On decreasing price of pins by Rs 4 / dozen a person buys 12 more pins Rs . 48. Find new price of pins per dozen.

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Define Variables
Let the original price of pins per dozen be \( x \) rupees.

### Step 2: Determine New Price
The new price after a decrease of Rs. 4 per dozen will be:
\[ x - 4 \text{ rupees} \]

### Step 3: Calculate the Number of Pins Bought
The number of pins bought at the original price is:
\[ \text{Number of pins} = \frac{48}{x} \]
At the new price, the number of pins bought is:
\[ \text{Number of pins} = \frac{48}{x - 4} \]

### Step 4: Set Up the Equation
According to the problem, the person buys 12 more pins at the new price:
\[ \frac{48}{x - 4} = \frac{48}{x} + 12 \]

### Step 5: Clear the Denominators
To eliminate the fractions, multiply through by \( x(x - 4) \):
\[ 48x = 48(x - 4) + 12x(x - 4) \]

### Step 6: Simplify the Equation
Expanding both sides:
\[ 48x = 48x - 192 + 12x^2 - 48x \]
This simplifies to:
\[ 0 = 12x^2 - 192 \]

### Step 7: Rearrange the Equation
Rearranging gives us:
\[ 12x^2 - 192 = 0 \]

### Step 8: Factor the Equation
Dividing the entire equation by 12:
\[ x^2 - 16 = 0 \]
This factors to:
\[ (x - 4)(x + 4) = 0 \]

### Step 9: Solve for \( x \)
Setting each factor to zero gives:
\[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \]
\[ x + 4 = 0 \quad \Rightarrow \quad x = -4 \quad (\text{not valid since price cannot be negative}) \]

### Step 10: Find New Price
The original price was \( x = 16 \) rupees per dozen. Thus, the new price is:
\[ x - 4 = 16 - 4 = 12 \text{ rupees per dozen} \]

### Final Answer
The new price of pins per dozen is **Rs. 12**.

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