Final answer:
To have $3000 in 15 years at an 8% interest rate compounded monthly, one would need to initially deposit approximately $886.49. This calculation is done using the compound interest formula rearranged to solve for the principal amount.
Step-by-step explanation:
To figure out how much you need to deposit now to have $3000 in 15 years at an 8% interest rate compounded monthly, we start with the compound interest formula:
a = p(1 + r/n)^(nt)
Where:
- a is the amount we want to have in the future, which is $3000.
- p is the principal amount, or the initial deposit we're solving for.
- r is the annual interest rate (in decimal form, 8% is 0.08).
- n is the number of times interest is compounded per year, which is 12 for monthly compounding.
- t is the number of years, which is 15.
We rearrange the formula to solve for p:
p = a / (1 + r/n)^(nt)
Plugging in our values:
p = 3000 / (1 + 0.08/12)^(12*15)
Now we do the calculation:
p = 3000 / (1 + 0.0066667)^(180)
p = 3000 / (1.0066667)^(180)
p ≈ 3000 / 3.386035
p ≈ 886.49
You would need to deposit approximately $886.49 now in order to have $3000 in the account in 15 years with an 8% interest rate compounded monthly.