A man bought a certain number of chairs for Rs 10,000. He kept one for his own use and sold the rest at the rate Rs 50 more than he gave for one chair. Besides getting his own chair for nothing, he made a profit of Rs 450. How many chairs did he buy ?

Answer

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

### Step 1: Define the variables
Let the number of chairs bought by the man be \( x \).

### Step 2: Calculate the cost price of each chair
The total cost of the chairs is Rs 10,000. Therefore, the cost price (CP) of each chair is:
\[ \text{CP of each chair} = \frac{10000}{x} \]

### Step 3: Determine the number of chairs sold
Since the man kept one chair for his own use, the number of chairs he sold is:
\[ \text{Number of chairs sold} = x - 1 \]

### Step 4: Calculate the selling price of each chair
He sold the remaining chairs at a price Rs 50 more than the cost price of one chair. Therefore, the selling price (SP) of each chair is:
\[ \text{SP of each chair} = \frac{10000}{x} + 50 \]

### Step 5: Calculate the total selling price
The total selling price for the chairs sold is:
\[ \text{Total SP} = \text{Number of chairs sold} \times \text{SP of each chair} = (x - 1) \left( \frac{10000}{x} + 50 \right) \]

### Step 6: Set up the profit equation
The profit made by the man is given as Rs 450. The profit can also be calculated as:
\[ \text{Profit} = \text{Total SP} - \text{Total CP} \]
Substituting the known values:
\[ 450 = (x - 1) \left( \frac{10000}{x} + 50 \right) - 10000 \]

### Step 7: Simplify the equation
Expanding the equation:
\[ 450 = (x - 1) \left( \frac{10000}{x} + 50 \right) - 10000 \]
\[ 450 = (x - 1) \left( \frac{10000 + 50x}{x} \right) - 10000 \]
\[ 450 = \frac{(x - 1)(10000 + 50x)}{x} - 10000 \]

Multiply through by \( x \) to eliminate the fraction:
\[ 450x = (x - 1)(10000 + 50x) - 10000x \]

### Step 8: Expand and rearrange
Expanding the left side:
\[ 450x = 10000x - 10000 + 50x^2 - 50x - 10000x \]
\[ 450x = 50x^2 - 10000 \]

Rearranging gives:
\[ 50x^2 - 450x - 10000 = 0 \]

### Step 9: Simplify the quadratic equation
Dividing the entire equation by 50:
\[ x^2 - 9x - 200 = 0 \]

### Step 10: Factor the quadratic equation
Factoring the quadratic:
\[ (x - 20)(x + 10) = 0 \]

### Step 11: Solve for \( x \)
Setting each factor to zero gives:
\[ x - 20 = 0 \quad \text{or} \quad x + 10 = 0 \]
Thus,
\[ x = 20 \quad \text{or} \quad x = -10 \]

Since the number of chairs cannot be negative, we have:
\[ x = 20 \]

### Conclusion
The man bought **20 chairs**.