Due to 25% fall in the rate of eggs, one can buy 2 dozen eggs more than before by investing Rs 162. Then the orignal rate per dozen of the eggs is

Options

A. Rs 22
B. Rs 24
C. Rs 27
D. Rs 30

Answer

To solve the problem step by step, we will analyze the information given and find the original rate per dozen of eggs.

### Step 1: Understand the problem
We know that due to a 25% fall in the rate of eggs, one can buy 2 dozen eggs more than before by investing Rs 162. We need to find the original rate per dozen of eggs.

### Step 2: Define the original price
Let the original price per dozen of eggs be Rs. \( x \).

### Step 3: Calculate the new price after the fall
Since the price falls by 25%, the new price per dozen will be:
\[ \text{New Price} = x - 0.25x = 0.75x \]

### Step 4: Determine the number of dozens bought before and after the price fall
Before the price fall, the number of dozens that could be bought with Rs 162 is:
\[ \text{Dozens before} = \frac{162}{x} \]
After the price fall, the number of dozens that can be bought is:
\[ \text{Dozens after} = \frac{162}{0.75x} \]

### Step 5: Set up the equation based on the increase in dozens
According to the problem, the difference in the number of dozens bought before and after the price fall is 2 dozen:
\[ \frac{162}{0.75x} - \frac{162}{x} = 2 \]

### Step 6: Solve the equation
To solve the equation, first find a common denominator:
\[ \frac{162}{0.75x} - \frac{162}{x} = 2 \]
This can be rewritten as:
\[ \frac{162x - 162 \cdot 0.75x}{0.75x^2} = 2 \]
Simplifying the numerator:
\[ \frac{162x - 121.5x}{0.75x^2} = 2 \]
\[ \frac{40.5x}{0.75x^2} = 2 \]
Cross-multiplying gives:
\[ 40.5x = 2 \cdot 0.75x^2 \]
\[ 40.5x = 1.5x^2 \]
Rearranging the equation:
\[ 1.5x^2 - 40.5x = 0 \]
Factoring out \( x \):
\[ x(1.5x - 40.5) = 0 \]
This gives us two solutions, \( x = 0 \) or \( 1.5x = 40.5 \).

### Step 7: Find the value of \( x \)
Solving \( 1.5x = 40.5 \):
\[ x = \frac{40.5}{1.5} = 27 \]

### Conclusion
The original rate per dozen of the eggs is Rs. 27.