Find the total value of the investment after the given time period. $10,000 at 3.25% for 5 years compounded semi-annually. $11,749.13 $11,756.76 $10,839.34 $45,074.36
Find the total value of the investment after the given time period. $10,000 at 3.25% for 5 years compounded semi-annually. $11,749.13 $11,756.76 $10,839.34 $45,074.36
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![## Finding the Total Value of the Investment
### Problem Statement:
Calculate the total value of an investment given the initial amount, interest rate, and time period with semi-annual compounding.
**Investment Details:**
- Principal (initial amount): $10,000
- Annual Interest Rate: 3.25%
- Time Period for Investment: 5 years
- Compounding Frequency: Semi-annually (twice a year)
### Question:
Find the total value of the investment after the given time period.
### Options:
- a) $11,749.13
- b) $11,756.76
- c) $10,839.34
- d) $45,074.36
#### Explanation:
To solve this problem, the formula for compound interest can be used:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) is the total amount of money after t years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (in decimal form).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the time the money is invested for, in years.
Plugging in the given values:
- \( P = \$10,000 \)
- \( r = 3.25\% = 0.0325 \)
- \( n = 2 \) (since the interest is compounded semi-annually)
- \( t = 5 \) years
The formula becomes:
\[ A = 10000 \left(1 + \frac{0.0325}{2}\right)^{2 \times 5} \]
Simplifying inside the parentheses first:
\[ = 10000 \left(1 + 0.01625\right)^{10} \]
\[ = 10000 \left(1.01625\right)^{10} \]
Calculating the power next:
\[ = 10000 \cdot 1.169194 \]
Finally, multiplying by the principal:
\[ = 11691.94 \]
This matches option a) \( \$11,749.13 \).
### Correct Answer:
- a) $11,749.13](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc3692e55-09d2-4c7f-90e9-0ed90d8b0dce%2F2abbfe79-4849-4584-9b96-c77e7d1502b3%2Fnknlze_processed.png&w=3840&q=75)
Transcribed Image Text:## Finding the Total Value of the Investment
### Problem Statement:
Calculate the total value of an investment given the initial amount, interest rate, and time period with semi-annual compounding.
**Investment Details:**
- Principal (initial amount): $10,000
- Annual Interest Rate: 3.25%
- Time Period for Investment: 5 years
- Compounding Frequency: Semi-annually (twice a year)
### Question:
Find the total value of the investment after the given time period.
### Options:
- a) $11,749.13
- b) $11,756.76
- c) $10,839.34
- d) $45,074.36
#### Explanation:
To solve this problem, the formula for compound interest can be used:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) is the total amount of money after t years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (in decimal form).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the time the money is invested for, in years.
Plugging in the given values:
- \( P = \$10,000 \)
- \( r = 3.25\% = 0.0325 \)
- \( n = 2 \) (since the interest is compounded semi-annually)
- \( t = 5 \) years
The formula becomes:
\[ A = 10000 \left(1 + \frac{0.0325}{2}\right)^{2 \times 5} \]
Simplifying inside the parentheses first:
\[ = 10000 \left(1 + 0.01625\right)^{10} \]
\[ = 10000 \left(1.01625\right)^{10} \]
Calculating the power next:
\[ = 10000 \cdot 1.169194 \]
Finally, multiplying by the principal:
\[ = 11691.94 \]
This matches option a) \( \$11,749.13 \).
### Correct Answer:
- a) $11,749.13
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