A man bought a certain number of chairs for ₹ 10000. He kept one for his own use and sold the rest at the rate ₹ 50 more than he gave for one chair. Besides getting his own chair for nothing, he made a profit of ₹ 450. How many chairs did he buy ?

Given,

Man bought a certain number of chairs for ₹ 10000.

Let no. of chairs bought be x.

∴ Cost of one chair = ₹ $\dfrac{10000}{x}$

S.P. of each chair = ₹ $\dfrac{10000}{x} + 50$

Since, he kept one chair for himself so he sold (x - 1) chairs.

Given,

⇒ Profit = ₹ 450

⇒ S.P. - C.P. = ₹ 450

$\Rightarrow (x - 1) \times \Big(\dfrac{10000}{x} + 50\Big) - 10000 = 450 \\[1em] \Rightarrow 10000 + 50x - \dfrac{10000}{x} - 50 - 10000 = 450 \\[1em] \Rightarrow 50x - \dfrac{10000}{x} = 450 + 50 \\[1em] \Rightarrow \dfrac{50x^2 - 10000}{x} = 500 \\[1em] \Rightarrow 50x^2 - 10000 = 500x \\[1em] \Rightarrow 50(x^2 - 200) = 500x \\[1em] \Rightarrow x^2 - 200 = 10x \\[1em] \Rightarrow x^2 - 10x - 200 = 0 \\[1em] \Rightarrow x^2 - 20x + 10x - 200 = 0 \\[1em] \Rightarrow x(x - 20) + 10(x - 20) = 0 \\[1em] \Rightarrow (x + 10)(x - 20) = 0 \\[1em] \Rightarrow x + 10 = 0 \text{ or } x - 20 = 0 \\[1em] \Rightarrow x = -10 \text{ or } x = 20.$

Since no. of chairs cannot be negative.

∴ x = 20.

Hence, no. of chairs bought = 20.