A sum of money lent out at simple interest amounts to `₹7440` in `3` years and to `₹9360` in `7` years. Find the sum of money and the rate of interest.

Answer

To solve the problem step by step, we will find the principal amount (sum of money) and the rate of interest using the information given.

### Step 1: Understand the problem
We know that:
- The amount after 3 years is ₹7440.
- The amount after 7 years is ₹9360.

We need to find the principal amount (P) and the rate of interest (R).

### Step 2: Write the formula for amount in terms of principal and simple interest
The formula for the amount (A) in terms of principal (P), rate of interest (R), and time (T) is:
\[ A = P + \text{SI} \]
Where:
\[ \text{SI} = \frac{P \times R \times T}{100} \]

### Step 3: Set up equations based on the information given
For the first scenario (after 3 years):
\[ 7440 = P + \frac{P \times R \times 3}{100} \]
This can be rearranged to:
\[ 7440 = P + \frac{3PR}{100} \] (Equation 1)

For the second scenario (after 7 years):
\[ 9360 = P + \frac{P \times R \times 7}{100} \]
This can be rearranged to:
\[ 9360 = P + \frac{7PR}{100} \] (Equation 2)

### Step 4: Subtract Equation 1 from Equation 2
Now, we will subtract Equation 1 from Equation 2 to eliminate P:
\[ 9360 - 7440 = \left(P + \frac{7PR}{100}\right) - \left(P + \frac{3PR}{100}\right) \]
This simplifies to:
\[ 1920 = \frac{7PR}{100} - \frac{3PR}{100} \]
\[ 1920 = \frac{4PR}{100} \]

### Step 5: Solve for PR
Now, we can solve for \( \frac{PR}{100} \):
\[ PR = 1920 \times \frac{100}{4} \]
\[ PR = 48000 \]
So, we have:
\[ \frac{PR}{100} = 480 \] (Equation 3)

### Step 6: Substitute back to find P
Now, we can substitute \( \frac{PR}{100} = 480 \) back into Equation 1:
\[ 7440 = P + 3 \times 480 \]
\[ 7440 = P + 1440 \]
Now, solve for P:
\[ P = 7440 - 1440 \]
\[ P = 6000 \]

### Step 7: Find the rate of interest R
Now that we have P, we can use Equation 3 to find R:
\[ \frac{6000 \times R}{100} = 480 \]
\[ 6000R = 48000 \]
\[ R = \frac{48000}{6000} \]
\[ R = 8 \]

### Final Results
- The principal amount (sum of money) is **₹6000**.
- The rate of interest is **8%**.