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? Click the icon to view some finance formulas. ... The amount of money that should be deposited is $. (Round up to the nearest cent.)
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? Click the icon to view some finance formulas. ... The amount of money that should be deposited is $. (Round up to the nearest cent.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
13. Answer these questions
![**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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffebfc829-3adc-4601-a5b5-bd5b3e3c23b0%2Fdb9250cf-b876-4b3f-8fb7-ee038d0550b6%2F7naigvl_processed.jpeg&w=3840&q=75)
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffebfc829-3adc-4601-a5b5-bd5b3e3c23b0%2Fdb9250cf-b876-4b3f-8fb7-ee038d0550b6%2Feinf9dd_processed.jpeg&w=3840&q=75)
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.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
