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?

Answer 1

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
$$\begin{gathered} \Rightarrow xy=180...\left(i\right) \\ \Rightarrow x=\frac{180}{y} \end{gathered}$$
Also, given that if 3 more pens are purchased for the same amount, the price per pen gets reduced by Rs 3
$\Rightarrow (y-3)(x+3)=180...\left(ii\right)$
Put the value of x in (ii) we get
$$\begin{gathered} \left(\frac{180}{x}-3\right)(x+3)=180 \\ \Rightarrow \left(\frac{180-3x}{x}\right)(x+3)=180 \\ \Rightarrow (180-3x)(x+3)=180x \\ \Rightarrow 180x+540-3x^2-9x=180x \\ \Rightarrow 3x^2+9x-540=0 \\ \Rightarrow x^2+3x-180=0 \\ \Rightarrow x^2+15x-12x-180=0 \\ \Rightarrow x(x+15)-12(x+15)=0 \\ \Rightarrow (x+15)(x-12)=0 \\ \Rightarrow x+15=0\mathrm{or}x-12=0 \\ \Rightarrow x=-15\mathrm{or}x=12 \end{gathered}$$
We ignore the negative value.
$\Rightarrow x=12$
Therefore, number of pens is 12.