A $20\%$ reduction in the price of an apple enables a man to buy $10$ apples more for ₹54. Then what is the reduced price of apples per dozen?

Total amount spent = ₹54.
Reduction in price = $20\%$ of ₹54 = $\frac{20}{100}\times 54=\text{₹}10.8$.

This saving of ₹10.8 allows him to buy $10$ extra apples.

$$\text{Reduced price of 10 apples}=\text{₹}10.8$$

$$\text{Reduced price of 1 apple}=\frac{10.8}{10}=\text{₹}1.08$$

We need to find the reduced price per dozen ($12$ apples):

$$\text{Price per dozen}=1.08\times 12=\text{₹}12.96$$