Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### Compound Interest Calculations
#### Example Question:
6. What amount deposited today in an account paying 4% compounded semiannually would grow to $100,000 in 10 years?
This type of question involves calculating the present value needed to achieve a future value with compound interest. We'll use the formula for compound interest to solve this problem.
#### Formula:
\[ P = \frac{A}{(1 + \frac{r}{n})^{nt}} \]
Where:
- \( P \) = principal amount (initial deposit)
- \( A \) = amount of money accumulated after n years, including interest.
- \( r \) = annual interest rate (decimal)
- \( n \) = number of times the interest is compounded per year
- \( t \) = number of years the money is invested or borrowed for
In this scenario:
- \( A = \$100,000 \)
- \( r = 0.04 \) (4% annual interest rate)
- \( n = 2 \) (compounded semiannually)
- \( t = 10 \) years
Plugging in the values:
\[ P = \frac{100,000}{(1 + \frac{0.04}{2})^{2 \times 10}} \]
Run the calculations to find the initial deposit amount needed for the desired future value.](https://content.bartleby.com/qna-images/question/ac0e0cf7-9820-475e-b9ae-d17ddf0c1fbd/4b202f02-d785-4f62-9594-b6879d883396/znuuind.jpeg)
Transcribed Image Text:### Compound Interest Calculations
#### Example Question:
6. What amount deposited today in an account paying 4% compounded semiannually would grow to $100,000 in 10 years?
This type of question involves calculating the present value needed to achieve a future value with compound interest. We'll use the formula for compound interest to solve this problem.
#### Formula:
\[ P = \frac{A}{(1 + \frac{r}{n})^{nt}} \]
Where:
- \( P \) = principal amount (initial deposit)
- \( A \) = amount of money accumulated after n years, including interest.
- \( r \) = annual interest rate (decimal)
- \( n \) = number of times the interest is compounded per year
- \( t \) = number of years the money is invested or borrowed for
In this scenario:
- \( A = \$100,000 \)
- \( r = 0.04 \) (4% annual interest rate)
- \( n = 2 \) (compounded semiannually)
- \( t = 10 \) years
Plugging in the values:
\[ P = \frac{100,000}{(1 + \frac{0.04}{2})^{2 \times 10}} \]
Run the calculations to find the initial deposit amount needed for the desired future value.
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