To determine the final price of the shoes after the discount and subsequent increase, we can break down the calculations step-by-step.

1. Initial Reduction:

   • Original price of the shoes: $150
   • The discount was 1/5 of $150.

   $\text{Discount amount} = \frac{1}{5} \times 150 = 30$

   $\text{Reduced price} = 150 - 30 = 120$

2. Subsequent Increase:

   • The price was then increased by one quarter (1/4) of the reduced price of $120.

   $\text{Increase amount} = \frac{1}{4} \times 120 = 30$

   $\text{Final price} = 120 + 30 = 150$

So, the final price of the shoes is $150.

Would you like more details on any part of the process?

Here are some related questions for further practice:

1. What would the price be if the initial reduction were 1/3 instead of 1/5?
2. If a second increase were 20% instead of 1/4, what would the final price be?
3. How much would the shoes cost if the initial discount was $40 instead?
4. What would happen if both the reduction and the increase were the same percentage?
5. How would the final price change if the increase applied to the original price?

Tip: To solve percentage problems, always perform calculations in steps, ensuring each change is applied to the correct price value.