# A man buys a number of pens for ₹ 180. If he had bought 3 more pens for the same amount, each pen would have cost him ₹ 3 less.

Question:

A man buys a number of pens for ₹ 180. If he had bought 3 more pens for the same amount, each pen would have cost him ₹ 3 less. How many pens did he buy?

Solution:

Let number of pens bought = x
Let the price per pen = Rs y
It is given that the price of x pens is Rs 180

$\Rightarrow x y=180 \quad \ldots(i)$

$\Rightarrow x=\frac{180}{y}$

Also, given that if 3 more pens are purchased for the same amount, the price per pen gets reduced by Rs 3

$\Rightarrow(y-3)(x+3)=180 \quad \ldots(i i)$

Put the value of x in (ii) we get

$\left(\frac{180}{x}-3\right)(x+3)=180$

$\Rightarrow\left(\frac{180-3 x}{x}\right)(x+3)=180$

$\Rightarrow(180-3 x)(x+3)=180 x$

$\Rightarrow 180 x+540-3 x^{2}-9 x=180 x$

$\Rightarrow 3 x^{2}+9 x-540=0$

$\Rightarrow x^{2}+3 x-180=0$

$\Rightarrow x^{2}+15 x-12 x-180=0$

$\Rightarrow x(x+15)-12(x+15)=0$

$\Rightarrow(x+15)(x-12)=0$

$\Rightarrow x+15=0$ or $x-12=0$

$\Rightarrow x=-15$ or $x=12$

We ignore the negative value.

$\Rightarrow x=12$

Therefore, number of pens is 12.