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(i) [tex]$C_l = P(1 + \frac{R}{100})$[/tex]
(ii) [tex]$C_i = P\left[\left(1 + \frac{R}{200}\right)^T - 1\right]$[/tex]
(iii) [tex]$C_i = P\left[\left(1 + \frac{R}{100}\right)^T - 1\right]$[/tex]
(iv) [tex]$C_l = P\left(1 + \frac{R}{100}\right)^T - 1$[/tex]

(a) साधारण व्याज र चक्रीय व्याज पत्ता लगाउनुहोस्। वार्षिक [tex]$20\%$[/tex] ब्याजदरले 2 वर्षमा हुने वार्षिक मिश्रित ब्याज र अर्धवार्षिक मिश्रित ब्याज पत्ता लगाउनुहोस्।

Find the simple interest and compound interest on a sum of money at an annual interest rate of [tex]$20\%$[/tex] over 2 years, compounded annually and semi-annually.

(i) Write down the formula for calculating the yearly compound interest.
(ii) उक्त धनराशि पत्ता लगाउनुहोस्। Find the sum of money.
(iii) अर्धवार्षिक र वार्षिक मिश्रित ब्याजको अन्तर पत्ता लगाउनुहोस्। Find the difference between half-yearly and yearly compound interest.

Note: वार्षिक चक्रीय व्याजदरमा कुनै रकमको मिश्रधन 2 वर्ष र 3 वर्षमा क्रमशः र. 23,328 र र. [tex]$25,194.24$[/tex] हुन्छ। The compound interest on a sum of money over 2 years and 3 years is Rs. 23,328 and Rs. [tex]$25,194.24$[/tex] respectively.

(a) वार्षिक चक्रीय मिश्रधन पत्ता लगाउने सूत्र लेख्नुहोस्। Write down the formula to find the annual compound interest.



Answer :

GinnyAnswer
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

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