A sum of money, invested at compound interest, amounts to Rs 16,500 in 1 year and to Rs 19,965 in 3 years. Find the rate per cent and the original sum of money invested.

Answer

To solve the problem step by step, we will use the formula for compound interest and the information provided about the amounts after 1 year and 3 years.

### Step 1: Identify the given information
- Amount after 1 year (A1) = Rs 16,500
- Amount after 3 years (A3) = Rs 19,965

### Step 2: Set up the equations using the compound interest formula
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Where:
- \( A \) = Amount after time \( T \)
- \( P \) = Principal amount (initial investment)
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

For the first year:
\[ A_1 = P \left(1 + \frac{R}{100}\right)^1 = 16500 \]
This simplifies to:
\[ A_1 = P \left(1 + \frac{R}{100}\right) = 16500 \] (Equation 1)

For the third year:
\[ A_3 = P \left(1 + \frac{R}{100}\right)^3 = 19965 \]
This simplifies to:
\[ A_3 = P \left(1 + \frac{R}{100}\right)^3 = 19965 \] (Equation 2)

### Step 3: Express \( P \) from Equation 1
From Equation 1, we can express \( P \):
\[ P = \frac{16500}{1 + \frac{R}{100}} \]

### Step 4: Substitute \( P \) in Equation 2
Substituting \( P \) from Equation 1 into Equation 2:
\[ 19965 = \left(\frac{16500}{1 + \frac{R}{100}}\right) \left(1 + \frac{R}{100}\right)^3 \]

### Step 5: Simplify the equation
This simplifies to:
\[ 19965 = 16500 \cdot \frac{(1 + \frac{R}{100})^3}{(1 + \frac{R}{100})} \]
\[ 19965 = 16500 \cdot (1 + \frac{R}{100})^2 \]

### Step 6: Isolate \( (1 + \frac{R}{100})^2 \)
Dividing both sides by 16500:
\[ \frac{19965}{16500} = (1 + \frac{R}{100})^2 \]
Calculating the left side:
\[ 1.2 = (1 + \frac{R}{100})^2 \]

### Step 7: Take the square root
Taking the square root of both sides:
\[ \sqrt{1.2} = 1 + \frac{R}{100} \]
Calculating \( \sqrt{1.2} \):
\[ \sqrt{1.2} \approx 1.095 \]
So:
\[ 1.095 = 1 + \frac{R}{100} \]

### Step 8: Solve for \( R \)
Subtracting 1 from both sides:
\[ 0.095 = \frac{R}{100} \]
Multiplying by 100:
\[ R = 9.5 \]

### Step 9: Find the original principal amount \( P \)
Substituting \( R \) back into the equation for \( P \):
\[ P = \frac{16500}{1 + \frac{9.5}{100}} \]
Calculating:
\[ P = \frac{16500}{1.095} \approx 15000 \]

### Final Result
- Rate of interest \( R \) = 9.5%
- Original sum of money invested \( P \) = Rs 15,000