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?
(b) How much interest will be earned?
(c) If he can only deposit $16,000 now, how short of $32,000 will he be?
(a) Using the formula for compound interest:
A = P(1 + r/n)^(nt), where
A = amount after 5 years = $32,000
P = principal (amount to be deposited today)
r = annual interest rate = 6.4%
n = number of times compounded per year = 4 (quarterly)
t = time in years = 5
32,000 = P(1 + 0.064/4)^(4*5)
32,000 = P(1.016)^20
P = 22,139.48
Therefore, Gordon Freeman should deposit $22,139.48 today to have $32,000 in 5 years.
(b) The total interest earned can be calculated as:
Total Interest = A - P
Total Interest = $32,000 - $22,139.48
Total Interest = $9,860.52
Therefore, the total interest earned will be $9,860.52.
(c) If Gordon Freeman can only deposit $16,000 now, we need to calculate how much he will be short of $32,000 in 5 years:
A = P(1 + r/n)^(nt)
32,000 = 16,000(1 + 0.064/4)^(4*5)
32,000 = 16,000(1.016)^20
32,000 = 16,000(1.367)
32,000 - 16,000 = 16,000
Therefore, he will be short $16,000.
To calculate the amount Gordon Freeman should deposit today, we can use the formula for compound interest:
a) A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment ($32,000)
P = the initial deposit
r = the annual interest rate (6.4% or 0.064)
n = the number of times the interest is compounded per year (quarterly, so 4)
t = the number of years (5)
Plugging in the values, we can rearrange the formula to solve for P:
32,000 = P(1 + 0.064/4)^(4*5)
Simplifying the equation:
32,000 = P(1 + 0.016)^20
Divide both sides by (1 + 0.016)^20:
P = 32,000 / (1 + 0.016)^20
Using a calculator, we find:
P ≈ $24,265.35
Therefore, Gordon Freeman should deposit approximately $24,265.35 today.
b) To calculate the interest earned, we can subtract the initial deposit from the future value:
Interest earned = Future value - Initial deposit
Interest earned = $32,000 - $24,265.35
Interest earned ≈ $7,734.65
Therefore, Gordon Freeman will earn approximately $7,734.65 in interest.
c) If he can only deposit $16,000 now, we can calculate the shortfall:
Shortfall = Future value - Initial deposit
Shortfall = $32,000 - $16,000
Shortfall = $16,000
Therefore, Gordon Freeman will be short $16,000 of his target amount.