A sum of money lent out at simple interest amounts to `₹2880` in `2` years and to `₹3600` in `5` years. Find the sum of money and the rate of interest.

Answer

To solve the problem step by step, we will follow the approach outlined in the video transcript.

### Step 1: Understand the Problem
We need to find the principal amount (P) and the rate of interest (R) given that:
- The amount after 2 years is ₹2880.
- The amount after 5 years is ₹3600.

### Step 2: Set Up the Equations
Using the formula for the amount in simple interest:
\[ \text{Amount} = \text{Principal} + \text{Simple Interest} \]
The simple interest formula is:
\[ \text{Simple Interest} = \frac{P \times R \times T}{100} \]

1. For 2 years:
\[ 2880 = P + \frac{P \times R \times 2}{100} \]
This can be rewritten as:
\[ 2880 = P + \frac{2PR}{100} \] (Equation 1)

2. For 5 years:
\[ 3600 = P + \frac{P \times R \times 5}{100} \]
This can be rewritten as:
\[ 3600 = P + \frac{5PR}{100} \] (Equation 2)

### Step 3: Subtract the Equations
Now, we will subtract Equation 1 from Equation 2 to eliminate P:
\[ 3600 - 2880 = \left(P + \frac{5PR}{100}\right) - \left(P + \frac{2PR}{100}\right) \]
This simplifies to:
\[ 720 = \frac{5PR}{100} - \frac{2PR}{100} \]
\[ 720 = \frac{3PR}{100} \]

### Step 4: Solve for PR
Now we can solve for \( \frac{PR}{100} \):
\[ \frac{3PR}{100} = 720 \]
\[ PR = 720 \times \frac{100}{3} \]
\[ PR = 24000 \] (Equation 3)

### Step 5: Substitute PR into One of the Equations
Now we can use Equation 2 to find P:
\[ 3600 = P + \frac{5 \times 24000}{100} \]
\[ 3600 = P + 1200 \]
\[ P = 3600 - 1200 \]
\[ P = 2400 \]

### Step 6: Find the Rate of Interest R
Now substitute P back into Equation 3 to find R:
\[ 2400R = 24000 \]
\[ R = \frac{24000}{2400} \]
\[ R = 10\% \]

### Final Answers
- The principal amount (P) is ₹2400.
- The rate of interest (R) is 10%.