35. The Tanners have received an $8000 gift from one of their parents to invest in their child's college education. They estimate that they will need $20,000 in 12 years to achieve their educational goals for their child. What interest rate compounded semiannually would the Tanners need to achieve this goal?
35. The Tanners have received an $8000 gift from one of their parents to invest in their child's college education. They estimate that they will need $20,000 in 12 years to achieve their educational goals for their child. What interest rate compounded semiannually would the Tanners need to achieve this goal?
Essentials Of Investments
11th Edition
ISBN:9781260013924
Author:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Publisher:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Chapter1: Investments: Background And Issues
Section: Chapter Questions
Problem 1PS
Related questions
Question
![**Investment Calculation Exercise**
**Problem Statement:**
The Tanners have received an $8,000 gift from one of their parents to invest in their child’s college education. They estimate that they will need $20,000 in 12 years to achieve their educational goals for their child. What interest rate compounded semiannually would the Tanners need to achieve this goal?
---
To solve this problem, we need to apply the formula for compound interest compounded semiannually. The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount ($8,000).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times interest is compounded per year (2 for semiannual).
- \( t \) is the time the money is invested for in years.
Given \( A = $20,000 \), \( P = $8,000 \), \( t = 12 \) years, and \( n = 2 \),
we need to solve for \( r \).
\[
20,000 = 8,000 \left(1 + \frac{r}{2}\right)^{2 \times 12}
\]
Simplify and solve for \( r \):
\[
20,000 = 8,000 \left(1 + \frac{r}{2}\right)^{24}
\]
Divide both sides by 8,000:
\[
2.5 = \left(1 + \frac{r}{2}\right)^{24}
\]
Taking the 24th root of both sides:
\[
\left(2.5\right)^{\frac{1}{24}} = 1 + \frac{r}{2}
\]
Subtracting 1 from both sides:
\[
\left(2.5\right)^{\frac{1}{24}} - 1 = \frac{r}{2}
\]
Finally, multiply by 2:
\[
r = 2 \left(\left(2.5\right)^{\frac{1}{24}} - 1\right)
\]
Using a calculator to evaluate:
After calculating, you will find that \( r \approx 0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F56784505-0743-408a-9ee7-26834241f533%2F17f7f0bf-8572-4b50-a212-58223d27545d%2Fxj08n09o_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Investment Calculation Exercise**
**Problem Statement:**
The Tanners have received an $8,000 gift from one of their parents to invest in their child’s college education. They estimate that they will need $20,000 in 12 years to achieve their educational goals for their child. What interest rate compounded semiannually would the Tanners need to achieve this goal?
---
To solve this problem, we need to apply the formula for compound interest compounded semiannually. The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount ($8,000).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times interest is compounded per year (2 for semiannual).
- \( t \) is the time the money is invested for in years.
Given \( A = $20,000 \), \( P = $8,000 \), \( t = 12 \) years, and \( n = 2 \),
we need to solve for \( r \).
\[
20,000 = 8,000 \left(1 + \frac{r}{2}\right)^{2 \times 12}
\]
Simplify and solve for \( r \):
\[
20,000 = 8,000 \left(1 + \frac{r}{2}\right)^{24}
\]
Divide both sides by 8,000:
\[
2.5 = \left(1 + \frac{r}{2}\right)^{24}
\]
Taking the 24th root of both sides:
\[
\left(2.5\right)^{\frac{1}{24}} = 1 + \frac{r}{2}
\]
Subtracting 1 from both sides:
\[
\left(2.5\right)^{\frac{1}{24}} - 1 = \frac{r}{2}
\]
Finally, multiply by 2:
\[
r = 2 \left(\left(2.5\right)^{\frac{1}{24}} - 1\right)
\]
Using a calculator to evaluate:
After calculating, you will find that \( r \approx 0
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