Now, isolate ( log 1r ) by dividing both sides by 5:
( log 1r frac{log 1.15}{5} approx 0.01213956807 )
To find ( 1r ), we take the antilogarithm (or power of 10) of both sides:
( 1r 10^{0.01213956807} approx 1.0283467221 )
Thus, the annual growth rate ( r ) is approximately 0.0283467221, or more simply put, 2.83%.
Conclusion: Practical Applications of Compound Growth Rates
Understanding and calculating annual compound growth rates can be crucial for various fields, including finance, economics, and urban planning. For a city like the one discussed, this calculation can help in planning future infrastructure, resource allocation, and policy-making based on anticipated growth patterns.
By using modern tools like spreadsheets and financial calculators, these calculations become more accessible. However, knowing the underlying mathematics is still essential for validating results and ensuring accurate forecasting.