Question

True or False? In Exercises \(67-70\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If prices are rising at a rate of 0.5\(\%\) per month, then they are rising at a rate of 6\(\%\) per year.

Short answer

The statement is false. Prices are rising at an approximate rate of 6.17% per year, not 6%, due to compound interest.

Step-by-step solution

Understanding the Problem

We are given that the prices are rising at a rate of 0.5% per month. We need to find out if they would rise by 6% at the end of the year.

Calculation

Firstly, let's understand the calculation if it was simple interest. For simple interest, if something increases by 0.5% per month, we multiply by the number of months in a year to find the annual rate: \(0.5\% * 12 = 6\%\). However, this situation regards compound interest. Therefore, we need to take into account that after month 1 the total has increased, so in month 2 the increase is not only 0.5% of the original total, but 0.5% of the new total after month 1. This would keep going for all the 12 months of the year. So, we should calculate it this way: \((1 + 0.005)^{12} - 1\). This formula calculates the increase for each month and then subtracts the original total to find the total increase over the year.

Conclusion

By doing the calculation we find that the rate is not 6%, but rather about 6.17%. Therefore, the statement 'If prices are rising at a rate of 0.5% per month, then they are rising at a rate of 6% per year' is false. The prices are actually rising at a rate of about 6.17% annually when considering compound interest.

Key concepts

Monthly Interest Rate

Understanding the concept of a monthly interest rate is key in grasping how compound interest works. When we talk about a monthly interest rate, we mean the interest rate applied to the principal amount every month. The monthly interest rate can be calculated by dividing the annual interest rate by 12 (as there are 12 months in a year). However, this exercise focuses on a compound monthly interest rate of 0.5%, meaning that each month 0.5% is added to the new balance, not just the original principal. This means that the amount continues to grow by differing dollar amounts each month because it is also growing based on the previously accumulated interest.

Let’s break it down further with these key points:

• Each month's interest is calculated using the updated balance, including any past interest added.
• This creates a "compounding effect" where the interest earns its own interest over time.
• Understanding how this compounding works monthly can make drastic changes to annual percentages.

Annual Interest Rate

When examining annual interest rates, it's crucial to distinguish between the simplicity of just adding up monthly rates and the effect of compounding interest. If you were to evaluate these monthly rates by simply aggregating them in a straightforward manner, you'd conclude that the total annual interest rate equals the monthly interest rate multiplied by 12. For instance, if the monthly rate is 0.5%, multiplying by 12 gives an annual rate of 6%, but this assumes simple interest.

Since we are dealing with compound interest, the calculation is more complex. Using compound interest, the formula adjusts the principal at each month before applying the interest rate again. Therefore, the effective annual rate becomes higher than merely multiplying by 12. More formally, with a monthly rate of 0.5%, the correct approach would employ the formula \( (1+ rac{0.5}{100})^{12} - 1 \) to find the accurate annual growth rate based on monthly compounding.

• Annual rates derived from compound monthly rates tend to be higher due to repeated compounding.
• The higher effective annual rate is seen in real-world financial products, where compounding is standard practice.

Exponential Growth

Exponential growth is a fundamental concept in understanding how compound interest accumulates over time. Unlike linear growth, where changes happen at a steady rate, exponential growth means that the rate of change actually increases as time goes on. This happens because with each period, the interest not only applies to the initial principal but also applies to any interest accrued from prior periods.

In the context of this exercise, exponential growth is demonstrated by calculating \( (1 + 0.005)^{12} \). Each of these operations (raising to the power of 12) compiles the interest over a designated number of periods, here monthly, giving rise to a situation where each month's increment builds on the previous increments. This accumulation effect can end up increasing the principal at a much higher rate than one might initially expect.

Key takeaways include:

• Compounding leads to much faster growth over time due to the cumulative effect of interest on interest.
• An understanding of exponential growth is essential, especially in areas like finance, biology, and population studies.
• Visualizing exponential growth with graphs shows an upward curve that becomes steeper over time.