To understand what happens to [tex]$200 at 4% interest after 10 years, we will calculate the value using three different methods: Simple Interest, Compound Interest Annually, and Compound Interest Biannually.
### Simple Interest
Simple Interest is calculated using the formula:
\[ SI = P \times r \times t \]
Where:
- \( SI \) is the simple interest
- \( P \) is the principal amount (initial amount of money)
- \( r \) is the interest rate
- \( t \) is the time period in years
Given:
- \( P = 200 \)
- \( r = 4\% = 0.04 \)
- \( t = 10 \)
Plugging in these values:
\[ SI = 200 \times 0.04 \times 10 = 80 \]
The total amount after 10 years will be:
\[ Total (Simple) = Principal + Simple Interest = 200 + 80 = 280 \]
### Compound Interest Annually
Compound Interest Annually is calculated using the formula:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
For annual compounding, \( n = 1 \):
\[ A = P (1 + r)^{t} \]
Given:
- \( P = 200 \)
- \( r = 4\% = 0.04 \)
- \( t = 10 \)
- \( n = 1 \)
Plugging in these values:
\[ A = 200 \left(1 + 0.04\right)^{10} \approx 296.0488569836689 \]
### Compound Interest Biannually
Compound Interest Biannually is calculated using a similar formula, but compounded twice a year:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
For biannual compounding, \( n = 2 \):
\[ A = P \left(1 + \frac{0.04}{2}\right)^{2 \times 10} \]
Given:
- \( P = 200 \)
- \( r = 4\% = 0.04 \)
- \( t = 10 \)
- \( n = 2 \)
Plugging in these values:
\[ A = 200 \left(1 + \frac{0.04}{2}\right)^{2 \times 10} \approx 297.18947919567097 \]
### Summary
After 10 years, the values are as follows:
- Simple Interest: $[/tex]80, resulting in a total amount of [tex]$280.
- Compound Interest Annually: Approximately $[/tex]296.0488569836689.
- Compound Interest Biannually: Approximately $297.18947919567097.
Thus, the methods yield slightly different totals due to the compounding effect.
