Let's revisit the problem, considering the correct answer \( P_{2021} = 73000 \).

### Problem Restatement:
- Population in 2020: \( 100,000 \)
- Population decreased by \( y\% \) from 2020 to 2021.
- Population increased by \( x\% \) from 2021 to 2022.
- Population in 2022 was greater than in 2020.
- \( x - y = 10 \).

### Step 1: Expressing the population in 2021 and 2022
- Population in 2021:
  \[
  P_{2021} = 100000 \times \left(1 - \frac{y}{100}\right)
  \]
- Population in 2022:
  \[
  P_{2022} = P_{2021} \times \left(1 + \frac{x}{100}\right) = 100000 \times \left(1 - \frac{y}{100}\right) \times \left(1 + \frac{x}{100}\right)
  \]

### Step 2: Apply the condition that \( P_{2022} > 100000 \)
\[
100000 \times \left(1 - \frac{y}{100}\right) \times \left(1 + \frac{x}{100}\right) > 100000
\]
Simplify by dividing both sides by 100,000:
\[
\left(1 - \frac{y}{100}\right) \times \left(1 + \frac{x}{100}\right) > 1
\]

### Step 3: Simplify and expand the inequality
Expand the left side:
\[
1 - \frac{y}{100} + \frac{x}{100} - \frac{xy}{10000} > 1
\]
Subtract 1 from both sides:
\[
-\frac{y}{100} + \frac{x}{100} - \frac{xy}{10000} > 0
\]
Multiply the inequality by 100:
\[
-x + y > \frac{xy}{100}
\]

### Step 4: Utilize the condition \( x - y = 10 \)
Given \( x = y + 10 \), substitute this into the inequality:
\[
(y + 10) - y > \frac{y(y + 10)}{100}
\]
Simplify:
\[
10 > \frac{y(y + 10)}{100}
\]
Multiply by 100:
\[
1000 > y(y + 10)
\]

### Step 5: Find the smallest \( P_{2021} \)
Solve for \( y \) by testing integer values:

- If \( y = 20 \), then \( x = 30 \):
  \[
  20 \times 30 = 600 \quad \text{(which is less than 1000)}
  \]
  Calculate \( P_{2021} \):
  \[
  P_{2021} = 100000 \times \left(1 - \frac{20}{100}\right) = 100000 \times 0.8 = 80000
  \]

- If \( y = 27 \), then \( x = 37 \):
  \[
  27 \times 37 = 999 \quad \text{(which is less than 1000)}
  \]
  Calculate \( P_{2021} \):
  \[
  P_{2021} = 100000 \times \left(1 - \frac{27}{100}\right) = 100000 \times 0.73 = 73000
  \]

### Conclusion:
The correct smallest possible population in 2021 is indeed **73000** when \( y = 27 \) and \( x = 37 \).

Would you like more details on any step, or do you have other questions? Here are some related questions:

1. How would the population change if \( x - y \) was a different value?
2. What happens to \( P_{2022} \) if \( y \) increases or decreases by 1%?
3. Can you generalize the method to find the minimum \( P_{2021} \) for any \( x \) and \( y \)?
4. How does the quadratic relationship \( y(y + 10) \) affect the solution?
5. What are the implications if the town had a different starting population?

**Tip**: Always verify the steps by substituting back into the original problem to ensure the accuracy of the results.

Explore a population problem involving percentage changes and inequalities. Understand how to calculate population changes using percentage increases and decreases. Suitable for high school students studying algebra and percentage calculations.

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