Question: Question 3 of 12, Step 1 of 2 Hattie has recently inherited $8000, which she wants to deposit into…

Question 3 of 12, Step 1 of 2

Hattie has recently inherited $8000, which she wants to deposit into an IRA account. She has determined that her two best bets are an account that compounds semi-annually at an annual rate of 5.3% (Account 1) and an account that compounds annually at an annual rate of 2.6% (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 compare the compounded interest for each option after a given period. Let’s assume Hattie is interested in the amount after 1 year. Account 1: Compounds semi-annually at 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 (\$8000). - \( r \) is the annual interest rate (decimal) (5.3% = 0.053). - \( n \) is the number of times the interest is compounded per year (semi-annually = 2). - \( t \) is the time the money is invested for in years (1 year). Substituting the values, we get: \[ A = 8000 \left(1 + \frac{0.053}{2}\right)^{2 \times 1} \] \[ A = 8000 \left(1 + 0.0265\right)^{2} \] \[ A = 8000 \times (1.0265)^{2} \] \[ A = 8000 \times 1.05307025 \] \[ A \approx 8424.56 \] Account 2: Compounds annually at 2.6% Again using the compound interest formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Substituting the values for Account 2: \[ A = 8000 \left(1 + \frac{0.026}{1}\right)^{1 \times 1} \] \[ A = 8000 \times (1.026) \] \[ A = 8000 \times 1.026 \] \[ A = 8208 \] Conclusion: Account 1 will lead to approximately \$8424.56 after 1 year, whereas Account 2 will lead to \$8208. Therefore, Account 1 would pay Hattie more interest.