In a city, 40% of the people are illiterate and 60% are poor. Among the rich people, 10% are illiterate. What percentage of the total population are both poor and illiterate?

Introduction / Context:
This question is a percentage and set intersection problem involving two categories, poverty and literacy. We are given the percentages of illiterates and poor people and partial information about illiteracy among the rich. We must determine what fraction of the total population is both poor and illiterate. Such questions are very common in data interpretation and reasoning sections.

Given Data / Assumptions:
- 40 percent of the total population is illiterate.
- 60 percent of the total population is poor, so 40 percent is rich.
- Among the rich, 10 percent are illiterate.
- We are asked to find the percentage of the total population that is both poor and illiterate.

Concept / Approach:
We can treat the total population as 100 units for simplicity. Then the percentages can be interpreted directly as counts. First, we calculate how many rich people are illiterate using the given information. Then we subtract this number from the total illiterate population to obtain the number of poor illiterates. Finally, we express that number as a percentage of the total population.

Step-by-Step Solution:
Assume total population = 100 units. Illiterates = 40% of 100 = 40 people. Poor people = 60% of 100 = 60 people. Therefore, rich people = 100 - 60 = 40 people. Among the rich, 10% are illiterate. Illiterate rich = 10% of 40 = 4 people. Total illiterates = 40 people, of which 4 are rich and illiterate. Hence, poor illiterates = total illiterates - illiterate rich = 40 - 4 = 36 people. As a percentage of the total population, this is 36 out of 100, that is 36%.

Verification / Alternative check:
We can also think in terms of a table: rows for poor and rich, columns for literate and illiterate. Rich illiterate are 4, rich literate are 36. Total illiterates are 40, so poor illiterates must be 36. Poor literates then are 60 - 36 = 24. All numbers are consistent and add up: rich (4 + 36) = 40, poor (36 + 24) = 60, illiterates (36 + 4) = 40, literates (24 + 36) = 60. This confirms that 36 out of 100 are poor and illiterate.

Why Other Options Are Wrong:
60: This would mean all poor people are illiterate, which contradicts the information about rich illiterates and the total illiterates.
50 and 40: These do not satisfy the breakdown of illiterates between rich and poor when checked with the given data.
64: This would exceed the total number of poor people and is impossible.

Common Pitfalls:
A frequent mistake is to add or subtract percentages without a clear reference to the total population, or to think that 40% illiterate and 60% poor means something like 40% of 60% without understanding the overlap. The safest way is to assume a convenient total population, such as 100, and then work out the actual counts in each intersecting group. Drawing a small table often helps to avoid confusion.

Final Answer:
The percentage of the total population that is both poor and illiterate is 36 percent.