When an article is sold at Rs 144, the gain percent is equal to cost price numerically. Find the cost price ?

Options

A. Rs 80
B. Rs 70
C. Rs 60
D. Rs 90

Answer

To solve the problem step by step, we need to find the cost price (CP) of an article given that when sold at Rs. 144, the gain percent is equal to the cost price numerically.

### Step 1: Define the Variables
Let the cost price be \( CP = X \).

### Step 2: Understand the Gain Percent
According to the problem, the gain percent is equal to the cost price numerically. Therefore, the gain percent is \( X\% \).

### Step 3: Use the Selling Price Formula
The formula for selling price (SP) in terms of cost price (CP) and gain percent is:
\[ SP = CP \times \left(1 + \frac{Gain\%}{100}\right) \]
Substituting the values we have:
\[ 144 = X \times \left(1 + \frac{X}{100}\right) \]

### Step 4: Simplify the Equation
This can be rewritten as:
\[ 144 = X \times \left(\frac{100 + X}{100}\right) \]
Multiplying both sides by 100 to eliminate the fraction:
\[ 14400 = X(100 + X) \]
This simplifies to:
\[ 14400 = 100X + X^2 \]

### Step 5: Rearrange the Equation
Rearranging gives us a standard quadratic equation:
\[ X^2 + 100X - 14400 = 0 \]

### Step 6: Factor the Quadratic Equation
We need to factor the quadratic equation. We look for two numbers that multiply to \(-14400\) and add to \(100\). The numbers are \(180\) and \(-80\). Thus, we can write:
\[ X^2 + 180X - 80X - 14400 = 0 \]
Grouping terms:
\[ X(X + 180) - 80(X + 180) = 0 \]
Factoring out \((X + 180)\):
\[ (X + 180)(X - 80) = 0 \]

### Step 7: Solve for X
Setting each factor to zero gives:
1. \( X + 180 = 0 \) → \( X = -180 \) (not valid since cost price cannot be negative)
2. \( X - 80 = 0 \) → \( X = 80 \)

### Conclusion
The cost price (CP) of the article is \( Rs. 80 \).