Statement Half of the villagers of a certain village have their own houses.…
2023
Statement
Half of the villagers of a certain village have their own houses. One-fifth of the villagers cultivate paddy. One-third of the villagers are literate. Four-fifth of the villagers is below twenty-five. Then, which one of the following is certainly true?
I. All the villagers who have their own houses are literate
II. Some villagers under twenty-five are literate
III. A quarter of the villages who have their own houses cultivate paddy
IV. Half of the villagers who cultivate paddy are literate
- A.
I follows
- B.
II follows
- C.
III follows
- D.
IV follows
Attempted by 41 students.
Show answer & explanation
Correct answer: B
Conclusion: Only the statement that some villagers under twenty-five are literate certainly follows.
Given proportions: homeowners = 1/2, paddy cultivators = 1/5, literate = 1/3, below 25 = 4/5.
Why some under twenty-five are literate: The 25-and-over group is 1 - 4/5 = 1/5. Literates are 1/3, which is greater than 1/5, so at least 1/3 - 1/5 = 2/15 of the population must be both literate and under twenty-five. Hence this statement certainly follows.
Why all homeowners being literate is not certain: Homeowners are 1/2 but literates are only 1/3, so there are fewer literates than homeowners. It is therefore impossible to guarantee that every homeowner is literate.
Why a quarter of homeowners cultivating paddy is not certain: Paddy cultivators are 1/5 and homeowners are 1/2. The overlap (homeowners who cultivate paddy) could be anywhere from 0 to (1/5)/(1/2) = 2/5 of homeowners, so it need not be 1/4.
Why half of paddy cultivators being literate is not certain: The intersection of paddy (1/5) and literate (1/3) can range from 0 to min(1/5,1/3)=1/5. Thus the fraction of paddy cultivators who are literate could vary from 0% to 100%, so it is not necessarily 50%.
Final answer: Only the statement that some villagers under twenty-five are literate certainly follows.