Final answer:
To have $10,000 in ten years with an annual compound interest rate of 10%, a person would need to deposit approximately $3855.43.
Step-by-step explanation:
To find out how much money needs to be deposited in a bank account with an annual compound interest rate of 10% to have $10,000 in ten years, we use the formula for compound interest:
A = P(1 + rac{r}{n})^{nt}
Where:
- A is the amount of money accumulated after n years, including interest.
- P is the principal amount (the initial amount of money).
- r is the annual interest rate (decimal).
- n is the number of times that interest is compounded per year.
- t is the time the money is invested for, in years.
Since the interest is compounded annually, n will be 1. The formula simplifies to:
A = P(1 + r)^t
Now, let's solve for P since we want to find the initial deposit that will grow to $10,000:
$10,000 = P(1 + 0.10)^{10}
Thus:
P = rac{$10,000}{(1 + 0.10)^{10}}
Calculating the denominator:
P = rac{$10,000}{(1.10)^{10}}
P = rac{$10,000}{2.59374}
P ≈ $3855.43
You would need to deposit approximately $3855.43 to have $10,000 in ten years at a 10% interest rate compounded annually.