A trader bought a number of articles for Rs900, five were damaged and he sold each of the rest at Rs2 more than what he paid for it. If on the whole he gains Rs80, find the number of articles bought.

Answer

To solve the problem, let's break it down step by step.

### Step 1: Define the Variables
Let the number of articles bought be \( x \). The total cost of the articles is given as Rs 900.

### Step 2: Calculate the Cost of One Article
The cost of one article can be expressed as:
\[ \text{Cost of one article} = \frac{900}{x} \]

### Step 3: Determine the Number of Articles Sold
Since 5 articles were damaged, the number of articles sold is:
\[ \text{Number of articles sold} = x - 5 \]

### Step 4: Selling Price of Each Article
Each of the remaining articles is sold at Rs 2 more than the cost price. Therefore, the selling price of each article is:
\[ \text{Selling price of each article} = \frac{900}{x} + 2 \]

### Step 5: Calculate Total Selling Price
The total selling price for the articles sold is:
\[ \text{Total Selling Price} = (x - 5) \left( \frac{900}{x} + 2 \right) \]

### Step 6: Set Up the Equation for Total Gain
The trader gains Rs 80 on the whole transaction. Therefore, the total selling price can also be expressed as:
\[ \text{Total Selling Price} = \text{Total Cost} + \text{Gain} = 900 + 80 = 980 \]

### Step 7: Equate the Total Selling Price to 980
Now, we can set up the equation:
\[ (x - 5) \left( \frac{900}{x} + 2 \right) = 980 \]

### Step 8: Expand the Equation
Expanding the left side:
\[ (x - 5) \left( \frac{900}{x} + 2 \right) = (x - 5) \cdot \frac{900}{x} + (x - 5) \cdot 2 \]
\[ = \frac{900(x - 5)}{x} + 2(x - 5) \]
\[ = \frac{900x - 4500}{x} + 2x - 10 \]

### Step 9: Combine Terms
Combining the terms gives:
\[ \frac{900x - 4500 + 2x^2 - 10x}{x} = 980 \]
Multiplying through by \( x \) to eliminate the fraction:
\[ 900x - 4500 + 2x^2 - 10x = 980x \]
\[ 2x^2 - 90x - 4500 = 0 \]

### Step 10: Simplify the Quadratic Equation
Dividing the entire equation by 2:
\[ x^2 - 45x - 2250 = 0 \]

### Step 11: Factor the Quadratic Equation
To factor the quadratic, we need two numbers that multiply to -2250 and add to -45. The factors are -75 and +30:
\[ (x - 75)(x + 30) = 0 \]

### Step 12: Solve for \( x \)
Setting each factor to zero gives:
1. \( x - 75 = 0 \) → \( x = 75 \)
2. \( x + 30 = 0 \) → \( x = -30 \) (not valid since the number of articles cannot be negative)

Thus, the number of articles bought is:
\[ \boxed{75} \]