How much must be deposited today into the following account in order to have $35,000 in 6 years for a down payment on a house? Assume no additional deposits are made.
An account with annual compounding and an APR of 7%
35000 = a (1.07)^6
so
a = 35000 / (1.07)^6
To calculate the amount that must be deposited today, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = Final amount ($35,000)
P = Principal amount (initial deposit)
r = Annual interest rate (APR as a decimal = 7% = 0.07)
n = Number of times interest is compounded per year (assuming annual compounding)
t = Number of years (6 years)
Plugging in the values into the formula, we get:
$35,000 = P(1 + 0.07/1)^(1*6)
Now, let's solve for P:
$35,000 = P(1 + 0.07)^(6)
$35,000 = P(1.07)^(6)
Divide both sides of the equation by (1.07)^(6):
P = $35,000 / (1.07)^(6)
Using a calculator, we can calculate P:
P ≈ $25,220.74
Therefore, approximately $25,220.74 must be deposited today to have $35,000 in 6 years for a down payment on a house.
To determine the amount that needs to be deposited today, we can use the formula for future value of a single sum:
FV = PV * (1 + r/n)^(n*t)
Where:
FV = Future Value
PV = Present Value (amount to be deposited today)
r = Annual interest rate (expressed as a decimal)
n = Number of compounding periods per year
t = Number of years
In this case, we have:
FV = $35,000
r = 7% (0.07 as a decimal)
n = 1 (annual compounding)
t = 6 years
Plugging these values into the formula, we have:
$35,000 = PV * (1 + 0.07/1)^(1*6)
Now, let's solve for PV:
PV = $35,000 / (1 + 0.07)^6
Calculating this using a calculator or spreadsheet:
PV = $35,000 / (1.07)^6
PV ≈ $23,651.05
Therefore, in order to have $35,000 in 6 years for a down payment on a house, you would need to deposit approximately $23,651.05 into the account today.