Gordon Freeman wants to have $32,000 in 5 years in order to purchase a new car.
(a) How much should he deposit today in an account earning 6.4%, compounded quarterly,
to have the required amount in 5 years?
To calculate how much Gordon Freeman should deposit today, we can use the formula for the future value of an investment with compound interest:
A = P(1 + r/n)^(nt)
where:
A is the future value of the investment
P is the principal amount (the amount to be deposited today)
r is the annual interest rate (in decimal form)
n is the number of times the interest is compounded per year
t is the number of years
In this case, the future value A is $32,000, the annual interest rate r is 6.4% (or 0.064 in decimal form), the number of times the interest is compounded per year n is 4 (quarterly), and the number of years t is 5.
Plugging these values into the formula, we get:
32,000 = P(1 + 0.064/4)^(4*5)
To solve for P, we need to isolate it on one side of the equation.
Dividing both sides of the equation by (1 + 0.064/4)^(4*5), we get:
P = 32,000 / (1 + 0.064/4)^(4*5)
Using a calculator, we can compute the value of P to be approximately $24,099.60.
Therefore, Gordon Freeman should deposit $24,099.60 today into the account earning 6.4%, compounded quarterly, in order to have $32,000 in 5 years.
We can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where A is the future value, P is the present value (or principal), r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the number of years.
In this case, we want to find P, so we can rearrange the formula:
P = A/(1 + r/n)^(nt)
Plugging in the values we have:
A = $32,000
r = 6.4% = 0.064 (decimal)
n = 4 (compounded quarterly)
t = 5 years
P = 32000/(1 + 0.064/4)^(4*5)
P = $24,358.19
Therefore, Gordon should deposit $24,358.19 today in order to have $32,000 in 5 years, assuming he earns 6.4% interest compounded quarterly.