Comprehension
Help Distress (HD) - Volunteer Project Information
Help Distress (HD) is an NGO involved in providing assistance to people suffering from natural disasters. Currently, it has 37 volunteers. They are involved in three projects:
- Tsunami Relief (TR) in Tamil Nadu
- Flood Relief (FR) in Maharashtra
- Earthquake Relief (ER) in Gujarat
Each volunteer working with Help Distress has to be involved in at least one relief work project.
Given Information:
- A maximum number of volunteers are involved in the FR project. Among them, the number of volunteers involved in FR project alone is equal to the volunteers having additional involvement in the ER project.
- The number of volunteers involved in the ER project alone is double the number of volunteers involved in all the three projects.
- 17 volunteers are involved in the TR project.
- The number of volunteers involved in the TR project alone is one less than the number of volunteers involved in ER project alone.
- 10 volunteers involved in the TR project are also involved in at least one more project.
Question: 1
Based on the information given above, the minimum number of volunteers involved in both FR and TR projects, but not in the ER project is
Based on the information given above, the minimum number of volunteers involved in both FR and TR projects, but not in the ER project is
Show Hint
Use Venn diagram with algebraic equations to handle overlapping set problems involving “only” and “at least” constraints.
Updated On: Jul 31, 2025
- 1
- 3
- 4
- 5
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The Correct Option is B
Solution and Explanation
Let $FR$ = set of volunteers in Flood Relief, $TR$ = set of volunteers in Tsunami Relief, and $ER$ = set of volunteers in Earthquake Relief. Total volunteers = 37.
Given:
- Maximum volunteers are in $FR$.
- In $FR$ project alone = In $FR \cap ER$ (but not in $TR$).
- $ER$ alone = twice the volunteers in all three projects.
- $TR$ total = 17.
- $TR$ alone = one less than $ER$ alone.
- 10 in $TR$ are also in at least one more project.
From these, set equations lead to the minimum $FR \cap TR$ but not $ER$ = \(\mathbf{3}\).
- In $FR$ project alone = In $FR \cap ER$ (but not in $TR$).
- $ER$ alone = twice the volunteers in all three projects.
- $TR$ total = 17.
- $TR$ alone = one less than $ER$ alone.
- 10 in $TR$ are also in at least one more project.
From these, set equations lead to the minimum $FR \cap TR$ but not $ER$ = \(\mathbf{3}\).
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Question: 2
Which of the following additional information would enable to find the exact number of volunteers involved in various projects?
Which of the following additional information would enable to find the exact number of volunteers involved in various projects?
Show Hint
For set problems, knowing the count of exactly-one-category members often completes the system of equations for unique solution.
Updated On: Jul 31, 2025
- Twenty volunteers are involved in FR.
- Four volunteers are involved in all the three projects.
- Twenty one volunteers are involved in exactly one project.
- No need for any additional information.
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The Correct Option is C
Solution and Explanation
Knowing the total number of people in exactly one project allows solving for all “only” regions in the Venn diagram. This, combined with totals in each project, determines all overlaps uniquely. Options (1) and (2) do not suffice as they leave multiple feasible configurations.
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Question: 3
After some time, the volunteers who were involved in all the three projects were asked to withdraw from one project. As a result, one of the volunteers opted out of the TR project, and one opted out of the ER project, while the remaining ones involved in all the three projects opted out of the FR project. Which of the following statements, then, necessarily follows?
After some time, the volunteers who were involved in all the three projects were asked to withdraw from one project. As a result, one of the volunteers opted out of the TR project, and one opted out of the ER project, while the remaining ones involved in all the three projects opted out of the FR project. Which of the following statements, then, necessarily follows?
Show Hint
Track the change in each set separately when members are removed or shifted to see relative changes in sizes.
Updated On: Jul 31, 2025
- The lowest number of volunteers is now in TR project.
- More volunteers are now in FR project as compared to ER project.
- More volunteers are now in TR project as compared to ER project.
- None of the above.
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The Correct Option is C
Solution and Explanation
After redistribution, TR loses only one volunteer from the all-three set, while ER loses one from all-three set and one from ER alone, making ER’s reduction larger. Hence TR ends up with more members than ER.
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Question: 4
After the withdrawal of volunteers, some new volunteers joined the NGO. Each one of them was allotted only one project in such a way that, the number of volunteers working in one project alone for each of the three projects became identical. At that point, it was also found that the number of volunteers involved in FR and ER projects was the same as the number of volunteers involved in TR and ER projects. Which of the projects now has the highest number of volunteers?
After the withdrawal of volunteers, some new volunteers joined the NGO. Each one of them was allotted only one project in such a way that, the number of volunteers working in one project alone for each of the three projects became identical. At that point, it was also found that the number of volunteers involved in FR and ER projects was the same as the number of volunteers involved in TR and ER projects. Which of the projects now has the highest number of volunteers?
Show Hint
In equalization problems, the project with initially highest overlaps often remains the largest when “only” values are balanced.
Updated On: Jul 31, 2025
- ER
- FR
- TR
- Cannot be determined
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The Correct Option is B
Solution and Explanation
Equalizing “only” counts across all projects while keeping overlaps fixed results in FR retaining the largest total because FR already had the largest overlaps and gains equal share in the only-section increase.
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