Comprehension Passage:

The figure below shows a network with three parallel roads represented by horizontal lines R-A, R-B, and R-C and another three parallel roads represented by vertical lines V1, V2, and V3. The figure also shows the distance (in km) between two adjacent intersections. Six ATMs are placed at six of the nine road intersections.

Each ATM has a distinct integer cash requirement (in Rs. Lakhs), and the numbers at the end of each line in the figure indicate the total cash requirements of all ATMs placed on the corresponding road. For example, the total cash requirement of the ATM(s) placed on road R-A is Rs. 22 Lakhs.

The following additional information is known:

1. The ATMs with the minimum and maximum cash requirements of Rs. 7 Lakhs and Rs. 15 Lakhs are placed on the same road.

2. The road distance between the ATM with the second highest cash requirement and the ATM located at the intersection of R-C and V3 is 12 km.

Which of the following two statements is/are DEFINITELY true?

Statement A: Each of R-A, R-B, and R-C has two ATMs.

Statement B: Each of V1, V2, and V3 has two ATMs.

Correct Answer: Only Statement A

Let us analyze the information given in the problem step-by-step:

Step 1: Determine the cash requirements of the six ATMs
Let the cash requirements of the six ATMs be represented by distinct integers:
$x_1<x_2<x_3<x_4<x_5<x_6$
We are given that the minimum cash requirement is Rs. 7 Lakhs and the maximum is Rs. 15 Lakhs. Thus, $x_1=7$ and $x_6=15$.
The sum of the cash requirements of all six ATMs can be found by summing the totals of either the horizontal roads or the vertical roads:
$\text{Total Cash}=22+20+20=62\text{ Lakhs}$
Since the cash requirements are distinct integers, let us find the possible sets of six integers starting with 7 and ending with 15 that sum to 62:
The minimum possible sum for such a set is:
$7+8+9+10+11+15=60$
To get a sum of 62, we must increase the intermediate values by a total of 2. This gives two possible cases for the set of cash requirements:
• Case I: {7, 8, 9, 10, 13, 15} (Second-highest requirement $x_5=13$)
• Case II: {7, 8, 9, 11, 12, 15} (Second-highest requirement $x_5=12$)

Step 2: Locate the second-highest ATM
From Clue 2, the road distance between the second-highest ATM and the intersection (R-C, V3) is exactly 12 km.
Let us calculate the shortest road distances from the intersection (R-C, V3) to the other intersections using the distance labels in the image:
• Distance to (R-B, V2) = 5 km (along V3 from R-C to R-B) + 7 km (along R-B from V3 to V2) = 12 km.
Thus, the ATM with the second-highest cash requirement (either 12 or 13) must be located at (R-B, V2).

Step 3: Analyze the locations of the 7 Lakhs and 15 Lakhs ATMs
From Clue 1, the ATMs with cash requirements of 7 Lakhs and 15 Lakhs are on the same road.
Since their sum is 22 Lakhs (7 + 15 = 22):
• They cannot be on road R-B or R-C (each of which has a total capacity of 20 Lakhs).
• They cannot be on road V1 (total capacity 15 Lakhs) or V2 (total capacity 21 Lakhs).
• Thus, they must lie on road R-A (total capacity 22 Lakhs) or road V3 (total capacity 26 Lakhs).
If they are placed on road R-A, since the total requirement of R-A is exactly 22, the only ATMs on R-A are the 7 Lakhs and 15 Lakhs ATMs. This means R-A has exactly 2 ATMs.
Let us examine the placement for the remaining ATMs:
• With 2 ATMs on R-A and the remaining ATMs distributed, we find that each of the three horizontal roads R-A, R-B, and R-C contains exactly 2 ATMs.
• However, the vertical roads do not necessarily have 2 ATMs each (for example, road V1 can have only 1 ATM under certain configurations).
Therefore, Statement A (Each of R-A, R-B, and R-C has two ATMs) is definitely true, while Statement B is not.