A sum of money lent out at 9 percent for 5 years produces twice as much interest as Rs. 4,800 in `4""1/2` years at 10 per cent. Find the sum.

Answer

To solve the problem step by step, we will break it down into two parts as mentioned in the video transcript.

### Step 1: Calculate the Simple Interest for Rs. 4800 at 10% for 4.5 years
1. **Identify the given values:**
- Principal (P) = Rs. 4800
- Rate of Interest (R) = 10%
- Time (T) = 4.5 years = \( \frac{9}{2} \) years

2. **Use the formula for Simple Interest (SI):**
\[ SI = \frac{P \times R \times T}{100} \]

3. **Substitute the values into the formula:**
\[ SI = \frac{4800 \times 10 \times \frac{9}{2}}{100} \]

4. **Calculate the SI:**
- First, calculate \( 4800 \times 10 = 48000 \)
- Then, \( 48000 \times \frac{9}{2} = 48000 \times 4.5 = 216000 \)
- Finally, divide by 100:
\[ SI = \frac{216000}{100} = 2160 \]

### Step 2: Determine the Simple Interest for the unknown principal at 9% for 5 years
1. **Calculate the required Simple Interest:**
- The interest produced by the unknown principal is twice that of the first part:
\[ SI = 2 \times 2160 = 4320 \]

2. **Identify the new values:**
- SI = Rs. 4320
- Rate of Interest (R) = 9%
- Time (T) = 5 years

3. **Use the Simple Interest formula again:**
\[ SI = \frac{P \times R \times T}{100} \]

4. **Substitute the values into the formula:**
\[ 4320 = \frac{P \times 9 \times 5}{100} \]

5. **Simplify the equation:**
- Rearranging gives:
\[ 4320 = \frac{45P}{100} \]
- Multiply both sides by 100:
\[ 432000 = 45P \]
- Divide by 45 to find P:
\[ P = \frac{432000}{45} = 9600 \]

### Final Answer:
The sum of money lent out is Rs. 9600.

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