Certainly! Let's break down the problem and find the total amount in Rui Feng's account after 6 years under different compounding conditions.
Given:
- Principal ([tex]$P$[/tex]): [tex]$15,000
- Annual Interest Rate ($[/tex]r[tex]$): 5.68%
- Number of Years ($[/tex]t[tex]$): 6 years
We will use the compound interest formula:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( A \) = 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 the interest is compounded per year.
- \( t \) = number of years the money is invested.
### Part (a): Compounded Monthly
For monthly compounding:
- The interest is compounded 12 times a year (\( n = 12 \)).
- The annual interest rate \( r \) is 5.68%, which is 0.0568 in decimal form.
Plugging the values into the compound interest formula:
\[ A_{\text{monthly}} = 15000 \left(1 + \frac{0.0568}{12}\right)^{12 \times 6} \]
Using this formula, we find:
\[ A_{\text{monthly}} = 15000 \left(1 + \frac{0.0568}{12}\right)^{72} \]
After solving this expression, the total amount in the account after 6 years, if the interest is compounded monthly, is approximately:
\[ A_{\text{monthly}} \approx 21,074.13 \]
### Part (b): Compounded Half-yearly
For half-yearly compounding:
- The interest is compounded 2 times a year (\( n = 2 \)).
- The annual interest rate \( r \) remains 0.0568 in decimal form.
Plugging the values into the formula:
\[ A_{\text{half-yearly}} = 15000 \left(1 + \frac{0.0568}{2}\right)^{2 \times 6} \]
Using this formula, we find:
\[ A_{\text{half-yearly}} = 15000 \left(1 + \frac{0.0568}{2}\right)^{12} \]
After solving this expression, the total amount in the account after 6 years, if the interest is compounded half-yearly, is approximately:
\[ A_{\text{half-yearly}} \approx 20,991.14 \]
### Summary:
1. The total amount in Rui Feng's account after 6 years with monthly compounding is approximately $[/tex]21,074.13.
2. The total amount in Rui Feng's account after 6 years with half-yearly compounding is approximately $20,991.14.
