Welcome to Westonci.ca, the Q&A platform where your questions are met with detailed answers from experienced experts. Get expert answers to your questions quickly and accurately from our dedicated community of professionals. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
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%.
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.
