The price of an article was increase by `P%`. Letter the new price was decrease by `P%`. If the latest price was Re. 1, the original price was-

Options

A. Rs. 1
B. Rs. `((1-P^2)/(100))`
C. Rs. `((10000)/(10000-P^2))`
D. Rs. `((sqrt(1-P^2))/(100))`

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Define the Original Price
Let the original price of the article be \( x \).

### Step 2: Calculate the Increased Price
The price is increased by \( P\% \). The new price after the increase can be calculated as:
\[ \text{Increased Price} = x + \left( \frac{P}{100} \times x \right) = x \left( 1 + \frac{P}{100} \right) = x \left( \frac{100 + P}{100} \right) \]

### Step 3: Calculate the Decreased Price
This new price is then decreased by \( P\% \). The price after the decrease can be calculated as:
\[ \text{Decreased Price} = \text{Increased Price} - \left( \frac{P}{100} \times \text{Increased Price} \right) = \text{Increased Price} \left( 1 - \frac{P}{100} \right) \]
Substituting the increased price:
\[ \text{Decreased Price} = x \left( \frac{100 + P}{100} \right) \left( 1 - \frac{P}{100} \right) \]

### Step 4: Simplify the Expression
Now, we can simplify the expression:
\[ \text{Decreased Price} = x \left( \frac{100 + P}{100} \right) \left( \frac{100 - P}{100} \right) = x \left( \frac{(100 + P)(100 - P)}{10000} \right) \]
Using the difference of squares:
\[ (100 + P)(100 - P) = 10000 - P^2 \]
Thus, we have:
\[ \text{Decreased Price} = x \left( \frac{10000 - P^2}{10000} \right) \]

### Step 5: Set the Decreased Price Equal to 1
According to the problem, the final price after the decrease is Re. 1:
\[ x \left( \frac{10000 - P^2}{10000} \right) = 1 \]

### Step 6: Solve for the Original Price \( x \)
To find \( x \), we rearrange the equation:
\[ x = \frac{10000}{10000 - P^2} \]

### Final Answer
Thus, the original price of the article is:
\[ \boxed{\frac{10000}{10000 - P^2}} \]