Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Title: Calculating Initial Investment with Compound Interest**
**Question:**
How much money should be deposited today in an account that earns 7.5% compounded monthly so that it will accumulate to $15,000 in three years?
**Instructions:**
Click the icon to view some finance formulas.
**Input:**
The amount of money that should be deposited is $[ ]
(Round up to the nearest cent.)
---
**Explanation:**
To solve this, we use the formula for compound interest:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (initial deposit).
- \( r \) 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.
Substitute the known values into the formula and solve for \( P \):
- \( A = 15,000 \)
- \( r = 0.075 \)
- \( n = 12 \) (compounded monthly)
- \( t = 3 \)
This formula helps determine how much you need to deposit initially to reach your desired financial goal with compound interest.](https://content.bartleby.com/qna-images/question/febfc829-3adc-4601-a5b5-bd5b3e3c23b0/db9250cf-b876-4b3f-8fb7-ee038d0550b6/7naigvl_processed.jpeg)
Transcribed Image Text:**Title: Calculating Initial Investment with Compound Interest**
**Question:**
How much money should be deposited today in an account that earns 7.5% compounded monthly so that it will accumulate to $15,000 in three years?
**Instructions:**
Click the icon to view some finance formulas.
**Input:**
The amount of money that should be deposited is $[ ]
(Round up to the nearest cent.)
---
**Explanation:**
To solve this, we use the formula for compound interest:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (initial deposit).
- \( r \) 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.
Substitute the known values into the formula and solve for \( P \):
- \( A = 15,000 \)
- \( r = 0.075 \)
- \( n = 12 \) (compounded monthly)
- \( t = 3 \)
This formula helps determine how much you need to deposit initially to reach your desired financial goal with compound interest.
![**How much money should be deposited today in an account that earns 7.5% compounded monthly so that it will accumulate to $15,000 in three years?**
---
**Formulas**
In the provided formulas, A is the balance in the account after t years, P is the principal investment, r is the annual interest rate in decimal form, n is the number of compounding periods per year, and Y is the investment's effective annual yield in decimal form.
1. Future Value of Investment:
\[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \]
2. Present Value of Investment:
\[ P = \frac{A}{\left( 1 + \frac{r}{n} \right)^{nt}} \]
3. Continuous Compounding:
\[ A = Pe^{rt} \]
4. Effective Annual Yield:
\[ Y = \left( 1 + \frac{r}{n} \right)^n - 1 \]
These formulas are used to calculate various aspects of compound interest and investment growth over time, using different compounding methods.](https://content.bartleby.com/qna-images/question/febfc829-3adc-4601-a5b5-bd5b3e3c23b0/db9250cf-b876-4b3f-8fb7-ee038d0550b6/einf9dd_processed.jpeg)
Transcribed Image Text:**How much money should be deposited today in an account that earns 7.5% compounded monthly so that it will accumulate to $15,000 in three years?**
---
**Formulas**
In the provided formulas, A is the balance in the account after t years, P is the principal investment, r is the annual interest rate in decimal form, n is the number of compounding periods per year, and Y is the investment's effective annual yield in decimal form.
1. Future Value of Investment:
\[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \]
2. Present Value of Investment:
\[ P = \frac{A}{\left( 1 + \frac{r}{n} \right)^{nt}} \]
3. Continuous Compounding:
\[ A = Pe^{rt} \]
4. Effective Annual Yield:
\[ Y = \left( 1 + \frac{r}{n} \right)^n - 1 \]
These formulas are used to calculate various aspects of compound interest and investment growth over time, using different compounding methods.
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