The hotel manager divided the freshly vacated rooms for cleaning

Question

The hotel manager divided the freshly vacated rooms for cleaning, equally among 10 housekeepers with none left over. 10 rooms were then vacated ahead of time and in response, the manager redivided all freshly vacated rooms equally among 12 housekeepers, with no rooms left over. How many freshly vacated rooms were there before 10 rooms were vacated ahead of time?

(1) Before the 10 rooms were vacated ahead of time, there were more than 100 freshly vacated rooms.
(2) Before the 10 rooms were vacated ahead of time, there were fewer than 180 freshly vacated rooms.

(1) Before the 10 rooms were vacated ahead of time, there were more than 100 freshly vacated rooms.
(2) Before the 10 rooms were vacated ahead of time, there were fewer than 180 freshly vacated rooms.

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aditya1818 · Answer

This question is easy if you comprehend well

Since it was equally divided among 10 housekeeps, there the rooms has to be in the multiples of 10
and so it can be (10,20,30,40,50,60,......)

People left early and new vacant room were there. It was redivided among 12 housekeepers equally 
and so now new room count has to be (12,24,36,48,60.......)

After this I jumped to the Statements and clearly I and II both are insufficient for obvious reasons as rooms are be literally anything as per the above domain specified

combing them would still give us couple of options and not a definite answer

Answer is option (E)

egmat · Answer

Let's call the original number of freshly vacated rooms  x .

From the question stem, we get two key constraints: 
 Constraint 1:  x is divisible by  10  (rooms divided equally among 10 housekeepers, none left over)
 Constraint 2:  x + 10 is divisible by  12  (after 10 more rooms vacated, divided equally among 12 housekeepers)

Finding the possible values of x: 
Numbers divisible by 10: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180...

Now check which of these, when we add 10, give a number divisible by 12:
• x =  50  → 50 + 10 = 60 ✓ (divisible by 12)
• x =  110  → 110 + 10 = 120 ✓ (divisible by 12)
• x =  170  → 170 + 10 = 180 ✓ (divisible by 12)
• x =  230  → 230 + 10 = 240 ✓ (divisible by 12)

The pattern: x = 50, 110, 170, 230, 290... (increases by 60 each time - LCM of 10 and 12)

Statement 1:  Before the 10 rooms were vacated, there were more than 100 rooms.
So x > 100. Possible values:  110, 170, 230, 290...

Multiple values possible →  Not Sufficient

Statement 2:  Before the 10 rooms were vacated, there were fewer than 180 rooms.
So x < 180. Possible values:  50, 110, 170

Multiple values possible →  Not Sufficient

Combining Both Statements: 
100 < x < 180

Possible values:  110  and  170

Even with both constraints combined, we still have TWO possible values!

Case 1:  x = 110 rooms originally, then 120 rooms after (120 ÷ 12 = 10 rooms per housekeeper) ✓
 Case 2:  x = 170 rooms originally, then 180 rooms after (180 ÷ 12 = 15 rooms per housekeeper) ✓

Both cases satisfy ALL conditions, but give different answers.

Answer: E

Edskore · Answer

Great explanation above! One thing worth flagging for anyone who wants to internalize the logic quickly: this is a classic Divisibility + Number Properties DS pattern that the GMAT Focus Edition uses frequently in Word Problems.

The core of this question is recognizing that two divisibility constraints on x give you a periodic set of solutions — not just one or two random values. Once you spot that x must be divisible by 10 and x+10 must be divisible by 12, you're really solving for x = 50 (mod 60), which gives the sequence 50, 110, 170, 230... The LCM of 10 and 12 (= 60) controls the spacing.

The trap in Data Sufficiency here is thinking "two inequalities should pin down one value" — but they don't, because the feasible set {50, 110, 170...} has multiple values between 100 and 180, specifically 110 and 170. Always verify DS sufficiency by checking whether the constraints together produce a unique value, not just a reduced set.

Answer: E — neither statement alone nor together is sufficient.

ExpertsGlobal5 · Answer

Explanation:  
 
Let the initial number of vacated rooms be X.
Since the manager divided the rooms equally among 10 housekeepers with none left over: X is a multiple of 10.
The final number of vacated rooms after 10 more rooms were vacated = X + 10 
Since the manager divided (X + 10) rooms equally among 12 housekeepers with none left over: (X + 10) is a multiple of 12.
  We need to find whether the value of X     can be determined.  
 
 Statement (1) 
 
Since X > 100 and X is a multiple of 10, possible values of X are 110, 120, 130, 140, 150, 160, and so on.
However, since (X + 10) also needs to be a multiple of 12, the values of X that satisfy all the conditions are 110, 170, 230, and so on.
 
It is NOT possible to determine with certainty the value of X.   Hence, Statement (1) is insufficient   . 
 
 Statement (2) 
 
Since X < 180 and X is a multiple of 10, possible values of X are 170, 160, 150, 140, 130, 120, 110, and so on.
However, since (X + 10) also needs to be a multiple of 12, the values of X that satisfy all the conditions are 170, 110, and 50.
 
It is NOT possible to determine with certainty the value of X.   Hence, Statement (2) is insufficient   . 
 
As Statement (1) alone as well as Statement (2) alone is insufficient to answer the question, we need to now combine the two statements. 
 
 Statement (1) and Statement (2) combined  
 
The two statements combined give us X = 110 and X = 170 as two solutions that satisfy all the conditions.
 
It is NOT possible to determine with certainty the value of X.   Hence, Statement (1) and Statement (2) combined   are insufficient   . 
 
  E is the correct answer choice.

crimson_noise · Answer

Let total number of rooms before vacating be 't', let initial number of rooms assigned to each house keeper be a and number of rooms assigned to each housekeeper after 10 rooms were vacated be b
So, t=10a and t-10=12 b
or, 5a-5= 6b

Option 1:
T>100
Or, 10 a > 100
Or, a> 10
Not sufficient

Option 2:
T <180
Or, 10a <180
Or, a <18

Option 1+2:
10< a <18
So, a= 11,12,13,14,15,16,17
Now, 5a -5=6 b
Or, b= (5a-5)/6
Only possible value of a that makes (5a -5) divisible by 6 is 13
So, t=10a or t=130

The hotel manager divided the freshly vacated rooms for cleaning

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GMAT Data Sufficiency (DS) Questions

The hotel manager divided the freshly vacated rooms for cleaning

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