What is the maximum possible duration, in minutes, of calls from Priya to Rahul?
Answer and solution
Correct answer: D
Solution
Answer: D — 150
1. Priya makes 150 minutes of calls to Vertex friends in total. Therefore Priya→Rahul cannot exceed 150. Rahul also receives exactly 150 minutes from Vertex friends, so the same upper bound follows independently.
2. The upper bound must be shown attainable; merely quoting a row total is not enough. Set the Vertex-only calls to Rahul→Vikram=25, Rahul→Ananya=150, Priya→Rahul=150, Vikram→Priya=100 and Ananya→Vikram=350; all other Vertex-only directed calls are zero.
3. Their outgoing totals are Rahul 175, Priya 150, Vikram 100, Ananya 350. Incoming totals are Rahul 150, Priya 100, Vikram 375, Ananya 150, matching the table.
4. Complete the cross-network calls with Kabir→Rahul 250, Kabir→Priya 275, Zoya→Vikram 100, Zoya→Ananya 100; Rahul→Kabir 50, Priya→Kabir 100, Vikram→Kabir 75 and Vikram→Zoya 125. Also Kabir→Zoya 100 and Zoya→Kabir 50. All other calls are zero.
5. Every supplied table entry is met, nobody calls themselves, and Priya→Rahul=150. Thus the maximum is 150, option D. Individual outgoing and incoming totals need not be equal.
Verification: 5. Every supplied table entry is met, nobody calls themselves, and Priya→Rahul=150. Thus the maximum is 150, option D. Individual outgoing and incoming totals need not be equal.
1. Priya makes 150 minutes of calls to Vertex friends in total. Therefore Priya→Rahul cannot exceed 150. Rahul also receives exactly 150 minutes from Vertex friends, so the same upper bound follows independently.
2. The upper bound must be shown attainable; merely quoting a row total is not enough. Set the Vertex-only calls to Rahul→Vikram=25, Rahul→Ananya=150, Priya→Rahul=150, Vikram→Priya=100 and Ananya→Vikram=350; all other Vertex-only directed calls are zero.
3. Their outgoing totals are Rahul 175, Priya 150, Vikram 100, Ananya 350. Incoming totals are Rahul 150, Priya 100, Vikram 375, Ananya 150, matching the table.
4. Complete the cross-network calls with Kabir→Rahul 250, Kabir→Priya 275, Zoya→Vikram 100, Zoya→Ananya 100; Rahul→Kabir 50, Priya→Kabir 100, Vikram→Kabir 75 and Vikram→Zoya 125. Also Kabir→Zoya 100 and Zoya→Kabir 50. All other calls are zero.
5. Every supplied table entry is met, nobody calls themselves, and Priya→Rahul=150. Thus the maximum is 150, option D. Individual outgoing and incoming totals need not be equal.
G Strategy shortcut
Method: Bound + construction: min(Priya’s outgoing-to-Vertex 150, Rahul’s incoming-from-Vertex 150)=150. Exhibit a valid allocation attaining it.Verification: 5. Every supplied table entry is met, nobody calls themselves, and Priya→Rahul=150. Thus the maximum is 150, option D. Individual outgoing and incoming totals need not be equal.
G Strategy: Method: Bound + construction: min(Priya’s outgoing-to-Vertex 150, Rahul’s incoming-from-Vertex 150)=150. Exhibit a valid allocation attaining it. Verification: 5. Every supplied table entry is met, nobody calls themselves, and Priya→Rahul=150. Thus the maximum is 150, option D. Individual outgoing and incoming totals need not be equal.