Chapter 29: Problem 963

Why are slight differences in growth rate percentages so important?

Short Answer

Expert verified

Slight differences in growth rate percentages are important because they can lead to significant differences in values over time. As demonstrated in the example of two investment options with annual growth rates of 5% and 6%, after 10 years, the 1% difference in growth rate results in a substantial difference of $161.96 between the two investments. This concept is crucial when making decisions like comparing economic growth rates, evaluating population growth, or investing assets. Understanding the implications of different growth rates is essential for informed decision-making.

Step by step solution

Understand Growth Rate Percentage

Growth rate percentage measures the increase or decrease in a particular value over a specific period of time. It is expressed as a percentage and compares the final value to the initial value. Growth rates are commonly used in economics, finance, and population studies to understand how fast a value is growing or shrinking.

A Small Difference in Growth Rates

Let's consider an example to see how a small difference in growth rates can lead to significant differences over time. Assume we have an initial investment of $1000 and two different investment options: Option A has an annual growth rate of 5%, and Option B has an annual growth rate of 6%. We will calculate how much money we would have after 10 years with each option.

Calculate Growth using Option A

With Option A, our initial investment of $1000 will grow at an annual rate of 5%. Using the formula for compound interest, we can calculate the value of our investment after 10 years: Investment Value = P(1 + r/n)^(nt) Where P is the initial investment, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years. In our example, P = 1000, r = 0.05, n = 1 (assuming annual compounding), and t = 10. Investment Value = 1000(1 + 0.05/1)^(1*10) Investment Value = 1000(1.05)^10 Investment Value ≈ $1628.89

Calculate Growth using Option B

With Option B, our initial investment of $1000 will grow at an annual rate of 6%. Using the same formula for compound interest, we can calculate the value of our investment after 10 years: Investment Value = P(1 + r/n)^(nt) In this case, P = 1000, r = 0.06, n = 1, and t = 10. Investment Value = 1000(1 + 0.06/1)^(1*10) Investment Value = 1000(1.06)^10 Investment Value ≈ $1790.85

Comparing the Results

After 10 years, the value of our investment with Option A (5% growth rate) is $1628.89, while the value of our investment with Option B (6% growth rate) is $1790.85. This is a difference of $161.96, which is significant considering the initial investment of $1000.

Conclusion

Even slight differences in growth rate percentages can have a substantial impact on values over time, as demonstrated in our example. This is important to take into consideration when making investment decisions, comparing economic growth rates between countries, or evaluating population growth. A small increase in growth rate can lead to significantly higher values over time, making it essential to understand the implications of different growth rates when making decisions.

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