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CAT 2025Slot 3DILRQues & Sol

Data InterpretationHard

Data Set

Anu, Bijay, Chetan, Deepak, Eshan, and Faruq are six friends. Each of them uses a mobile number from exactly one of the two mobile operators - Xitel and Yocel. During the last month, the six friends made several calls to each other. Each call was made by one of these six friends to another. The table below summarizes the number of minutes of calls that each of the six made to (outgoing minutes) and received from (incoming minutes) these friends, grouped by the operators. Some of the entries are missing.

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It is known that the duration of calls from Faruq to Eshan was 200 minutes. Also, there were no calls from:
i. Bijay to Eshan,
ii. Chetan to Anu and Chetan to Deepak,
iii. Deepak to Bijay and Deepak to Faruq,
iv. Eshan to Chetan and Eshan to Deepak.

Question 1

What was the duration of calls (in minutes) from Bijay to Anu?

Solution

Setup: Since Anu and Bijay are the only two Xitel users, any outgoing minutes from one Xitel user to Xitel must be the incoming minutes for the other Xitel user. We use this symmetry to fill in missing values, then use cross-operator balance equations to find all unknowns.

Steps:

  1. Identify operator groups: Anu and Bijay use Xitel; Chetan, Deepak, Eshan, and Faruq use Yocel. Since there are exactly two Xitel users, any Xitel→Xitel call must be between Anu and Bijay only.
  2. Find b (Bijay's outgoing to Xitel = calls from Bijay to Anu): Anu's incoming from Xitel = 50 minutes. The only Xitel user who can call Anu is Bijay. Therefore, Bijay→Anu = 50 minutes, so b = 50.
  3. Verify with Bijay's incoming from Xitel (c): Bijay's incoming from Xitel must equal Anu's outgoing to Xitel = 100 minutes, confirming c = 100. Since Bijay→Anu = b = 50 minutes, the answer is confirmed.

Final Answer: 50

Question 2

What was the total duration of calls (in minutes) made by Anu to friends having mobile numbers from Operator Yocel?

Solution

Setup: Anu's outgoing minutes to Yocel represent all calls Anu made to Yocel users (Chetan, Deepak, Eshan, Faruq). This value (labeled 'a') is missing from the table and must be found using the cross-operator balance: total outgoing from Xitel users to Yocel = total incoming from Xitel for Yocel users.

Steps:

  1. Set up the cross-operator equation: All minutes flowing from Xitel to Yocel must equal all minutes received by Yocel users from Xitel. So: (Anu's outgoing to Yocel) + (Bijay's outgoing to Yocel) = Sum of incoming from Xitel for Chetan, Deepak, Eshan, Faruq.
  2. Plug in known values: Bijay's outgoing to Yocel = 200 (from table). Incoming from Xitel: Chetan = 250, Deepak = 275, Eshan = 100, Faruq = 100. Sum = 725.
  3. Solve for a: a + 200 = 725 → a = 525. Anu's total calls to Yocel users = 525 minutes.

Final Answer: 525

Question 3

What was the total duration of calls (in minutes) made by Faruq to friends having mobile numbers from Operator Yocel?

Solution

Setup: Faruq's outgoing minutes to Yocel (labeled 'e') represent all calls Faruq made to Yocel users (Chetan, Deepak, Eshan). We find 'e' using the within-Yocel balance: total outgoing from Yocel users to Yocel = total incoming from Yocel for Yocel users.

Steps:

  1. Set up the within-Yocel equation: All Yocel→Yocel call minutes are internal. So: Σ(outgoing to Yocel by Yocel users) = Σ(incoming from Yocel for Yocel users). Left side: Chetan (175) + Deepak (150) + Eshan (100) + Faruq (e) = 425 + e.
  2. Calculate right side: Incoming from Yocel: Chetan = 150, Deepak = 100, Eshan = 375, Faruq = 150. Sum = 775.
  3. Solve for e: 425 + e = 775 → e = 350. Faruq's total calls to Yocel users = 350 minutes.

Final Answer: 350

Question 4

What was the duration of calls (in minutes) from Deepak to Chetan?

Solution

Setup: After completing the full table (a=525, b=50, c=100, d=200, e=350), we use the given zero-call constraints to isolate individual call durations. Chetan's incoming from Yocel and Deepak's incoming from Yocel are the key anchor points, combined with the known Faruq→Eshan = 200 minutes.

Steps:

  1. Find Faruq→Deepak using Deepak's incoming from Yocel: Deepak's incoming from Yocel = 100 minutes. Yocel users who can call Deepak are Chetan, Eshan, and Faruq. Given constraints: Chetan→Deepak = 0 and Eshan→Deepak = 0. Therefore, all 100 minutes come from Faruq: Faruq→Deepak = 100.
  2. Find Faruq→Chetan using Faruq's total outgoing to Yocel: Faruq's outgoing to Yocel = 350 minutes, split among Chetan, Deepak, and Eshan. Faruq→Eshan = 200 (given), Faruq→Deepak = 100 (from step 1). So Faruq→Chetan = 350 − 200 − 100 = 50.
  3. Find Deepak→Chetan using Chetan's incoming from Yocel: Chetan's incoming from Yocel = 150 minutes. Yocel callers to Chetan: Deepak, Eshan, Faruq. Given Eshan→Chetan = 0. So: (Deepak→Chetan) + (Faruq→Chetan) = 150 → (Deepak→Chetan) + 50 = 150 → Deepak→Chetan = 100.

Final Answer: 100 (Option A)