Bernie deposited $ 4000 into an account that pays 45%/a compounded quarterly during the first year. The interest rate on this account is then increased by 0.2% each year. Calculate the balance in Bernie's account after three years
To calculate the balance in Bernie's account after three years, we need to use the compound interest formula.
The formula for compound interest is:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment/loan, including interest
P = the principal investment amount (the initial deposit or loan amount)
r = annual interest rate (in decimal form)
n = number of times that interest is compounded per year
t = number of years the money is invested/borrowed for
In this case, the principal investment amount (P) is $4000, the annual interest rate (r) is 45% (0.45 in decimal form), and interest is compounded quarterly, so the number of times interest is compounded per year (n) is 4. The interest rate is then increased by 0.2% (0.002 in decimal form) each year, so for the second year, the interest rate is 45.2% (0.452 in decimal form), and for the third year, the interest rate is 45.4% (0.454 in decimal form).
Now let's calculate the balance in Bernie's account after three years:
For the first year:
A1 = 4000(1 + 0.45/4)^(4*1) = 4000(1.1125)^4
For the second year:
A2 = A1(1 + 0.452/4)^(4*1)
For the third year:
A3 = A2(1 + 0.454/4)^(4*1)
To calculate the final balance, we need to substitute the calculated values of A1 and A2 into the formula for A3:
A3 = A2(1 + 0.454/4)^(4*1)
A3 = (A1(1 + 0.452/4)^(4*1))(1 + 0.454/4)^(4*1)
Now, let's calculate the values step by step:
First, calculate A1:
A1 = 4000(1.1125)^4
Then, calculate A2:
A2 = A1(1 + 0.452/4)^(4*1)
Finally, calculate A3:
A3 = (A2)(1 + 0.454/4)^(4*1)
By plugging in the numbers and performing the calculations, you will find the balance in Bernie's account after three years.
To calculate the balance in Bernie's account after three years, we first need to find the value of the account after the first year, then the second year, and finally the third year. Here's how we can break it down step-by-step:
Step 1: Calculate the balance after the first year.
Given:
Principal amount (P) = $4000
Interest rate (r) = 45% = 0.45
Compounding period (n) = quarterly = 4 times in a year
Using the compound interest formula:
A = P(1 + r/n)^(n*t)
Where:
A = final amount
P = principal amount
r = interest rate
n = number of compounding periods per year
t = time in years
Substituting the values:
A1 = $4000(1 + 0.45/4)^(4*1)
A1 ≈ $4000(1.1125)^4
A1 ≈ $4000(1.57216)
A1 ≈ $6,288.64
Therefore, the balance after the first year is approximately $6,288.64.
Step 2: Calculate the balance after the second year.
Given:
Interest rate increment per year = 0.2% = 0.002 (increased by 0.2% each year)
Using the compound interest formula with an increased interest rate:
A2 = A1(1 + (r + r_increment)/n)^(n*t)
Where:
A2 = final amount after the second year
A1 = balance after the first year
r_increment = interest rate increment per year
Substituting the values:
A2 = $6,288.64(1 + (0.45 + 0.002)/4)^(4*1)
A2 ≈ $6,288.64(1.1141)^4
A2 ≈ $6,288.64(1.601758996)
A2 ≈ $10,052.47
Therefore, the balance after the second year is approximately $10,052.47.
Step 3: Calculate the balance after the third year.
Using the compound interest formula once again with an increased interest rate:
A3 = A2(1 + (r + r_increment)/n)^(n*t)
Where:
A3 = final amount after the third year
A2 = balance after the second year
Substituting the values:
A3 = $10,052.47(1 + (0.45 + 2*(0.002))/4)^(4*1)
A3 ≈ $10,052.47(1.1168)^4
A3 ≈ $10,052.47(1.659535797)
A3 ≈ $16,676.01
Therefore, the balance in Bernie's account after three years will be approximately $16,676.01.