In the given questions, two quantities are given. One as Quantity-I and another is Quantity-II. You have to determine relationship between these two quantities and choose the appropriate options as given below:
In village A, `20%` of the total population had gone to a formal school at some point of their lifetime. Out of the remaining, `75%` have never gone to any school and the remaining 80 were home schooled.
Quantities:
I. Total population of village A
II. Total population of village B, where `60%` are males and the remaining 160 are females.

Options

A. Quantity- I `gt` Quantity-II
B. Quantity- I `lt` Quantity-II
C. Quantity- I `le` Quantity-II
D. Quantity- I = Quantity-II

Answer

To solve the problem, we will analyze the information given for both villages A and B step by step.

### Step 1: Determine the total population of village A

Let the total population of village A be \( x \).

1. **Calculate the number of people who went to a formal school:**
- 20% of the total population went to a formal school.
- This can be calculated as:
\[ \text{People who went to school} = 0.20 \times x = 0.2x \]

2. **Calculate the remaining population:**
- The remaining population after accounting for those who went to school is:
\[ \text{Remaining population} = x - 0.2x = 0.8x \]

3. **Determine how many of the remaining population never went to school:**
- Out of the remaining \( 0.8x \), 75% never went to school:
\[ \text{People who never went to school} = 0.75 \times 0.8x = 0.6x \]

4. **Calculate the number of people who were homeschooled:**
- The remaining population after accounting for those who never went to school is:
\[ \text{Remaining after never went to school} = 0.8x - 0.6x = 0.2x \]
- We know from the problem that this remaining population equals 80 (the number of homeschooled children):
\[ 0.2x = 80 \]

5. **Solve for \( x \):**
- To find \( x \), divide both sides by 0.2:
\[ x = \frac{80}{0.2} = 400 \]

Thus, the total population of village A is **400**.

### Step 2: Determine the total population of village B

1. **Given information about village B:**
- 60% of the total population are males, and the remaining 160 are females.
- Let the total population of village B be \( y \).

2. **Calculate the percentage of females:**
- Since 60% are males, the percentage of females is:
\[ 100\% - 60\% = 40\% \]

3. **Set up the equation for females:**
- We know that 40% of the total population \( y \) equals 160:
\[ 0.40y = 160 \]

4. **Solve for \( y \):**
- To find \( y \), divide both sides by 0.40:
\[ y = \frac{160}{0.40} = 400 \]

Thus, the total population of village B is also **400**.

### Conclusion

Now we have:
- Quantity I (Total population of village A) = 400
- Quantity II (Total population of village B) = 400

Therefore, we can conclude that:
\[ \text{Quantity I} = \text{Quantity II} \]

### Final Answer
The relationship between the two quantities is that they are equal.