stuart daniels estimates that he will need 25,000 to set up a small business in 7 years. how much must Stuart invest now at 12%interest compounded quarterly to achieve his goal?
12/4 = 3% per period
7* 4 = 28 periods
1.03^28 = 2.288
so
2.288 x = 25,000
x = 10,927
To find out how much Stuart must invest now at 12% interest compounded quarterly, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value (amount needed for the small business) = $25,000
P = the principal amount (initial investment)
r = annual interest rate (12% = 0.12)
n = number of times the interest is compounded per year (quarterly = 4 times)
t = number of years before the amount is needed (7 years)
By substituting the known values into the formula, we can solve for P:
25,000 = P(1 + 0.12/4)^(4*7)
Now let's calculate the values step by step:
Step 1: Divide the annual interest rate by the number of times compounded per year:
0.12/4 = 0.03
Step 2: Multiply the number of times compounded per year by the number of years before the amount is needed:
4 * 7 = 28
Step 3: Calculate the exponential part:
(1 + 0.03)^(28) ≈ 2.8954
Step 4: Rearrange the formula to solve for P:
P = 25,000 / 2.8954 ≈ $8,642.84
Therefore, Stuart must invest approximately $8,642.84 now at a 12% interest rate compounded quarterly to achieve his goal of $25,000 in 7 years.