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.

Friend	Operator	Outgoing minutes to		Incoming minutes from
		Operator Xitel	Operator Yocel	Operator Xitel	Operator Yocel
Anu	Xitel	100		50	225
Bijay	Xitel		200		125
Chetan	Yocel	50	175	250	150
Deepak	Yocel	100	150	275	100
Eshan	Yocel		100	100	375
Faruq	Yocel	0		100	150

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.

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

The correct answer is 100.

To find the duration of calls from Deepak to Chetan, we can analyze the call records step-by-step using the given table and constraints. Let us denote the call duration from person P to person Q as $c(P,Q)$.

Based on their mobile operators, the friends are categorized as follows:
• Xitel Users (X): Anu (A), Bijay (B)
• Yocel Users (Y): Chetan (C), Deepak (D), Eshan (E), Faruq (F)

Thus, for any person, the outgoing minutes to Xitel ($\mathrm{OX}$) represents calls to {A, B}, and outgoing minutes to Yocel ($\mathrm{OY}$) represents calls to {C, D, E, F}. Similarly, incoming minutes from Yocel ($\mathrm{IY}$) represents calls received from {C, D, E, F}.

Let's use the provided information and constraints to determine the relevant call durations:

1. Determine Eshan's outgoing calls to Yocel users
From the table, Eshan's outgoing minutes to Yocel users is:

$\mathrm{OY}\left(E\right)=c(E,C)+c(E,D)+c(E,F)=100$

According to constraint (iv), there were no calls from Eshan to Chetan and Eshan to Deepak:
• $c(E,C)=0$
• $c(E,D)=0$
Substituting these values, we get:

$0+0+c(E,F)=100\Rightarrow c(E,F)=100$

2. Analyze Faruq's incoming calls from Yocel users
Faruq's incoming minutes from Yocel users is:

$\mathrm{IY}\left(F\right)=c(C,F)+c(D,F)+c(E,F)=150$

According to constraint (iii), there were no calls from Deepak to Faruq:
• $c(D,F)=0$
Substituting $c(D,F)=0$ and $c(E,F)=100$:

$c(C,F)+0+100=150\Rightarrow c(C,F)=50$

3. Analyze Chetan's outgoing calls to Yocel users
Chetan's outgoing minutes to Yocel users is:

$\mathrm{OY}\left(C\right)=c(C,D)+c(C,E)+c(C,F)=175$

According to constraint (ii), there were no calls from Chetan to Deepak:
• $c(C,D)=0$
Substituting $c(C,D)=0$ and $c(C,F)=50$:

$0+c(C,E)+50=175\Rightarrow c(C,E)=125$

4. Analyze Eshan's incoming calls from Yocel users
Eshan's incoming minutes from Yocel users is:

$\mathrm{IY}\left(E\right)=c(C,E)+c(D,E)+c(F,E)=375$

We are given that the duration of calls from Faruq to Eshan is $c(F,E)=200$ minutes.
Substituting $c(C,E)=125$ and $c(F,E)=200$:

$125+c(D,E)+200=375$

$325+c(D,E)=375\Rightarrow c(D,E)=50$

5. Calculate the duration of calls from Deepak to Chetan
Deepak's outgoing minutes to Yocel users is:

$\mathrm{OY}\left(D\right)=c(D,C)+c(D,E)+c(D,F)=150$

We already know that $c(D,F)=0$ and we calculated $c(D,E)=50$.
Substituting these values:

$c(D,C)+50+0=150$

$c(D,C)=150-50=100$

Thus, the duration of calls from Deepak to Chetan was 100 minutes.