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The question asks about price differences between goods sold by Mr. A, Mr. B, Mr. C, and Mr. D, with specific relationships between their prices. Let's break it down:

- Mr. A sells his goods at a price **20% lower than Mr. B**. This means if Mr. B's price is \( P_B \), then Mr. A's price is:
  \[
  P_A = 0.8 \times P_B
  \]

- Mr. A's goods are **20% higher than Mr. C's** price. This means:
  \[
  P_A = 1.2 \times P_C
  \]

- Additionally, Mr. A's goods are **10% cheaper than Mr. D's**. So:
  \[
  P_A = 0.9 \times P_D
  \]

The goal is to find by what percentage Mr. B's goods are more expensive than Mr. D's.

### Step-by-Step Approach:

1. From the first condition, \( P_A = 0.8 \times P_B \), solving for \( P_B \):
   \[
   P_B = \frac{P_A}{0.8}
   \]

2. From the third condition, \( P_A = 0.9 \times P_D \), solving for \( P_D \):
   \[
   P_D = \frac{P_A}{0.9}
   \]

3. Now, the percentage difference between \( P_B \) and \( P_D \):
   \[
   \text{Percentage Difference} = \frac{P_B - P_D}{P_D} \times 100
   \]

   Substituting \( P_B = \frac{P_A}{0.8} \) and \( P_D = \frac{P_A}{0.9} \):
   \[
   \text{Percentage Difference} = \frac{\frac{P_A}{0.8} - \frac{P_A}{0.9}}{\frac{P_A}{0.9}} \times 100
   \]

4. Simplifying the equation:
   \[
   \text{Percentage Difference} = \frac{\frac{P_A}{0.8} - \frac{P_A}{0.9}}{\frac{P_A}{0.9}} \times 100 = \frac{\frac{9P_A - 8P_A}{7.2P_A}} \times 100 = \frac{P_A}{7.2P_A} \times 100
   \]
   \[
   = \frac{1}{0.125} = 12.5\%
   \]

Thus, Mr. B's goods are **12.5% more expensive** than Mr. D's.

The correct answer is **12.5%**.

Would you like further clarification or more details?

Here are 5 related questions to explore further:
1. How does changing the percentage differences affect the final comparison?
2. Can we solve this using algebraic symbols instead of numbers?
3. What if all percentage differences were reversed?
4. How do you compare prices when more variables are introduced?
5. How can we express these price comparisons graphically?

**Tip**: Always translate percentage relationships into algebraic equations to simplify comparisons and calculations.

Mr. A sells his goods at a price 20% lower than Mr. B but 20% higher than Mr. C. Additionally, Mr. A's goods are 10% cheaper than Mr. D's. By what percentage are Mr. B's goods more expensive than Mr. D's?

Learn how to compare prices between Mr. B and Mr. D based on percentage differences. This problem covers percentage calculations and algebraic concepts, perfect for Grades 10-12 students.

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