On July 1, 1996, Anna invested $2000 in an account that earned 6%/a compounded monthly. On July 1, 2001, she moved the total amount to a new account that paid 8%/a compounded quarterly. Determine the balance in her new account on January 1, 2008.
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To determine the balance in Anna's new account on January 1, 2008, we need to calculate the future value of her initial investment compounded quarterly over the given time period.
First, let's calculate the number of compounding periods for each account:
Account 1:
Time period: July 1, 1996 to July 1, 2001
Number of years: 5
Number of compounding periods per year: 12 (compounded monthly)
Total number of compounding periods: 5 * 12 = 60
Account 2:
Time period: July 1, 2001 to January 1, 2008
Number of years: 6.5
Number of compounding periods per year: 4 (compounded quarterly)
Total number of compounding periods: 6.5 * 4 = 26
Next, let's calculate the future value of the initial investment in each account:
Account 1:
Principal amount: $2000
Annual interest rate: 6%
Number of compounding periods: 60
Using the formula for compound interest:
Future Value = Principal * (1 + (interest rate / number of compounding periods))^(number of compounding periods * number of years)
Future Value = $2000 * (1 + (0.06 / 12))^(60 * 5)
Calculating this value gives us the balance of Account 1 on July 1, 2001.
Account 2:
Principal amount: Balance of Account 1 on July 1, 2001
Annual interest rate: 8%
Number of compounding periods: 26
Using the same compound interest formula:
Future Value = Principal * (1 + (interest rate / number of compounding periods))^(number of compounding periods * number of years)
Future Value = (Balance of Account 1 on July 1, 2001) * (1 + (0.08 / 4))^(26 * 6.5)
Calculating this value gives us the balance of Anna's new account on January 1, 2008.