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, FloodRelief (FR) in Maharashtra, and Earthquake Relief (ER) in Gujarat. Each volunteer working with Help Distress has to be involved in at least one relief work project.

[CAT 2005]

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:

(1) 1(2) 3
(3) 4(4) 5

Solution:



17 volunteers are involved in the TR project and 10 in TR are also involved in other projects. Thus, 7 volunteers are involved only in TR.
8 volunteers are involved in ER alone.
4 volunteers are involved in all the three projects.

Let x people be involved in FR alone.
Number of people involved in FR and ER but not TR = x – 4

Now, a + b + 4 = 10
a + b = 6

Also, 7 + a + b + 4 + x + x – 4 + 8 = 37
2x = 16 or x = 8
Number of Volunteers involved in FR > Number of Volunteers involved in TR

And Number of Volunteers involved in FR > Number of Volunteers involved in ER

16 + a > 17 and 16 + a > 16 + b or a > b
a and b can be (6, 0), (5, 1), (4, 2)
The minimum number of volunteers involved in both FR and TR projects, but not in the ER Project = minimum value of a = 4
Hence, option 3.
Question 2:
Which of the following additional information would enable to find the exact number of volunteers involved in various projects?

(1)Twenty volunteers are involved in FR.
(2)Four volunteers are involved in all the three projects.
(3)Twenty three volunteers are involved in exactly one project.
(4)No need for any additional information.

Solution:
We can obtain the information in options 2 and 3 from the initial data.
Based on the information given in the explanation to the first question, the information in option 1 will give us the value of a, which in turn will give us the value of b. Thus, option 1 would enable us to find the exact number of volunteers involved in various projects.
Hence, option 1.

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?

(1)The lowest number of volunteers is now in TR project.
(2)More volunteers are now in FR project as compared to ER project.
(3)More volunteers are now in TR project as compared to ER project.
(4)None of the above

Solution:



After the volunteers withdraw as mentioned, the number of volunteers working on different projects is as shown.
Number of volunteers working on TR = 7 + 6 + 3 = 16
Number of volunteers working on FR = 14 + a
Number of volunteers working on ER = 15 + b
Considering the possible values of a and b,
14 + a > 15 + b
More volunteers are now in FR than in ER
Hence, option 2.

Question 4:
After the withdrawal of volunteers, as indicated in the previous question, some new volunteers joined the NGO. Each one of them was allotted only one project in a manner such 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?
(1) ER(2) FR(3) TR
(4) Cannot be determined

Solution:
Let m volunteers be added to TR project and n be added to each of FR and ER projects.

Then, 7 + m = 8 + n
m = n + 1

Also, b + 2 = 5
b = 3 and a = 3

Number of volunteers working on TR = 7 + n + 1 + 4 + 5 = 17 + n
Number of volunteers working on FR = 17 + n
Number of volunteers working on ER = 18 + n

Thus, ER has the highest number of volunteers.
Hence, option 1.