A certain sum of money at compound interset amounts to Rs 6,600 in 1 year and to Rs 7,986 in 3 years. Find the sum and the rate percent.

Answer

To solve the problem step by step, we will use the formula for compound interest and the information provided in the question.

### Given:
- Amount after 1 year (A1) = Rs 6,600
- Amount after 3 years (A3) = Rs 7,986

### Step 1: Set up the equations
Let:
- P = Principal amount
- R = Rate of interest (in percent)

From the formula for compound interest, we have:
1. For 1 year:
\[ A_1 = P \left(1 + \frac{R}{100}\right)^1 = 6600 \]
This can be rewritten as:
\[ P \left(1 + \frac{R}{100}\right) = 6600 \quad \text{(Equation 1)} \]

2. For 3 years:
\[ A_3 = P \left(1 + \frac{R}{100}\right)^3 = 7986 \]
This can be rewritten as:
\[ P \left(1 + \frac{R}{100}\right)^3 = 7986 \quad \text{(Equation 2)} \]

### Step 2: Divide Equation 2 by Equation 1
To eliminate P, we divide Equation 2 by Equation 1:
\[ \frac{P \left(1 + \frac{R}{100}\right)^3}{P \left(1 + \frac{R}{100}\right)} = \frac{7986}{6600} \]
This simplifies to:
\[ \left(1 + \frac{R}{100}\right)^2 = \frac{7986}{6600} \]

### Step 3: Calculate the right-hand side
Calculating the fraction:
\[ \frac{7986}{6600} = 1.21 \]
So we have:
\[ \left(1 + \frac{R}{100}\right)^2 = 1.21 \]

### Step 4: Take the square root
Taking the square root of both sides:
\[ 1 + \frac{R}{100} = \sqrt{1.21} = 1.1 \]

### Step 5: Solve for R
Now, we can solve for R:
\[ \frac{R}{100} = 1.1 - 1 = 0.1 \]
Multiplying both sides by 100:
\[ R = 10\% \]

### Step 6: Find the Principal (P)
Now that we have R, we can substitute it back into Equation 1 to find P:
\[ P \left(1 + \frac{10}{100}\right) = 6600 \]
This simplifies to:
\[ P \left(1 + 0.1\right) = 6600 \]
\[ P \cdot 1.1 = 6600 \]
Now, divide both sides by 1.1:
\[ P = \frac{6600}{1.1} = 6000 \]

### Final Answers:
- Principal (P) = Rs 6000
- Rate of interest (R) = 10%