31.

A shopkeeper purchases a certain number of books for Rs. 960. If the cost per book was Rs. 8 less, the number of books that could be purchased for Rs. 960 would be 4
more. Write an equation, taking the original cost of each book to be Rs. x, and solve it to find the original cost of the books.

Let consider the original cost of each book = Rs. x

Therefore, Number of books brought for Rs. 960 = $\frac{960}{\mathrm{x}}$

As given,

If the cost per book was Rs. 8 less, the number of books that could be purchased for Rs. 960 would be 4 more.

$\therefore$ The cost of each book = Rs. (x – 8)

Number of books bought for Rs. 960 = $\frac{960}{\mathrm{x} - 8}$

From the given information, we have

$\frac{960}{\mathrm{x} - 8} - \frac{960}{\mathrm{x}} = 4$

$\Rightarrow \frac{960\mathrm{x} - 960\mathrm{x} + 960 \times 8}{\mathrm{x}(\mathrm{x} - 8)} = 4$

$\Rightarrow \mathrm{x}(\mathrm{x} - 8) = \frac{960 \times 8}{4}$

$\Rightarrow {\mathrm{x}}^2 - 8\mathrm{x} = 1920$

$\Rightarrow {\mathrm{x}}^2 - 8\mathrm{x} - 1920 = 0$

$\Rightarrow \mathrm{x}(\mathrm{x} - 48) +\hspace{0.17em}40(\mathrm{x} - 48) = 0$

$\Rightarrow (\mathrm{x} - 48)(\mathrm{x} + 40) = 0$

$\Rightarrow \mathrm{x} = 48 \mathrm{or} \mathrm{x} - 40$

But, as x can not be negative.

$\therefore$ x = 48

Thus, the original cost of each book is Rs. 48