If the price of an article is increased by Rs. 2 per dozen that at present the number of things available for Rs. 56 is 8 less than before. The price per dozen at present is:

Options

A. Rs. 14
B. Rs. 12
C. Rs. 10
D. Rs. 28

Answer

To solve the problem step by step, we will define the variables and set up the equation based on the information provided.

### Step 1: Define the Variables
Let the present price per dozen be \( X \).
Thus, the previous price per dozen would be \( X - 2 \) (since the price increased by Rs. 2).

### Step 2: Calculate the Number of Items Available
- The number of items available for Rs. 56 at the previous price:
\[ \text{Number of items} = \frac{56}{\text{Price per item}} = \frac{56}{\frac{X - 2}{12}} = \frac{56 \times 12}{X - 2} = \frac{672}{X - 2} \]

- The number of items available for Rs. 56 at the present price:
\[ \text{Number of items} = \frac{56}{\text{Present price per item}} = \frac{56}{\frac{X}{12}} = \frac{56 \times 12}{X} = \frac{672}{X} \]

### Step 3: Set Up the Equation
According to the problem, the number of items available at present is 8 less than before:
\[ \frac{672}{X} = \frac{672}{X - 2} - 8 \]

### Step 4: Clear the Fractions
To eliminate the fractions, we can multiply through by \( X(X - 2) \):
\[ 672(X - 2) = 672X - 8X(X - 2) \]

### Step 5: Expand and Rearrange
Expanding both sides:
\[ 672X - 1344 = 672X - 8X^2 + 16X \]
Now, simplify:
\[ -1344 = -8X^2 + 16X \]
Rearranging gives:
\[ 8X^2 - 16X - 1344 = 0 \]

### Step 6: Simplify the Equation
Dividing the entire equation by 8:
\[ X^2 - 2X - 168 = 0 \]

### Step 7: Factor the Quadratic Equation
We need to factor the quadratic equation:
\[ (X - 14)(X + 12) = 0 \]

### Step 8: Solve for \( X \)
Setting each factor to zero gives:
1. \( X - 14 = 0 \) → \( X = 14 \)
2. \( X + 12 = 0 \) → \( X = -12 \) (not valid since price cannot be negative)

Thus, the present price per dozen is:
\[ X = 14 \]

### Final Answer
The present price per dozen is **Rs. 14**.

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