Answer:
Let's first calculate the amount of interest that would be earned if the interest were compounded annually. The formula for the future value of a single sum is:
F = P * (1 + r/n)^(nt)
Where:
F is the future value
P is the principal (the initial deposit)
r is the annual interest rate
n is the number of compounding periods per year
t is the number of years
For our calculation, we have:
P = $10,000
r = 1.8% = 0.018
n = 1 (annual compounding)
t = 30
So, the future value of the account with annual compounding is:
F = $10,000 * (1 + 0.018/1)^(1 * 30) = $10,000 * (1.018)^30 = $21,784.08
Now, let's calculate the amount of interest that would be earned if the interest were compounded monthly. The formula for the future value of a single sum is the same, but we need to use the monthly compounding rate (r/12) instead of the annual rate and the number of months (12t) instead of the number of years:
F = P * (1 + r/n)^(nt)
Where:
F is the future value
P is the principal (the initial deposit)
r is the annual interest rate
n is the number of compounding periods per year
t is the number of years
For our calculation, we have:
P = $10,000
r = 1.8% = 0.018
n = 12 (monthly compounding)
t = 30
So, the future value of the account with monthly compounding is:
F = $10,000 * (1 + 0.018/12)^(12 * 30) = $10,000 * (1.0015)^360 = $22,254.51
The difference in the two future values is $22,254.51 - $21,784.08 = $470.43.
So, the account would be worth $470.43 more if interest were compounded monthly rather than annually over a period of 30 years. Round to the nearest dollar, the answer is $470.
Explanation: