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

The correct option is 350.

Step 1: Understand the Goal and Classify the Operators
We need to find the total duration of calls (in minutes) made by Faruq to friends who use mobile numbers from Operator Yocel.
First, let us categorize each friend based on their mobile operator as given in the table:
• Xitel Users: Anu, Bijay
• Yocel Users: Chetan, Deepak, Eshan, Faruq

Step 2: Understand the Sum of Calls within Yocel
Let us consider the calls that are both made and received by Yocel users (calls between Chetan, Deepak, Eshan, and Faruq).
Since a call within this group is made by one Yocel user to another Yocel user, the total outgoing call minutes to Yocel operators from Yocel users must be exactly equal to the total incoming call minutes from Yocel operators received by Yocel users.

Therefore, we can write the following equation:

$$\sum \text{Outgoing minutes to Yocel from Yocel users}=\sum \text{Incoming minutes from Yocel to Yocel users}$$

Step 3: Extract Values from the Table
From the table, we extract the outgoing minutes to Yocel (Operator Yocel column under Outgoing minutes) and incoming minutes from Yocel (Operator Yocel column under Incoming minutes) for each Yocel user:
• Chetan: Outgoing to Yocel = 175, Incoming from Yocel = 150
• Deepak: Outgoing to Yocel = 150, Incoming from Yocel = 100
• Eshan: Outgoing to Yocel = 100, Incoming from Yocel = 375
• Faruq: Outgoing to Yocel = $O_Y\left(F\right)$ (unknown), Incoming from Yocel = 150

Step 4: Solve the Equation
Summing the outgoing Yocel-to-Yocel minutes:
$$175+150+100+O_Y\left(F\right)=425+O_Y\left(F\right)$$
Summing the incoming Yocel-to-Yocel minutes:
$$150+100+375+150=775$$
Equating the two sums:
$$425+O_Y\left(F\right)=775$$
Solving for $O_Y\left(F\right)$:
$$O_Y\left(F\right)=775-425=350$$

Thus, the total duration of calls made by Faruq to friends using Yocel is 350 minutes.