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Sagot :
Sure, let's go through each part of the question step-by-step.
### Given Data:
- Principal amount, [tex]\(P = \text{Rs. }18,082\)[/tex]
- Annual rate of interest, [tex]\(R = 20\%\)[/tex]
- Time, [tex]\(T = 2\)[/tex] years
### Formulas Involved:
1. Yearly Compound Interest:
[tex]\[ CI_{\text{yearly}} = P \left( \left(1 + \frac{R}{100}\right)^T - 1 \right) \][/tex]
2. Half-Yearly Compound Interest:
[tex]\[ CI_{\text{half-yearly}} = P \left( \left(1 + \frac{R}{200}\right)^{2T} - 1 \right) \][/tex]
### Step-by-Step Solution:
#### (a) Calculation of Yearly Compound Interest:
Using the formula for yearly compound interest:
[tex]\[ CI_{\text{yearly}} = P \left( \left(1 + \frac{R}{100}\right)^T - 1 \right) \][/tex]
Substitute the given values into the formula:
[tex]\[ CI_{\text{yearly}} = 18082 \left( \left(1 + \frac{20}{100}\right)^2 - 1 \right) \][/tex]
After performing the calculation, we find that the yearly compound interest is:
[tex]\[ CI_{\text{yearly}} \approx \text{Rs. } 7956.08 \][/tex]
#### (b) Calculation of Half-Yearly Compound Interest:
Using the formula for half-yearly compound interest:
[tex]\[ CI_{\text{half-yearly}} = P \left( \left(1 + \frac{R}{200}\right)^{2T} - 1 \right) \][/tex]
Substitute the given values into the formula:
[tex]\[ CI_{\text{half-yearly}} = 18082 \left( \left(1 + \frac{20}{200}\right)^{4} - 1 \right) \][/tex]
After performing the calculation, we find that the half-yearly compound interest is:
[tex]\[ CI_{\text{half-yearly}} \approx \text{Rs. } 8391.86 \][/tex]
#### (c) Difference Between Half-Yearly and Yearly Compound Interest:
To find the difference between the half-yearly and yearly compound interest:
[tex]\[ \text{Difference} = CI_{\text{half-yearly}} - CI_{\text{yearly}} \][/tex]
Using the values obtained:
[tex]\[ \text{Difference} \approx 8391.86 - 7956.08 = \text{Rs. } 435.78 \][/tex]
### Conclusion:
1. Yearly Compound Interest after 2 years: Rs. 7956.08
2. Half-Yearly Compound Interest after 2 years: Rs. 8391.86
3. Difference between Half-Yearly and Yearly Compound Interest: Rs. 435.78
### Given Data:
- Principal amount, [tex]\(P = \text{Rs. }18,082\)[/tex]
- Annual rate of interest, [tex]\(R = 20\%\)[/tex]
- Time, [tex]\(T = 2\)[/tex] years
### Formulas Involved:
1. Yearly Compound Interest:
[tex]\[ CI_{\text{yearly}} = P \left( \left(1 + \frac{R}{100}\right)^T - 1 \right) \][/tex]
2. Half-Yearly Compound Interest:
[tex]\[ CI_{\text{half-yearly}} = P \left( \left(1 + \frac{R}{200}\right)^{2T} - 1 \right) \][/tex]
### Step-by-Step Solution:
#### (a) Calculation of Yearly Compound Interest:
Using the formula for yearly compound interest:
[tex]\[ CI_{\text{yearly}} = P \left( \left(1 + \frac{R}{100}\right)^T - 1 \right) \][/tex]
Substitute the given values into the formula:
[tex]\[ CI_{\text{yearly}} = 18082 \left( \left(1 + \frac{20}{100}\right)^2 - 1 \right) \][/tex]
After performing the calculation, we find that the yearly compound interest is:
[tex]\[ CI_{\text{yearly}} \approx \text{Rs. } 7956.08 \][/tex]
#### (b) Calculation of Half-Yearly Compound Interest:
Using the formula for half-yearly compound interest:
[tex]\[ CI_{\text{half-yearly}} = P \left( \left(1 + \frac{R}{200}\right)^{2T} - 1 \right) \][/tex]
Substitute the given values into the formula:
[tex]\[ CI_{\text{half-yearly}} = 18082 \left( \left(1 + \frac{20}{200}\right)^{4} - 1 \right) \][/tex]
After performing the calculation, we find that the half-yearly compound interest is:
[tex]\[ CI_{\text{half-yearly}} \approx \text{Rs. } 8391.86 \][/tex]
#### (c) Difference Between Half-Yearly and Yearly Compound Interest:
To find the difference between the half-yearly and yearly compound interest:
[tex]\[ \text{Difference} = CI_{\text{half-yearly}} - CI_{\text{yearly}} \][/tex]
Using the values obtained:
[tex]\[ \text{Difference} \approx 8391.86 - 7956.08 = \text{Rs. } 435.78 \][/tex]
### Conclusion:
1. Yearly Compound Interest after 2 years: Rs. 7956.08
2. Half-Yearly Compound Interest after 2 years: Rs. 8391.86
3. Difference between Half-Yearly and Yearly Compound Interest: Rs. 435.78
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