A certain sum of money amounts to Rs 6780 in 2 years and to Rs 7360.50 in `3(1)/(2)` years. Find the sum and the rate of interset.

Options

A. Rs 6006 , 6.4 Percent per annum
B. Rs 8006 , 6.4 Percent Per annum
C. Rs 5006 , 5 Percent Per annum
D. Rs 5505 , 5 Percent Per annum

Answer

To solve the problem step by step, we will first find the principal amount (sum of money) and then calculate the rate of interest.

### Step 1: Identify the amounts and the time periods
- Amount after 2 years (A1) = Rs 6780
- Amount after 3.5 years (A2) = Rs 7360.50
- Time period for A1 = 2 years
- Time period for A2 = 3.5 years

### Step 2: Calculate the interest earned in the time period from 2 years to 3.5 years
The difference in amounts gives us the interest earned during the period from 2 years to 3.5 years.

\[ \text{Interest} = A2 - A1 = 7360.50 - 6780 = 580.50 \]

### Step 3: Calculate the time difference
The time difference between 2 years and 3.5 years is:
\[ 3.5 - 2 = 1.5 \text{ years} \]

### Step 4: Calculate the rate of interest for the period of 1.5 years
Now we can find the interest for 1 year by dividing the interest by the time period:

\[ \text{Interest for 1 year} = \frac{580.50}{1.5} = 387 \]

### Step 5: Calculate the total interest for 2 years
Now we can find the total interest for the 2-year period:

\[ \text{Total Interest for 2 years} = 387 \times 2 = 774 \]

### Step 6: Calculate the principal amount (P)
Using the amount after 2 years and the total interest for 2 years, we can find the principal:

\[ P = A1 - \text{Total Interest for 2 years} = 6780 - 774 = 6006 \]

### Step 7: Calculate the rate of interest
Using the formula for simple interest:

\[ \text{Interest} = \frac{P \times R \times T}{100} \]

Where:
- Interest = 774 (for 2 years)
- P = 6006
- T = 2 years

We can rearrange the formula to find R:

\[ R = \frac{Interest \times 100}{P \times T} = \frac{774 \times 100}{6006 \times 2} \]

Calculating R:

\[ R = \frac{77400}{12012} \approx 6.44\% \]

### Conclusion
- Principal amount (sum) = Rs 6006
- Rate of interest = 6.44% per annum (approximately 6.4%)