Question

A shopkeeper buys a number of books for Rs. 80. If he had bought 4 more books for the same amount, each book would have cost Rs. 1 less. How many books did he buy?
[3 Marks]

Answer 1

Let the number of books bought be $x$.

Then,
Cost of $x$ books $=Rs.80$
$\Rightarrow$ Cost of one book $=Rs\frac{80}{x}$
[0.5 Marks]

If the number of books bought is $x+4$, then cost of one book $=Rs\frac{80}{x+4}$
[0.5 Marks]

It is given that the cost of one book is reduced by one rupee.

$\therefore \frac{80}{x}-\frac{80}{x+4}=1$ ​​​​​​​[0.5 Marks]

$\Rightarrow 80(\frac{1}{x}-\frac{1}{x+4})=1$

$\Rightarrow 80\left\{\frac{x+4-x}{x(x+4)}\right\}=1$

$\Rightarrow \frac{320}{x^2+4x}=1$

$\Rightarrow x^2+4x=320$

$\Rightarrow x^2+4x-320=0$ ​​​​​​​[0.5 Marks]

$\Rightarrow x^2+20x-16x-320=0$

$\Rightarrow x(x+20)-16(x+20)=0$

$\Rightarrow (x+20)(x-16)=0$ ​​​​​​​[0.5 Marks]

$\Rightarrow x=-20orx=16\Rightarrow x=16$ [$\because x$ cannot be negative] ​​​​​​​

Hence, the required number of books is 16. ​​​​​​​[0.5 Marks]