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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%.
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