Calculus: Early Transcendentals
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ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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![### Compound Interest Calculation Exercise
**Problem Statement:**
Harold invested a certain amount of money in an account earning 5.5% interest compounded **twice a month**. After 10 years, there was $9050 in the account. How much did he originally invest?
**Formula for Compound Interest:**
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
- \( A \) = the amount of money accumulated after n years, including interest.
- \( P \) = the principal amount (the initial amount of money)
- \( r \) = annual interest rate (decimal)
- \( n \) = number of times that interest is compounded per year
- \( t \) = the number of years the money is invested for
**Given Data:**
- Final Amount (\( A \)) = $9050
- Annual Interest Rate (\( r \)) = 5.5% or 0.055
- Compounding Frequency (\( n \)) = twice a month = 24 times a year
- Time (\( t \)) = 10 years
**To Find:**
- The original investment (\( P \))
**Solution:**
We need to rearrange the compound interest formula to solve for \( P \):
\[ P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}} \]
**Inserting the Values:**
\[ P = \frac{9050}{\left(1 + \frac{0.055}{24}\right)^{24 \times 10}} \]
**Calculate \( P \) and Round Answer to 2 Decimal Places:**
\[ \text{The original investment was } \$ \boxed{\text{ \_\_\_\_\_ }} \text{. Round answer to 2 decimal places} \]
**Interactive Calculation:**
Use the formula above and a calculator to find the precise value for \( P \).
This exercise helps students understand the concept of compound interest and how it is applied in financial scenarios.](https://content.bartleby.com/qna-images/question/54e8c682-be67-4ef0-bd1d-cf8b2e543b76/e4e1a3c4-e1f5-4a88-9de2-1236a5fc8132/g4lkjuq_processed.png)
Transcribed Image Text:### Compound Interest Calculation Exercise
**Problem Statement:**
Harold invested a certain amount of money in an account earning 5.5% interest compounded **twice a month**. After 10 years, there was $9050 in the account. How much did he originally invest?
**Formula for Compound Interest:**
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
- \( A \) = the amount of money accumulated after n years, including interest.
- \( P \) = the principal amount (the initial amount of money)
- \( r \) = annual interest rate (decimal)
- \( n \) = number of times that interest is compounded per year
- \( t \) = the number of years the money is invested for
**Given Data:**
- Final Amount (\( A \)) = $9050
- Annual Interest Rate (\( r \)) = 5.5% or 0.055
- Compounding Frequency (\( n \)) = twice a month = 24 times a year
- Time (\( t \)) = 10 years
**To Find:**
- The original investment (\( P \))
**Solution:**
We need to rearrange the compound interest formula to solve for \( P \):
\[ P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}} \]
**Inserting the Values:**
\[ P = \frac{9050}{\left(1 + \frac{0.055}{24}\right)^{24 \times 10}} \]
**Calculate \( P \) and Round Answer to 2 Decimal Places:**
\[ \text{The original investment was } \$ \boxed{\text{ \_\_\_\_\_ }} \text{. Round answer to 2 decimal places} \]
**Interactive Calculation:**
Use the formula above and a calculator to find the precise value for \( P \).
This exercise helps students understand the concept of compound interest and how it is applied in financial scenarios.
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Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.Similar questions
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