Mr. Sharma borrowed a certain sum of money at 10% per annum compounded annually. If by paying Rs 19,360 at the end of the second year and Rs 31.944 at the end of the third year he clears the debt, find the sum borrowed by him.

Answer

To solve the problem step by step, we will follow the logic provided in the video transcript and break it down into clear steps.

### Step 1: Define the Variables
Let the sum borrowed by Mr. Sharma be \( x \).

### Step 2: Calculate the Amount after 2 Years
The amount after 2 years at 10% per annum compounded annually can be calculated using the formula:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \( P = x \) (the principal amount)
- \( r = 10 \) (the rate of interest)
- \( n = 2 \) (the number of years)

Thus, the amount after 2 years is:
\[ A = x \left(1 + \frac{10}{100}\right)^2 = x \left(1.1\right)^2 = x \cdot 1.21 = \frac{121x}{100} \]

### Step 3: Calculate the Remaining Amount after the First Payment
Mr. Sharma pays Rs 19,360 at the end of the second year. The remaining amount after this payment is:
\[ \text{Remaining Amount} = \frac{121x}{100} - 19,360 \]

### Step 4: Calculate the Amount after 1 More Year (3rd Year)
Now, we need to calculate the amount after one more year (the third year) on the remaining amount. The interest for the third year is calculated on the remaining amount:
\[ \text{Amount after 3 years} = \left(\frac{121x}{100} - 19,360\right) \left(1 + \frac{10}{100}\right) = \left(\frac{121x}{100} - 19,360\right) \cdot 1.1 \]

### Step 5: Set Up the Equation for the Second Payment
Mr. Sharma pays Rs 31,944 at the end of the third year to clear his debt. Therefore, we can set up the equation:
\[ \left(\frac{121x}{100} - 19,360\right) \cdot 1.1 = 31,944 \]

### Step 6: Solve the Equation
Expanding the equation:
\[ \frac{121x}{100} \cdot 1.1 - 19,360 \cdot 1.1 = 31,944 \]
\[ \frac{121x \cdot 1.1}{100} - 21,296 = 31,944 \]
Adding 21,296 to both sides:
\[ \frac{121x \cdot 1.1}{100} = 31,944 + 21,296 \]
\[ \frac{121x \cdot 1.1}{100} = 53,240 \]
Multiplying both sides by 100:
\[ 121x \cdot 1.1 = 5,324,000 \]
Dividing both sides by 1.1:
\[ 121x = \frac{5,324,000}{1.1} = 4,840,000 \]
Finally, dividing both sides by 121:
\[ x = \frac{4,840,000}{121} = 40,000 \]

### Conclusion
The sum borrowed by Mr. Sharma is Rs 40,000.

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