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Sagot :
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
### 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
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