Due to a reduction of 50 paise/dozen in the price of pens, Sunita buy dozen more pen for `Rs.78` then the present price of pen/dozen ?

Options

A. Rs. 5/dozen
B. Rs. 4/dozen
C. Rs. 6/dozen
D. Rs. 3/dozen

Answer

To solve the problem step by step, let's define the variables and analyze the situation:

1. **Define the variables:**
Let the present price of pens per dozen be \( P \) (in rupees).

2. **Understand the price reduction:**
The price of pens has been reduced by 50 paise (or 0.50 rupees) per dozen. Therefore, the new price of pens per dozen is:
\[ P - 0.50 \]

3. **Calculate the number of dozens bought at the old price:**
If Sunita buys pens at the old price \( P \), the number of dozens she can buy for Rs. 78 is:
\[ \text{Number of dozens at old price} = \frac{78}{P} \]

4. **Calculate the number of dozens bought at the new price:**
At the new price \( P - 0.50 \), the number of dozens she can buy for Rs. 78 is:
\[ \text{Number of dozens at new price} = \frac{78}{P - 0.50} \]

5. **Set up the equation based on the information given:**
According to the problem, due to the price reduction, Sunita can buy 1 dozen more pens. Therefore, we can set up the equation:
\[ \frac{78}{P - 0.50} = \frac{78}{P} + 1 \]

6. **Clear the fractions by multiplying through by \( P(P - 0.50) \):**
\[ 78P = 78(P - 0.50) + P(P - 0.50) \]

7. **Expand and simplify the equation:**
\[ 78P = 78P - 39 + P^2 - 0.50P \]
\[ 0 = P^2 - 0.50P - 39 \]

8. **Rearrange the equation:**
\[ P^2 - 0.50P - 39 = 0 \]

9. **Solve the quadratic equation using the quadratic formula:**
The quadratic formula is given by:
\[ P = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -0.50, c = -39 \):
\[ P = \frac{-(-0.50) \pm \sqrt{(-0.50)^2 - 4 \cdot 1 \cdot (-39)}}{2 \cdot 1} \]
\[ P = \frac{0.50 \pm \sqrt{0.25 + 156}}{2} \]
\[ P = \frac{0.50 \pm \sqrt{156.25}}{2} \]
\[ P = \frac{0.50 \pm 12.5}{2} \]

10. **Calculate the two possible values for \( P \):**
\[ P = \frac{13}{2} = 6.5 \quad \text{(valid price)} \]
\[ P = \frac{-12}{2} = -6 \quad \text{(not valid)} \]

11. **Conclusion:**
The present price of pens per dozen is \( \text{Rs. } 6.5 \).