A sum of money, invested at compound interest, amounts to Rs 19,360 in 2 years and to Rs 23,425.60 in 4 years. Find the rate percent and the original sum of money.

Answer

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

### Step 1: Understand the given information
We are given:
- Amount after 2 years (A1) = Rs 19,360
- Amount after 4 years (A2) = Rs 23,425.60

### Step 2: Use the compound interest formula
The compound interest formula is:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
Where:
- \( A \) = Amount
- \( P \) = Principal (original sum of money)
- \( r \) = Rate of interest (in percentage)
- \( t \) = Time (in years)

### Step 3: Set up the equations
Let the principal amount be \( P = x \).

For the first case (after 2 years):
\[ A_1 = x \left(1 + \frac{r}{100}\right)^2 \]
Substituting the value of \( A_1 \):
\[ 19360 = x \left(1 + \frac{r}{100}\right)^2 \] (Equation 1)

For the second case (after 4 years):
\[ A_2 = x \left(1 + \frac{r}{100}\right)^4 \]
Substituting the value of \( A_2 \):
\[ 23425.60 = x \left(1 + \frac{r}{100}\right)^4 \] (Equation 2)

### Step 4: Divide Equation 2 by Equation 1
Dividing Equation 2 by Equation 1 to eliminate \( x \):
\[ \frac{23425.60}{19360} = \frac{x \left(1 + \frac{r}{100}\right)^4}{x \left(1 + \frac{r}{100}\right)^2} \]
This simplifies to:
\[ \frac{23425.60}{19360} = \left(1 + \frac{r}{100}\right)^{4-2} \]
\[ \frac{23425.60}{19360} = \left(1 + \frac{r}{100}\right)^2 \]

### Step 5: Calculate the left side
Calculating the left side:
\[ \frac{23425.60}{19360} \approx 1.21 \]
Thus, we have:
\[ 1.21 = \left(1 + \frac{r}{100}\right)^2 \]

### Step 6: Take the square root
Taking the square root of both sides:
\[ \sqrt{1.21} = 1 + \frac{r}{100} \]
\[ 1.1 = 1 + \frac{r}{100} \]

### Step 7: Solve for \( r \)
Subtracting 1 from both sides:
\[ 0.1 = \frac{r}{100} \]
Multiplying both sides by 100:
\[ r = 10\% \]

### Step 8: Substitute \( r \) back to find \( x \)
Now, substitute \( r = 10\% \) back into Equation 1 to find \( x \):
\[ 19360 = x \left(1 + \frac{10}{100}\right)^2 \]
\[ 19360 = x \left(1.1\right)^2 \]
\[ 19360 = x \cdot 1.21 \]
Now, solving for \( x \):
\[ x = \frac{19360}{1.21} \approx 16000 \]

### Final Answer
- The rate percent \( r = 10\% \)
- The original sum of money \( P = 16000 \)