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 given amounts at different times.

### Step 1: Set Up the Equations
Let the principal amount be \( P \) and the rate of interest be \( r \% \).

From the problem, we know:
- The amount after 2 years is \( A_2 = 19,360 \)
- The amount after 4 years is \( A_4 = 23,425.60 \)

Using the compound interest formula:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]

We can write two equations based on the amounts given:

1. For 2 years:
\[ 19,360 = P \left(1 + \frac{r}{100}\right)^2 \quad \text{(Equation 1)} \]

2. For 4 years:
\[ 23,425.60 = P \left(1 + \frac{r}{100}\right)^4 \quad \text{(Equation 2)} \]

### Step 2: Divide the Equations
Now, we will divide Equation 2 by Equation 1 to eliminate \( P \):
\[ \frac{23,425.60}{19,360} = \frac{P \left(1 + \frac{r}{100}\right)^4}{P \left(1 + \frac{r}{100}\right)^2} \]
This simplifies to:
\[ \frac{23,425.60}{19,360} = \left(1 + \frac{r}{100}\right)^{4-2} \]
\[ \frac{23,425.60}{19,360} = \left(1 + \frac{r}{100}\right)^2 \]

### Step 3: Calculate the Left Side
Now, let's calculate \( \frac{23,425.60}{19,360} \):
\[ \frac{23,425.60}{19,360} = 1.21 \]
Thus, we have:
\[ \left(1 + \frac{r}{100}\right)^2 = 1.21 \]

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

### Step 5: Solve for \( r \)
Now, we can solve for \( r \):
\[ \frac{r}{100} = 1.1 - 1 \]
\[ \frac{r}{100} = 0.1 \]
\[ r = 10 \]

So, the rate of interest is \( 10\% \).

### Step 6: Find the Principal Amount
Now, we can substitute \( r \) back into Equation 1 to find \( P \):
\[ 19,360 = P \left(1 + \frac{10}{100}\right)^2 \]
\[ 19,360 = P \left(1.1\right)^2 \]
\[ 19,360 = P \times 1.21 \]
\[ P = \frac{19,360}{1.21} \]
Calculating \( P \):
\[ P = 16,000 \]

### Final Answer
The original sum of money (principal) is \( Rs. 16,000 \) and the rate of interest is \( 10\% \).

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