How much must be deposited today into the following account in order to have $ 25,000 in 8 years for a down payment on a house? Assume no additional deposits are made.
An account with annual compounding and an APR of 5%
$______ should be deposited today
To find the amount that should be deposited today, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value (in this case, $25,000)
P = the principal amount (the deposit we're trying to find)
r = the annual interest rate (5% or 0.05)
n = the number of times interest is compounded per year (assuming annual compounding, n = 1)
t = the number of years (8)
Plugging in the values:
$25,000 = P(1 + 0.05/1)^(1*8)
Simplifying:
$25,000 = P(1.05)^8
To solve for P, we divide both sides by (1.05)^8:
P = $25,000 / (1.05)^8
Calculating this on a calculator gives us:
P ≈ $18,440.00
So, approximately $18,440 should be deposited today to have $25,000 in 8 years for a down payment on a house.
To calculate the amount that should be deposited today, we can use the future value formula for compound interest:
Future Value = Present Value × (1 + (Interest Rate ÷ Compounding Periods))^(Compounding Periods × Time)
In this case, the future value we want to achieve is $25,000, the interest rate (APR) is 5%, and the time is 8 years. The compounding periods for this account are not mentioned, so we will assume it is compounded annually.
Plugging these values into the formula, we get:
$25,000 = Present Value × (1 + (0.05 ÷ 1))^(1 × 8)
To solve for the present value, we need to isolate it:
Present Value = $25,000 ÷ (1.05^8)
Using a calculator, we get:
Present Value ≈ $18,357.89
Therefore, approximately $18,357.89 should be deposited today into the account in order to have $25,000 after 8 years for a down payment on a house.
1.05^8 = 1.477455
25,000 / 1.477455 = 16,921