Calculate the amount and the compound interest on:
(i) रु 4,600 in 2 years when the rates of interest of successive years are 10% and 12% respectively.
(ii) रु 16,000 in 3 years, when the rates of the interest for successive years are 10%, 14% and 15% respectively.

Explanation

Let's solve the problem step by step.

### Part (i): Calculate the amount and compound interest on रु 4,600 in 2 years with interest rates of 10% and 12%.

1. **Identify the Principal (P) and Rates of Interest (R1 and R2)**:
- Principal (P) = रु 4,600
- Rate for Year 1 (R1) = 10%
- Rate for Year 2 (R2) = 12%

2. **Calculate the Amount after Year 1**:
- The formula for the amount after the first year is:
\[ A_1 = P \times \left(1 + \frac{R1}{100}\right) \]
- Substitute the values:
\[ A_1 = 4600 \times \left(1 + \frac{10}{100}\right) = 4600 \times 1.10 = 5060 \]

3. **Calculate the Amount after Year 2**:
- Now, use the amount from Year 1 as the principal for Year 2:
\[ A_2 = A_1 \times \left(1 + \frac{R2}{100}\right) \]
- Substitute the values:
\[ A_2 = 5060 \times \left(1 + \frac{12}{100}\right) = 5060 \times 1.12 = 5679.20 \]

4. **Calculate the Compound Interest**:
- Compound Interest (CI) is given by:
\[ CI = A_2 - P \]
- Substitute the values:
\[ CI = 5679.20 - 4600 = 1079.20 \]

### Summary for Part (i):
- Amount after 2 years = रु 5679.20
- Compound Interest = रु 1079.20

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### Part (ii): Calculate the amount and compound interest on रु 16,000 in 3 years with interest rates of 10%, 14%, and 15%.

1. **Identify the Principal (P) and Rates of Interest (R1, R2, and R3)**:
- Principal (P) = रु 16,000
- Rate for Year 1 (R1) = 10%
- Rate for Year 2 (R2) = 14%
- Rate for Year 3 (R3) = 15%

2. **Calculate the Amount after Year 1**:
- The formula for the amount after the first year is:
\[ A_1 = P \times \left(1 + \frac{R1}{100}\right) \]
- Substitute the values:
\[ A_1 = 16000 \times \left(1 + \frac{10}{100}\right) = 16000 \times 1.10 = 17600 \]

3. **Calculate the Amount after Year 2**:
- Use the amount from Year 1 as the principal for Year 2:
\[ A_2 = A_1 \times \left(1 + \frac{R2}{100}\right) \]
- Substitute the values:
\[ A_2 = 17600 \times \left(1 + \frac{14}{100}\right) = 17600 \times 1.14 = 20064 \]

4. **Calculate the Amount after Year 3**:
- Use the amount from Year 2 as the principal for Year 3:
\[ A_3 = A_2 \times \left(1 + \frac{R3}{100}\right) \]
- Substitute the values:
\[ A_3 = 20064 \times \left(1 + \frac{15}{100}\right) = 20064 \times 1.15 = 23073.60 \]

5. **Calculate the Compound Interest**:
- Compound Interest (CI) is given by:
\[ CI = A_3 - P \]
- Substitute the values:
\[ CI = 23073.60 - 16000 = 7073.60 \]

### Summary for Part (ii):
- Amount after 3 years = रु 23073.60
- Compound Interest = रु 7073.60

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