A hotel had 180 rooms numbered from 1 to 180. The rooms numbered from

Occupied room numbers = 180-25 +1 = 156
Occupied room numbers assigned to Kris = (180-27)/3 + 1 = 153/3 +1 = 51 +1= 52
Occupied room numbers assigned to Michel = 180-28/4 +1= 152/4 +1 = 38 +1 =39

L.C.M. of 3 and 4 = 12. Occupied room numbered Multiple of 12 is common to Kris and Michel.

Occupied room number common to both = 180-36/12 +1 = 13.
Unassigned occupied room number = 156-(91-26) = 256 – 65 = 91(B)

In total, 180 Hotel rooms -- numbered from 1-180.
Rooms from 25-180 ---occupied.
--> 180-25+1= 156 occupied rooms

Assigned to:
Kris --rooms multiples of 3 (range from 25-180)
Michel -- rooms multiples of 4 (range from 25-180)
George -- rooms common for Kris and Michel, also all unassigned occupied rooms.
--> how many rooms were assigned to George?

Kris: \(a_1 =27\), \(a_n=180\)
--> \(n= \frac{(180-27)}{3} +1= 52\)

Michel: \(a_1 =28\), \(a_n=180 \)
--> \(n= \frac{(180-28)}{4 }+1= 39\)

-->Rooms common for Kris and Michel:
\(a_1 =36\), \(a_n=180\)
--> \(n= \frac{(180-36)}{12} +1= 13\)

Finally, 52-13 = 39 rooms for Kris.
39-13 =26 rooms for Michel.
--> From occupied 156 rooms, 156 - 39-26= 91 rooms were assigned to George.

The answer is B.

We know that there are 180 rooms in total and that room number 25 to 180 are occupied. Chris was assigned occupied rooms numbered 3x and Michel was assigned occupied rooms numbered 4x, where x is an integer from 25 to 180.
George was assigned all other occupied rooms not assigned to Michel or Chris plus the rooms that were common to both Chris and Michel.

Rooms assigned to George = Number of unoccupied rooms - Rooms assigned to Chris - Rooms assigned to Michel + 2 * Rooms common to Michel and Chris
Number of unoccupied rooms = 180-25+1 = 156
Rooms assigned to Chris = (180-27)/3 = 153/3 = 51
Rooms assigned to Michel = (180-28)/4 = 152/4 = 38
Rooms common to Chris and Michel = 2*(180-36)/12 = 288/12 = 24 [multiply by 2 because the common rooms are counted twice i.e. one each in rooms assigned to Michel and Chris].
Rooms assigned to George = 156-51-38+24 = 91

The answer is B.

Ans: B

Full=156 Vacant= 24
multiple of 3 =((180-3)/3)+1=60, assigned to Kris=60- vacant rooms(3,6,9,12,15,18,21,24)=60-8=52
multiple of 4 =((180-4)/3)+1=45, assigned to Michel=60- vacant rooms(4,8,12,16,20,24)=45-6=39
total=52+39=91 assigned
multiple of both 3,4 or 12=((180-12)/12)+1=15, assigned to third=15-vacant rooms(12,24)=15-2=13

total-common=91-13=78 assigned
unassigned=156-78=78

assigned to third= 78+13=91

Rooms by K are 27, 30, 33, (36),..., (180). So, total no of rooms by K including those common for K and M: 180 = 27+(n-1)*3 --> n=52

Rooms by M are 28, 32, (36),..., (180). Total no of rooms by M including those common for K and M: 180 = 28+(n-1)*4 --> n=39

Total no of rooms common for K and M (nc): 180=36+(nc-1)*12 --> nc=13. Meanwhile,
Total occupied room = 180-25+1 = 156

Total occupied room by G = 156 - (52+39) +13*2 = 91

FINAL ANSWER IS (B)

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The number of rooms numbered multiple of 3 from 25 to 180 is: \(\frac{(180-27)}{3} + 1 = 52\)
The number of rooms numbered multiple of 4 from 25 to 180 is: \(\frac{(180-28)}{4} + 1 = 39\)
The number of rooms numbered multiple of 12 from 25 to 180 is: \(\frac{(180-36)}{12} + 1 = 13\)
The number of rooms were assigned to George is: \(\frac{(180-25)}{1} + 1 - 52 -39 + 2 * 13 = 91\)

=> Choice B

total rooms which were divided ; 180-24 ; 156
Kris had ; 156/3 ; 52 rooms
Michael had ; 156/4 ; 39 rooms
common ; LCM ; 12; 156/12 ; 13 rooms

Total Rooms assigned to Chris and Michel ; 91 - 13 = 78

George will have (156 - 78) + 13 = 91
IMO B

A hotel had 180 rooms numbered from 1 to 180. The rooms numbered from 25 to 180 were occupied and the rest were vacant. Two attendants, Kris and Michel, were given the duty of room service of occupied rooms. Each room numbered multiple of 3 was assigned to Kris and each room numbered multiple of 4 was given to Michel. If the rooms which were common for Kris and Michel, and rest of the unassigned occupied rooms were assigned to a third attendant named George, how many rooms were assigned to George?

A. 82
B. 91
C. 98
D. 99
E. 100

Given; A hotel had 180 rooms numbered from 1 to 180. The rooms numbered from 25 to 180 were occupied and the rest were vacant. Two attendants, Kris and Michel, were given the duty of room service of occupied rooms. Each room numbered multiple of 3 was assigned to Kris and each room numbered multiple of 4 was given to Michel.

Asked: If the rooms which were common for Kris and Michel, and rest of the unassigned occupied rooms were assigned to a third attendant named George, how many rooms were assigned to George?

Total occupied rooms = 180 - 25 + 1 = 156
Rooms assigned to Kris + common rooms to Kris and Michel = {27,30,33,...180} = (180-27}/3 +1 = 52
Rooms assigned to Michel + common rooms to Kris and Michel = {28,32,,...180} = (180-28}/4 +1 = 39
Rooms common to Kris and Michel = {36,48,...180} = (180-36)/12 + 1 = 13
Rooms assigned to Kris alone = 52 -13 = 39
Rooms assigned to Michel alone = 39 - 13 = 26

Rooms assigned to George = 13 + (156-39-26-13) = 156-65 = 91

IMO B

K attends rooms that are multiple of 3. Between 25 to 180, the series is 27,30,33,36,....180. Total 52 (I have used the formula of AP series).
M attends rooms that are multiple of 4. Between 25 to 180, the series is 28,32,46,40,....180. Total 39.
In the above two series, common terms are 36,48,....180. Total 13.
Now, total rooms 180-25+1=156.
Only K attends 52-13=39, and only M attends 39-13=26. So, G attends 13+(156-39-26-13)=91 (B).