Welcome to Westonci.ca, the place where your questions are answered by a community of knowledgeable contributors. Get quick and reliable solutions to your questions from a community of experienced professionals on our platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
Let's find out what happens to a principal amount of [tex]$200 when it is invested at an annual interest rate of 4% for 10 years using different calculation methods: simple interest, compound interest annually, and compound interest biannually.
### Simple Interest
The formula for calculating simple interest is:
\[ A = P(1 + rt) \]
where:
- \( P \) is the principal amount
- \( r \) is the annual interest rate
- \( t \) is the time in years
Given:
- \( P = 200 \)
- \( r = 0.04 \)
- \( t = 10 \)
Substitute these values into the formula:
\[ A = 200 (1 + 0.04 \times 10) \]
\[ A = 200 (1 + 0.4) \]
\[ A = 200 \times 1.4 \]
\[ A = 280.0 \]
So, the amount after 10 years with simple interest is $[/tex]280.0.
### Compound Interest (Compounded Annually)
The formula for calculating compound interest compounded annually is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
where:
- [tex]\( P \)[/tex] is the principal amount
- [tex]\( r \)[/tex] is the annual interest rate
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year (annually, [tex]\( n = 1 \)[/tex])
- [tex]\( t \)[/tex] is the time in years
Given:
- [tex]\( P = 200 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
- [tex]\( n = 1 \)[/tex]
- [tex]\( t = 10 \)[/tex]
Substitute these values into the formula:
[tex]\[ A = 200 \left(1 + \frac{0.04}{1}\right)^{1 \times 10} \][/tex]
[tex]\[ A = 200 (1 + 0.04)^{10} \][/tex]
[tex]\[ A = 200 \times 1.04^{10} \][/tex]
[tex]\[ A \approx 296.0488569836689 \][/tex]
So, the amount after 10 years with compound interest compounded annually is approximately [tex]$296.05. ### Compound Interest (Compounded Biannually) The formula for calculating compound interest compounded biannually is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - \( P \) is the principal amount - \( r \) is the annual interest rate - \( n \) is the number of times the interest is compounded per year (biannually, \( n = 2 \)) - \( t \) is the time in years Given: - \( P = 200 \) - \( r = 0.04 \) - \( n = 2 \) - \( t = 10 \) Substitute these values into the formula: \[ A = 200 \left(1 + \frac{0.04}{2}\right)^{2 \times 10} \] \[ A = 200 (1 + 0.02)^{20} \] \[ A = 200 \times 1.02^{20} \] \[ A \approx 297.18947919567097 \] So, the amount after 10 years with compound interest compounded biannually is approximately $[/tex]297.19.
### Summary
- Simple Interest: [tex]$280.0 - Compound Interest (Compounded Annually): $[/tex]296.05
- Compound Interest (Compounded Biannually): [tex]$297.19 These calculations show how the principal amount of $[/tex]200 grows differently under each interest calculation method over 10 years at an annual interest rate of 4%.
### Compound Interest (Compounded Annually)
The formula for calculating compound interest compounded annually is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
where:
- [tex]\( P \)[/tex] is the principal amount
- [tex]\( r \)[/tex] is the annual interest rate
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year (annually, [tex]\( n = 1 \)[/tex])
- [tex]\( t \)[/tex] is the time in years
Given:
- [tex]\( P = 200 \)[/tex]
- [tex]\( r = 0.04 \)[/tex]
- [tex]\( n = 1 \)[/tex]
- [tex]\( t = 10 \)[/tex]
Substitute these values into the formula:
[tex]\[ A = 200 \left(1 + \frac{0.04}{1}\right)^{1 \times 10} \][/tex]
[tex]\[ A = 200 (1 + 0.04)^{10} \][/tex]
[tex]\[ A = 200 \times 1.04^{10} \][/tex]
[tex]\[ A \approx 296.0488569836689 \][/tex]
So, the amount after 10 years with compound interest compounded annually is approximately [tex]$296.05. ### Compound Interest (Compounded Biannually) The formula for calculating compound interest compounded biannually is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - \( P \) is the principal amount - \( r \) is the annual interest rate - \( n \) is the number of times the interest is compounded per year (biannually, \( n = 2 \)) - \( t \) is the time in years Given: - \( P = 200 \) - \( r = 0.04 \) - \( n = 2 \) - \( t = 10 \) Substitute these values into the formula: \[ A = 200 \left(1 + \frac{0.04}{2}\right)^{2 \times 10} \] \[ A = 200 (1 + 0.02)^{20} \] \[ A = 200 \times 1.02^{20} \] \[ A \approx 297.18947919567097 \] So, the amount after 10 years with compound interest compounded biannually is approximately $[/tex]297.19.
### Summary
- Simple Interest: [tex]$280.0 - Compound Interest (Compounded Annually): $[/tex]296.05
- Compound Interest (Compounded Biannually): [tex]$297.19 These calculations show how the principal amount of $[/tex]200 grows differently under each interest calculation method over 10 years at an annual interest rate of 4%.
We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.
