Approximately how much principal would need to be placed into an account earning 3.575% interest compounded quarterly so that it has an accumulated value of $68,000 at the end of 30 years?
.03575/4 = .0089375 per quarter year
30 * 4 = 120 compoundings
V = Vi (1.0089375)^120
68,000 = Vi (2.90878 )
Vi = 23,377.45
To calculate the principal needed to achieve an accumulated value of $68,000 at the end of 30 years with an interest rate of 3.575% compounded quarterly, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = accumulated value
P = principal
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = number of years
In this case, we have:
A = $68,000
r = 3.575% = 0.03575
n = 4 (quarterly compounding)
t = 30
Using these values, we can solve for P:
$68,000 = P(1 + 0.03575/4)^(4*30)
Simplifying,
$68,000 = P(1.0089375)^120
$68,000 = P(3.17218616356)
Dividing both sides by 3.17218616356,
P ≈ $21,429.55
Therefore, the approximate principal needed to be placed into the account is $21,429.55.
To find the principal amount needed to accumulate $68,000 at the end of 30 years with an interest rate of 3.575% compounded quarterly, we can use the formula for compound interest:
A = P(1 + r/n)^(n*t)
Where:
A is the accumulated value,
P is the principal amount,
r is the interest rate,
n is the number of compounding periods per year, and
t is the number of years.
In this case, we know that the accumulated value (A) is $68,000, the interest rate (r) is 3.575%, and the number of compounding periods per year (n) is 4 (since it is compounded quarterly). We need to find the principal amount (P) and the number of years (t) is 30.
To find P, we can rearrange the formula:
P = A / (1 + r/n)^(n*t)
Now let's substitute the given values into the formula and solve for P:
P = $68,000 / (1 + 0.03575/4)^(4*30)
Simplifying the expression inside the brackets:
P = $68,000 / (1 + 0.0089375)^(120)
Calculating the value inside the brackets:
P = $68,000 / (1.0089375)^(120)
Using a calculator or a spreadsheet, raise 1.0089375 to the power of 120 and calculate the result:
P ≈ $16,165.12
Therefore, approximately $16,165.12 principal would need to be placed into the account to accumulate $68,000 at the end of 30 years with an interest rate of 3.575% compounded quarterly.