Lesson 17: Different Compounding Intervals Let's find out what happens when we repeatedly apply the same percent increase at different intervals of time. 17.1: Returns Over Three Years Earlier, you learned about a bank account that had an initial balance of \( \$ 1,000 \) and earned 1\% monthly interest. Each month, the interest was added to the account and no other deposits or withdrawals were made. To calculate the account balance in dollars after 3 years, Elena wrote: \( 1,000 \cdot(1.01)^{36} \) and Tyler wrote: \( 1,000 \cdot\left((1.01)^{12}\right)^{3} \). Discuss with a partner: 1. Why do Elena's expression and Tyler's expression both represent the account balance correctly? 2. Kiran said, "The account balance is about \( 1,000 \cdot(1.1268)^{3} \)." Do you agree? Why or why not?

To analyze the expressions provided by Elena, Tyler, and Kiran, let's break down the problem step by step. ### Known Conditions - Initial balance: \( P = 1000 \) - Monthly interest rate: \( r = 0.01 \) (1%) - Time period: 3 years, which is equivalent to \( t = 36 \) months (since \( 3 \times 12 = 36 \)) ### 1. Analyzing Elena's Expression Elena wrote: \[ B = 1000 \cdot (1.01)^{36} \] This expression represents the account balance after 36 months. The term \( (1.01)^{36} \) indicates that the interest is compounded monthly for 36 months. Each month, the account earns 1% interest on the current balance, and this is compounded over the entire period. ### 2. Analyzing Tyler's Expression Tyler wrote: \[ B = 1000 \cdot \left((1.01)^{12}\right)^{3} \] This expression can be interpreted as follows: - \( (1.01)^{12} \) calculates the total balance after one year (12 months) of compounding monthly interest. - Raising this to the power of 3, \( \left((1.01)^{12}\right)^{3} \), means that this yearly balance is then compounded for 3 years. Using the property of exponents, we can simplify Tyler's expression: \[ \left((1.01)^{12}\right)^{3} = (1.01)^{36} \] Thus, Tyler's expression is equivalent to Elena's expression, and both correctly represent the account balance after 3 years. ### 3. Analyzing Kiran's Statement Kiran said: \[ B \approx 1000 \cdot (1.1268)^{3} \] To evaluate Kiran's statement, we need to understand what \( 1.1268 \) represents. If we calculate the balance after 3 years using the monthly compounding formula, we can find the actual value of \( (1.01)^{36} \). Let's calculate \( (1.01)^{36} \): \[ (1.01)^{36} \approx 1.4308 \] Thus, the correct expression for the balance after 3 years is: \[ B = 1000 \cdot 1.4308 \approx 1430.80 \] Now, let's evaluate \( (1.1268)^{3} \): \[ (1.1268)^{3} \approx 1.4308 \] This means Kiran's expression is also correct in terms of the final balance, but the interpretation of \( 1.1268 \) as a single compounding factor over 3 years is misleading. The value \( 1.1268 \) does not represent the effective annual rate for monthly compounding; it is simply a misinterpretation of the compounding process. ### Conclusion 1. Both Elena's and Tyler's expressions correctly represent the account balance after 3 years. 2. Kiran's expression is approximately correct in terms of the final balance, but it misrepresents the nature of compounding interest over time. The correct approach is to use the monthly compounding formula as shown in Elena's and Tyler's expressions.