Let's solve this problem step-by-step.
### Step 1: Identify the Given Information
- Loan amount (Principal, P): [tex]$22,000
- Annual interest rate (R): 5.2%
- Loan duration: 24 months
- Compounding frequency: Semiannually
### Step 2: Convert the Annual Interest Rate to Decimal Form
The annual interest rate given is 5.2%. To convert this to a decimal for calculation purposes:
\[ \text{Annual interest rate} = \frac{5.2}{100} = 0.052 \]
### Step 3: Determine the Number of Compounding Periods per Year
Since the interest is compounded semiannually, this means it is compounded twice per year:
\[ \text{Number of compounding periods per year} = 2 \]
### Step 4: Calculate the Total Number of Compounding Periods for the Loan Term
The loan duration is 24 months. Since there are 12 months in a year, the loan term in years is:
\[ \text{Loan term in years} = \frac{24}{12} = 2 \]
Given that the interest is compounded semiannually (twice a year), the total number of compounding periods over the loan term is:
\[ \text{Total compounding periods} = 2 \text{ years} \times 2 \text{ periods per year} = 4 \]
### Step 5: Calculate the Interest Rate per Compounding Period
To find the interest rate per compounding period when compounded semiannually:
\[ \text{Period interest rate} = \frac{\text{Annual interest rate}}{\text{Periods per year}} = \frac{0.052}{2} = 0.026 \]
### Step 6: Calculate the Total Amount to be Paid Using the Compound Interest Formula
The compound interest formula is:
\[ A = P (1 + r/n)^{nt} \]
Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial loan amount).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times the interest is compounded per year.
- \( t \) is the time the money is invested or borrowed for, in years.
Plugging in the values:
\[ A = 22000 \times (1 + 0.026)^{4} \]
\[ A = 22000 \times (1.026)^{4} \]
\[ A = 22000 \times 1.108127 \]
\[ A = 24378.788741472003 \]
### Step 7: Calculate the Interest Charged on the Loan
To find the interest charged, we subtract the principal from the total amount paid:
\[ \text{Interest charged} = A - P \]
\[ \text{Interest charged} = 24378.788741472003 - 22000 \]
\[ \text{Interest charged} = 2378.7887414720026 \]
### Conclusion
The interest charged on the loan is approximately $[/tex]2378.79
