A shopkeeper purchases certain number of books for ₹ $960$. If the cost per book was ₹ $8$ less, the number of books that could be purchased for ₹ $960$ would be $4$ more. Write an equation, taking the original cost of each book to be ₹ $x$. Solve the equation to find the original cost of the books.
Let the original cost of each book $= $ ₹ $x$
Amount for which books purchased $= $ ₹ $960$
$\therefore \;$ Original number of books purchased $= \dfrac{960}{x}$
New cost of each book $= $ ₹ $\left(x - 8\right)$
New number of books that can be bought for ₹ $960$ $= \dfrac{960}{x - 8}$
As per question,
new number of books bought $= $ original number of books bought $+ \; 4$
i.e. $\;$ $\dfrac{960}{x - 8} = \dfrac{960}{x} + 4$
i.e. $\;$ $240 \left[\dfrac{1}{x - 8} - \dfrac{1}{x}\right] = 1$
i.e. $\;$ $240 \left[\dfrac{x - x + 8}{x^2 - 8x}\right] = 1$
i.e. $\;$ $x^2 - 8x - 1920 = 0$
i.e. $\;$ $x^2 - 48x + 40x - 1920 = 0$
i.e. $\;$ $x \left(x - 48\right) + 40 \left(x - 48\right) = 0$
i.e. $\;$ $\left(x - 48\right) \left(x + 40\right) = 0$
i.e. $\;$ $x - 48 = 0$ $\;$ or $\;$ $x + 40 = 0$
i.e. $\;$ $x = 48$ $\;$ or $\;$ $x = -40$
$\because \;$ cost of books cannot be negative,
$\therefore \;$ original cost of each book $= $ ₹ $x =$ ₹ $48$