If Rs. 3,750 amount to Rs. 4,620 in 3 years at simple interest. Find :
(i) the rate of interest.
the amount of Rs. 7,500 in `5""1/2` years at the same rate of interest.

Answer

To solve the problem step by step, let's break it down into two parts as given in the question.

### Part (i): Finding the Rate of Interest

1. **Identify the given values:**
- Principal (P) = Rs. 3,750
- Amount (A) = Rs. 4,620
- Time (T) = 3 years

2. **Calculate the Interest (I):**
\[ I = A - P = 4620 - 3750 = 870 \text{ Rs.} \]

3. **Use the Simple Interest formula:**
\[ I = \frac{P \times R \times T}{100} \]
Where:
- \( I \) = Interest
- \( P \) = Principal
- \( R \) = Rate of Interest (in %)
- \( T \) = Time (in years)

4. **Substituting the known values into the formula:**
\[ 870 = \frac{3750 \times R \times 3}{100} \]

5. **Rearranging the equation to find R:**
\[ R = \frac{870 \times 100}{3750 \times 3} \]

6. **Calculating R:**
\[ R = \frac{87000}{11250} = \frac{116}{15} \text{ percent per annum} \]

7. **Convert to mixed fraction:**
\[ R = 7 \frac{11}{15} \text{ percent per annum} \]

### Part (ii): Finding the Amount for Rs. 7,500 in 5.5 years

1. **Identify the new values:**
- New Principal (P) = Rs. 7,500
- Time (T) = 5.5 years = \( \frac{11}{2} \) years
- Rate of Interest (R) = \( 7 \frac{11}{15} \) percent per annum

2. **Convert R to an improper fraction for calculation:**
\[ R = \frac{116}{15} \text{ percent} \]

3. **Calculate the Interest (I) using the Simple Interest formula:**
\[ I = \frac{P \times R \times T}{100} \]
\[ I = \frac{7500 \times \frac{116}{15} \times \frac{11}{2}}{100} \]

4. **Calculating I:**
\[ I = \frac{7500 \times 116 \times 11}{15 \times 2 \times 100} \]
\[ I = \frac{7500 \times 1276}{3000} = \frac{9570000}{3000} = 3190 \text{ Rs.} \]

5. **Calculate the Amount (A):**
\[ A = P + I = 7500 + 3190 = 10690 \text{ Rs.} \]

### Final Answers:
- (i) Rate of Interest = \( 7 \frac{11}{15} \) percent per annum
- (ii) Amount for Rs. 7,500 in 5.5 years = Rs. 10,690

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