If you deposit $200 every month into an account which pays 4.1% interest, compounded monthly, how much money will you have in your account after 12 years?
Round your answer to the nearest whole dollar.
$
Suppose you were to deposit a lump sum today, instead of making monthly deposits. How much would you need to deposit today in order to end up with the same amount in the account?
Round your answer here to the nearest whole dollar.
$
To calculate the amount of money you will have in your account after 12 years with monthly deposits, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount
P = the principal amount (the initial deposit)
r = annual interest rate (in decimal form)
n = number of times interest is compounded per year
t = number of years
In this case, P = $200, r = 4.1% or 0.041, n = 12 (compounded monthly), and t = 12 years.
Using the formula, we have:
A = $200(1 + 0.041/12)^(12*12)
A = $200(1 + 0.00341667)^(144)
A ≈ $200(1.00341667)^144
A ≈ $200(1.574378778)
A ≈ $314.876
Therefore, after 12 years, you will have approximately $314.87 in your account.
Now, to calculate the lump sum deposit you would need to make today, we can use the formula for compound interest in reverse:
P = A/(1 + r/n)^(nt)
In this case, A = $314.87, r = 4.1% or 0.041, n = 12 (compounded monthly), and t = 12 years.
Using the formula, we have:
P = $314.87/(1 + 0.041/12)^(12*12)
P ≈ $314.87/(1 + 0.00341667)^(144)
P ≈ $314.87/(1.00341667)^144
P ≈ $314.87/1.574378778
P ≈ $199.99
Therefore, you would need to deposit approximately $200 today in order to end up with the same amount in the account after 12 years.