Logical Reasoning

Grids

Hard

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.

Figure for CAT 2025 2025 Slot 3 DILR question 11 (Logical Reasoning)

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 Anu to friends having mobile numbers from Operator Yocel?

Entered answer:

✅ Correct Answer: 525

Slide 1/11

The six friends call one another, so the working grid is a $6 \times 6$ table.

Rows are the caller, columns are the receiver, and each body cell holds the minutes of that one directional call.

A friend cannot call themselves, so the six diagonal cells are blocked.

The data set reports four totals for every friend:

Out-X $=$ minutes spent calling Xitel users,

Out-Y $=$ minutes spent calling Yocel users,

In-X $=$ minutes received from Xitel users,

In-Y $=$ minutes received from Yocel users.

These totals sit in the right border (outgoing) and bottom border (incoming).

From↓ To→	A	B	C	D	E	F	Out-X	Out-Y
A	—						0
B		—					50	0
C			—				50	0
D				—			200	100
E					—		0	0
F						—	350
In-X	50	0	200	125	0	200
In-Y	350	50	450	0	200	0

The blanks we ultimately want include Anu's Out-Y and Faruq's Out-Y, plus individual cells such as Bijay to Anu and Deepak to Chetan.

Facts to apply:

Faruq to Eshan $=200$ .

These calls are zero: Bijay→Eshan, Chetan→Anu, Chetan→Deepak, Deepak→Bijay, Deepak→Faruq, Eshan→Chetan, Eshan→Deepak.

To read Out-X / In-X and so on, we first need each friend's operator.

From↓ To→	A	B	C	D	E	F	Out-X	Out-Y
A	—						0
B		—					50	0
C			—				50	0
D				—			200	100
E					—		0	0
F					200	—	350
In-X	50	0	200	125	0	200
In-Y	350	50	450	0	200	0

Eshan's incoming is In-X $=0$ and In-Y $=200$ , a total of $200$ .

The one call we know reaching Eshan is Faruq to Eshan $=200$ , which already accounts for all of it.

Since it is recorded under In-Y, the caller Faruq is on Yocel.

Total minutes received by Xitel users $=\sum \text{Out-X}=0+50+50+200+0+350=650$ .

This must equal the combined incoming of the Xitel friends.

Each friend's incoming total (In-X $+$ In-Y) is $\{A=400,\ B=50,\ C=650,\ D=125,\ E=200,\ F=200\}$ .

Faruq is Yocel, so the Xitel friends are a subset of $\{A,B,C,D,E\}$ whose incoming totals add to $650$ : either $\{C\}$ or $\{A,B,E\}$ .

Total minutes made by Xitel users $=\sum \text{In-X}=50+0+200+125+0+200=575$ .

If Xitel $=\{C\}$ , Chetan's outgoing is only $50 \neq 575$ , which is impossible.

So Xitel $=\{A,B,E\}$ and Yocel $=\{C,D,F\}$ , and Faruq to Eshan $=200$ is entered.

From↓ To→	A	B	C	D	E	F	Out-X	Out-Y
A	—						0
B		—			0		50	0
C	0		—	0			50	0
D		0		—		0	200	100
E			0	0	—		0	0
F					200	—	350
In-X	50	0	200	125	0	200
In-Y	350	50	450	0	200	0

The seven listed no-call pairs are entered as $0$ :

Bijay→Eshan, Chetan→Anu, Chetan→Deepak, Deepak→Bijay, Deepak→Faruq, Eshan→Chetan, Eshan→Deepak.

From↓ To→	A	B	C	D	E	F	Out-X	Out-Y
A	—				0		0
B		—			0		50	0
C	0		—	0	0		50	0
D		0		—	0	0	200	100
E			0	0	—		0	0
F					200	—	350
In-X	50	0	200	125	0	200
In-Y	350	50	450	0	200	0

Eshan's In-Y $=200$ is entirely Faruq to Eshan, so no other Yocel friend called Eshan:

Chetan→Eshan $=0$ and Deepak→Eshan $=0$ .

Eshan's In-X $=0$ , so no Xitel friend called Eshan; Bijay→Eshan is already $0$ , hence Anu→Eshan $=0$ .

From↓ To→	A	B	C	D	E	F	Out-X	Out-Y
A	—				0		0
B	50	—	0	0	0	0	50	0
C	0		—	0	0		50	0
D		0		—	0	0	200	100
E			0	0	—		0	0
F					200	—	350
In-X	50	0	200	125	0	200
In-Y	350	50	450	0	200	0

Bijay's Out-Y $=0$ , so Bijay made no calls to the Yocel friends:

Bijay→Chetan $=0$ , Bijay→Deepak $=0$ , Bijay→Faruq $=0$ .

Bijay's Out-X $=50$ counts calls to the Xitel friends Anu and Eshan.

Bijay→Eshan $=0$ , so Bijay→Anu $=50$ .

From↓ To→	A	B	C	D	E	F	Out-X	Out-Y
A	—	0			0		0
B	50	—	0	0	0	0	50	0
C	0		—	0	0		50	0
D		0		—	0	0	200	100
E	0	0	0	0	—	0	0	0
F					200	—	350
In-X	50	0	200	125	0	200
In-Y	350	50	450	0	200	0

Eshan's Out-X $=0$ and Out-Y $=0$ , so Eshan made no calls at all:

Eshan→Anu $=0$ , Eshan→Bijay $=0$ , Eshan→Faruq $=0$ (Eshan→Chetan and Eshan→Deepak already $0$ ).

Anu's Out-X $=0$ counts calls to Bijay and Eshan; Anu→Eshan is already $0$ , so Anu→Bijay $=0$ .

From↓ To→	A	B	C	D	E	F	Out-X	Out-Y
A	—	0			0		0
B	50	—	0	0	0	0	50	0
C	0		—	0	0		50	0
D	200	0	100	—	0	0	200	100
E	0	0	0	0	—	0	0	0
F					200	—	350
In-X	50	0	200	125	0	200
In-Y	350	50	450	0	200	0

Deepak's Out-Y $=100$ counts calls to the Yocel friends Chetan and Faruq.

Deepak→Faruq $=0$ , so Deepak→Chetan $=100$ .

Deepak's Out-X $=200$ counts calls to the Xitel friends Anu, Bijay, Eshan.

Deepak→Bijay $=0$ and Deepak→Eshan $=0$ , so Deepak→Anu $=200$ .

From↓ To→	A	B	C	D	E	F	Out-X	Out-Y
A	—	0			0		0
B	50	—	0	0	0	0	50	0
C	0	50	—	0	0	0	50	0
D	200	0	100	—	0	0	200	100
E	0	0	0	0	—	0	0	0
F					200	—	350
In-X	50	0	200	125	0	200
In-Y	350	50	450	0	200	0

Chetan's Out-Y $=0$ counts calls to Deepak and Faruq; Chetan→Deepak is already $0$ , so Chetan→Faruq $=0$ .

Chetan's Out-X $=50$ counts calls to Anu, Bijay, Eshan; Chetan→Anu $=0$ and Chetan→Eshan $=0$ , so Chetan→Bijay $=50$ .

From↓ To→	A	B	C	D	E	F	Out-X	Out-Y
A	—	0	200	125	0	200	0
B	50	—	0	0	0	0	50	0
C	0	50	—	0	0	0	50	0
D	200	0	100	—	0	0	200	100
E	0	0	0	0	—	0	0	0
F					200	—	350
In-X	50	0	200	125	0	200
In-Y	350	50	450	0	200	0

Chetan's In-X $=200$ counts calls from Anu, Bijay, Eshan; Bijay→Chetan $=0$ and Eshan→Chetan $=0$ , so Anu→Chetan $=200$ .

Deepak's In-X $=125$ ; Bijay→Deepak $=0$ and Eshan→Deepak $=0$ , so Anu→Deepak $=125$ .

Faruq's In-X $=200$ ; Bijay→Faruq $=0$ and Eshan→Faruq $=0$ , so Anu→Faruq $=200$ .

From↓ To→	A	B	C	D	E	F	Out-X	Out-Y
A	—	0	200	125	0	200	0
B	50	—	0	0	0	0	50	0
C	0	50	—	0	0	0	50	0
D	200	0	100	—	0	0	200	100
E	0	0	0	0	—	0	0	0
F	150	0	350	0	200	—	350
In-X	50	0	200	125	0	200
In-Y	350	50	450	0	200	0

Deepak's In-Y $=0$ , so Faruq→Deepak $=0$ (Chetan→Deepak already $0$ ).

Chetan's In-Y $=450$ comes from Deepak and Faruq; Deepak→Chetan $=100$ , so Faruq→Chetan $=350$ .

Anu's In-Y $=350$ comes from Deepak and Faruq (Chetan→Anu $=0$ ); Deepak→Anu $=200$ , so Faruq→Anu $=150$ .

Bijay's In-Y $=50$ comes from Chetan and Faruq (Deepak→Bijay $=0$ ); Chetan→Bijay $=50$ , so Faruq→Bijay $=0$ .

From↓ To→	A	B	C	D	E	F	Out-X	Out-Y
A	—	0	200	125	0	200	0	525
B	50	—	0	0	0	0	50	0
C	0	50	—	0	0	0	50	0
D	200	0	100	—	0	0	200	100
E	0	0	0	0	—	0	0	0
F	150	0	350	0	200	—	350	350
In-X	50	0	200	125	0	200
In-Y	350	50	450	0	200	0

Anu's Out-Y $=$ Anu→Chetan $+$ Anu→Deepak $+$ Anu→Faruq $=200+125+200=525$ .

Faruq's Out-Y $=$ Faruq→Chetan $+$ Faruq→Deepak $=350+0=350$ .

The grid is now fully determined, and every border total matches its row or column sum.

Operators: Xitel $=\{$ Anu, Bijay, Eshan $\}$ , Yocel $=\{$ Chetan, Deepak, Faruq $\}$ .

Anu is on Xitel; the Yocel friends are Chetan, Deepak, Faruq.

From the completed grid: Anu→Chetan $=200$ , Anu→Deepak $=125$ , Anu→Faruq $=200$ .

Total $=200+125+200=525$ minutes.

Question 10

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

Question 12

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

Question 13

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

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

Solution

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