By a reduction of Rs. $1$ per kg in the price of sugar, Mohan can buy one kg more for Rs. $56$. Find the original price of sugar per kg.

Step 1: Determining the quadratic equation in $x$

Let the price of $1$ kg sugar that Mohan buys be Rs. $x$.

Given that, the total amount that Mohan has is Rs. $56$.

So, the quantity of sugar Mohan bought will be Rs. $\frac{56}{x}$.

It is given that if the price of sugar reduces by Rs. $1$ per kg, Mohan can buy $1$ kg more sugar for the same amount.

$$\begin{gathered} \therefore \frac{56}{x-1}-\frac{56}{x}=1 \\ \Rightarrow \frac{56x-56(x-1)}{x(x-1)}=1 \\ \Rightarrow 56x-56x+56=x(x-1) \\ \Rightarrow x^2-x-56=0 \end{gathered}$$

Step 2: Determining the original price of sugar per kg

By factorization method,

$x^2-x-56=0$

For splitting the middle term, i.e. $-1$ we need to find two numbers such that their sum is $-1$ and product is $-56$.

Such numbers can be $-8$ and $7$.

$$\begin{gathered} \Rightarrow x^2-8x+7x-56=0 \\ \Rightarrow x(x-8)+7(x-8)=0 \\ \Rightarrow (x-8)(x+7)=0 \\ \therefore x=-7,8 \end{gathered}$$

Since, the price of sugar cannot be negative.

$\therefore x=8$

Therefore, the original price of sugar is Rs. $8$ per kg.