You deposit $2,000 in an account that pays 8% interest compounded semiannually. After 4 years, the interest rate is increased to 8.28% compounded quarterly. What will be the value of the account after 4 more years? (Do not round until the final answer. Then, round to the nearest cent.)
Pt = Po(1+r)^n.
r = (8%/2) / 100% = 0.04 = Semi-annual
% rate expressed as a decimal.
n = 2 comp./yr * 4yrs = 8 Compounding
periods.
Pt = 2000(1.04)^8 = 2737.138101 after 4 yrs.
r = (8.28%/4) / 100% = 0.0207 = QPR
expressed as a decimal.
n = 4 comp./yr * 4 yrs.=16 Compounding periods.
Pt = 2737.138101(1.0207)^16 = $3798.98.
You deposit $1000 in an account that pays 6% interest compounded semiannually. After 3years, the interest rate is increased to 6.44% compounded quarterly. What will be the value of the account after a total of 6 years?
To solve this problem, we need to calculate the value of the account after 4 years with the initial 8% interest rate compounded semiannually, then calculate the value of the account after 4 more years with the increased 8.28% interest rate compounded quarterly and add these two values together.
Step 1: Calculate the value of the account after 4 years at 8% interest compounded semiannually.
First, let's calculate the number of compounding periods over 4 years, as compounding is done semiannually:
Number of compounding periods = 4 years * 2 semiannual periods = 8 semiannual periods
Next, let's calculate the interest rate per semiannual period:
Interest rate per semiannual period = 8% / 2 = 4% = 0.04 in decimal form
Now, we can calculate the value of the account after 4 years using the formula for compound interest:
Value = Principal * (1 + Interest rate)^Number of compounding periods
Value = $2,000 * (1 + 0.04)^8
Value ≈ $2,000 * (1.04)^8
Value ≈ $2,000 * 1.36049
Value ≈ $2,720.98 (rounded to the nearest cent)
Step 2: Calculate the value of the account after 4 more years at 8.28% interest compounded quarterly.
The second part of the problem asks for the value of the account after 4 more years with the interest rate increased to 8.28% compounded quarterly.
First, let's calculate the number of compounding periods over 4 years, as compounding is now done quarterly:
Number of compounding periods = 4 years * 4 quarterly periods = 16 quarterly periods
Next, let's calculate the interest rate per quarterly period:
Interest rate per quarterly period = 8.28% / 4 = 2.07% = 0.0207 in decimal form
Now, we can calculate the value of the account after 4 more years using the formula for compound interest:
Value = Principal * (1 + Interest rate)^Number of compounding periods
Value = $2,720.98 * (1 + 0.0207)^16
Value ≈ $2,720.98 * (1.0207)^16
Value ≈ $2,720.98 * 1.36995
Value ≈ $3,727.81 (rounded to the nearest cent)
Step 3: Add the values from step 1 and step 2 to find the total value of the account after 4 more years.
Total value = Value after 4 years + Value after 4 more years
Total value ≈ $2,720.98 + $3,727.81
Total value ≈ $6,448.79 (rounded to the nearest cent)
Therefore, the value of the account after 4 more years will be approximately $6,448.79.
To find the value of the account after 4 more years, we first need to calculate the value of the account after the initial 4 years at an interest rate of 8% compounded semiannually.
The formula to calculate the future value (FV) of an account with compound interest is:
FV = P * (1 + r/n)^(n*t)
Where:
FV = future value
P = principal amount (initial deposit)
r = annual interest rate (as a decimal)
n = number of times the interest is compounded per year
t = number of years
For the first 4 years at an 8% interest rate compounded semiannually:
P = $2,000
r = 8% = 0.08
n = 2 (semiannually compounded)
t = 4
Plugging these values into the formula, we get:
FV = $2,000 * (1 + 0.08/2)^(2*4)
= $2,000 * (1 + 0.04)^8
= $2,000 * (1.04)^8
≈ $2,000 * 1.36048928
≈ $2,720.98
So, after the initial 4 years, the value of the account will be approximately $2,720.98.
Next, we need to calculate the value of the account for the next 4 years, but with the interest rate increased to 8.28% compounded quarterly. We will use the same formula, but with the new values.
For the next 4 years at an 8.28% interest rate compounded quarterly:
P = $2,720.98 (value after the initial 4 years)
r = 8.28% = 0.0828
n = 4 (quarterly compounded)
t = 4
Plugging these values into the formula, we get:
FV = $2,720.98 * (1 + 0.0828/4)^(4*4)
= $2,720.98 * (1 + 0.0207)^16
≈ $2,720.98 * (1.0207)^16
≈ $2,720.98 * 1.3732372
≈ $3,740.44
Therefore, the value of the account after 4 more years will be approximately $3,740.44.