To find out how much you need to deposit now to reach a target amount in the future with compounded interest, you can use the formula for compound interest:
\[ A = P(1 + r)^n \]
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 years the money is invested or borrowed.
In this case, we want to solve for \( P \) (the initial deposit):
\[ P = \frac{A}{(1 + r)^n} \]
Given:
- \( A = 10,000 \)
- \( r = 0.08 \) (8% as a decimal)
- The time from January 1, 1999, to December 31, 2003, is 5 years.
So, \( n = 5 \).
Now, plugging in the values:
\[ P = \frac{10,000}{(1 + 0.08)^5} \] \[ P = \frac{10,000}{(1.08)^5} \] \[ (1.08)^5 \approx 1.469328 \]
Now, calculate \( P \):
\[ P = \frac{10,000}{1.469328} \approx 6805.97 \]
Therefore, you must deposit approximately $6,805.97 on January 1, 1999, to have a balance of $10,000 on December 31, 2003.
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