Question: Hattie has recently inherited $8300, which she wants to deposit into an IRA account. She has…

Hattie has recently inherited $8300, which she wants to deposit into an IRA account. She has determined that her two best bets are an account that compounds annually at an annual rate of 5.3% (Account 1) and an account that compounds semi-annually at an annual rate of 2.7% (Account 2).

Step 1 of 2: Which account would pay Hattie more interest?

Answer

Account 1 Account 2

Solution

To determine which account would pay Hattie more interest, we need to calculate the amount of interest each account would generate over the same period and compare them. Let’s assume Hattie will keep her money in the accounts for one year for simplicity. For Account 1, which compounds annually at an annual rate of 5.3%: The formula for compound interest is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \(A\) is the amount of money accumulated after n years, including interest. - \(P\) is the principal amount ($8300). - \(r\) is the annual interest rate (5.3% or 0.053). - \(n\) is the number of times that interest is compounded per year (1 for annually). - \(t\) is the time the money is invested for in years (1 year for this example). Substituting these values for Account 1: \[ A_1 = 8300 \left(1 + \frac{0.053}{1}\right)^{1 \times 1} = 8300 \times 1.053 \] Calculating \(A_1\): \[ A_1 = 8743.9 \] For Account 2, which compounds semi-annually at an annual rate of 2.7%: Substitute the values into the compound interest formula: \[ A_2 = 8300 \left(1 + \frac{0.027}{2}\right)^{2 \times 1} = 8300 \times \left(1 + 0.0135\right)^2 \] Calculate inside the parenthesis: \[ A_2 = 8300 \times 1.0135^2 \] Calculate \(1.0135^2\): \[ 1.0135^2 = 1.02728225 \] Then calculate \(A_2\): \[ A_2 = 8300 \times 1.02728225 = 8526.444575 \] Account 1: $8743.9 Account 2: $8526.44 Since Account 1 has a higher accumulated amount after one year, it would pay Hattie more interest. Therefore, Account 1 is the better option.