What sum of money borrowed on 24th May will amount to Rs. `10,210*20` on 17th October of the same year at 5 percent per annum simple interest.

Answer

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

### Step 1: Identify the given values
- Amount (A) = Rs. 10,210.20
- Rate of interest (R) = 5% per annum
- Time (T) = From 24th May to 17th October

### Step 2: Calculate the time in days
- From 24th May to 17th October, the number of days can be calculated as follows:
- May: 7 days (from 24th to 31st)
- June: 30 days
- July: 31 days
- August: 31 days
- September: 30 days
- October: 17 days
- Total days = 7 + 30 + 31 + 31 + 30 + 17 = 146 days

### Step 3: Convert time into years
- Since the rate is per annum, we convert days into years:
\[ T = \frac{146}{365} \text{ years} \]

### Step 4: Use the formula for Simple Interest
- The formula for the amount (A) in terms of principal (P), rate (R), and time (T) is:
\[ A = P + \text{SI} \]
where Simple Interest (SI) is given by:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
- Therefore, we can rewrite the equation as:
\[ A = P + \frac{P \times R \times T}{100} \]
\[ A = P \left(1 + \frac{R \times T}{100}\right) \]

### Step 5: Substitute the known values into the equation
- Substitute A, R, and T into the equation:
\[ 10,210.20 = P \left(1 + \frac{5 \times \frac{146}{365}}{100}\right) \]

### Step 6: Simplify the equation
- Calculate \( \frac{5 \times \frac{146}{365}}{100} \):
\[ \frac{5 \times 146}{36500} = \frac{730}{36500} = \frac{1}{50} \]
- Thus, the equation becomes:
\[ 10,210.20 = P \left(1 + \frac{1}{50}\right) \]
\[ 10,210.20 = P \left(\frac{51}{50}\right) \]

### Step 7: Solve for P
- Rearranging gives:
\[ P = 10,210.20 \times \frac{50}{51} \]
- Now, calculate P:
\[ P = 10,210.20 \times \frac{50}{51} \approx 10,000 \]

### Final Answer
- The principal amount borrowed is Rs. 10,000.

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