$60$ kg

This problem is best solved by working backwards from the final amount of rice.

1. Analyse Tuesday's sale: The shopkeeper sold $\frac{1}{2}$ of the remaining rice and was left with $20$ kg. This means the $20$ kg is the other half ($\frac{1}{2}$) that was not sold. Therefore, the amount of rice before Tuesday's sale was $20\times 2=40$ kg.
2. Analyse Monday's sale: This $40$ kg was the amount remaining after selling $\frac{1}{3}$ of the initial stock on Monday. If $\frac{1}{3}$ was sold, the remaining fraction was $1-\frac{1}{3}=\frac{2}{3}$.
3. Find the initial amount: So, $\frac{2}{3}$ of the initial amount is equal to $40$ kg.
   • If $\frac{2}{3}$ of the total is $40$ kg, then $\frac{1}{3}$ of the total is $40÷2=20$ kg.
   • The full initial amount ($\frac{3}{3}$) is $20\times 3=60$ kg.

Therefore, the initial amount of rice was $60$ kg.

• $40$ kg is incorrect because this was the amount before Tuesday's sale, not the initial amount.
• $120$ kg is incorrect. This error comes from incorrectly adding the fractions sold ($\frac{1}{3}+\frac{1}{2}=\frac{5}{6}$) without considering the 'remaining' part, implying the leftover $20$ kg is $\frac{1}{6}$ of the total.

Ok