A deposit of $2,000 earns interest at a rate of 14% compounded quarterly. After two and a half years the interest rate changes to 13.5% compounded monthly. How much is in the account after six years?
To calculate the amount in the account after six years, we need to break down the problem into two parts:
1. The first part is calculating the amount after two and a half years at an interest rate of 14% compounded quarterly.
2. The second part is calculating the amount after the remaining three and a half years at an interest rate of 13.5% compounded monthly.
Let's start with the first part:
1. Calculate the number of compounding periods for two and a half years at a quarterly compounding rate:
Number of periods = 2.5 years * 4 quarters per year = 10 quarters
2. Calculate the amount after two and a half years using the formula for compound interest:
A = P * (1 + r/n)^(n*t)
Where:
A = Amount after the given time period
P = Principal amount (initial deposit)
r = Annual interest rate (as a decimal)
n = Number of compounding periods per year
t = Number of years
In this case:
P = $2,000
r = 14% = 0.14
n = 4 (compounded quarterly)
t = 2.5 years
Plugging in the values:
A = $2,000 * (1 + 0.14/4)^(4*2.5)
= $2,000 * (1 + 0.035)^(10)
= $2,000 * (1.035)^(10)
≈ $2,955.76
Now, let's move on to the second part:
1. Calculate the number of compounding periods for the remaining three and a half years at a monthly compounding rate:
Number of periods = 3.5 years * 12 months per year = 42 months
2. Calculate the amount after three and a half years using the same compound interest formula:
A = P * (1 + r/n)^(n*t)
Where:
A = Amount after the given time period
P = Principal amount (amount after the first part)
r = Annual interest rate (as a decimal)
n = Number of compounding periods per year
t = Number of years
In this case:
P = $2,955.76 (amount after two and a half years)
r = 13.5% = 0.135
n = 12 (compounded monthly)
t = 3.5 years
Plugging in the values:
A = $2,955.76 * (1 + 0.135/12)^(12*3.5)
≈ $2,955.76 * (1.01125)^(42)
≈ $4,269.35
Therefore, the amount in the account after six years would be approximately $4,269.35.