S.P. of a pen is Rs 56. If value of its C.P. and profit% is equal then find its C.P. ?

Options

A. 30
B. 40
C. 46
D. 50

Answer

To solve the problem step by step, we will follow the mathematical approach based on the information provided.

### Step 1: Understand the problem
We know that the Selling Price (S.P.) of the pen is Rs. 56. We need to find the Cost Price (C.P.) such that the value of C.P. is equal to the profit percentage.

### Step 2: Set up the equation
Let the Cost Price (C.P.) be denoted as \( X \). Since the profit percentage is equal to the C.P., we can say that the profit percentage is also \( X \).

### Step 3: Use the formula for Selling Price
The formula for Selling Price (S.P.) is given by:
\[ S.P. = C.P. + \text{Profit} \]
Where Profit can be expressed as:
\[ \text{Profit} = \frac{\text{Profit Percentage}}{100} \times C.P. \]
Substituting the values, we have:
\[ S.P. = X + \frac{X}{100} \times X \]
This simplifies to:
\[ S.P. = X + \frac{X^2}{100} \]

### Step 4: Substitute the known S.P.
Since we know that S.P. is Rs. 56, we can write:
\[ 56 = X + \frac{X^2}{100} \]

### Step 5: Rearrange the equation
To eliminate the fraction, multiply the entire equation by 100:
\[ 5600 = 100X + X^2 \]
Rearranging gives us:
\[ X^2 + 100X - 5600 = 0 \]

### Step 6: Solve the quadratic equation
Now we will use the quadratic formula to solve for \( X \):
\[ X = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = 100 \), and \( c = -5600 \).

Calculating the discriminant:
\[ b^2 - 4ac = 100^2 - 4 \times 1 \times (-5600) = 10000 + 22400 = 32400 \]
Now, taking the square root:
\[ \sqrt{32400} = 180 \]

Now substituting back into the quadratic formula:
\[ X = \frac{-100 \pm 180}{2} \]
Calculating the two possible values:
1. \( X = \frac{80}{2} = 40 \)
2. \( X = \frac{-280}{2} = -140 \) (not a valid solution since C.P. cannot be negative)

### Step 7: Conclusion
Thus, the Cost Price (C.P.) of the pen is Rs. 40.