12 Days of Christmas GMAT Competition - Day 7: A factory produced 8

Question

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A factory produced 12 shipments of bottles, with each shipment containing 15 bottles. If the average number of defective bottles per shipment was 1.5, did any shipment contain at least 4 defective bottles?

(1) One-third of the shipments had no defective items, and one-third of the shipments had exactly 1 defective item each.

(2) Two-thirds of the shipments had at most 1 defective item each.

Bunuel · Accepted Answer

GMAT Club's Official Explanation:

A factory produced 12 shipments of bottles, with each shipment containing 15 bottles. If the average number of defective bottles per shipment was 1.5, did any shipment contain at least 4 defective bottles?

An average of 1.5 defective bottles per shipment for 12 shipments implies a total of 1.5 * 12 = 18 defective bottles.

(1) One-third of the shipments had no defective items, and one-third of the shipments had exactly 1 defective item each.

This means that 4 shipments had 0 defective items, and another 4 shipments had exactly 1 defective item each. Together, these 8 shipments accounted for 0 * 4 + 1 * 4 = 4 defective items, leaving 18 - 4 = 14 defective items to be distributed among the remaining 4 shipments.

Can each of these 4 shipments have fewer than 4 defective items? No, because if each shipment had fewer than 4 defective items, the maximum total would be 3 * 4 = 12, which is less than 14. Therefore, at least one of these 4 shipments must have at least 4 defective items. Sufficient.

(2) Two-thirds of the shipments had at most 1 defective item each.

This implies that 8 shipments had 1 or fewer defective items each. If those 8 shipments had exactly 1 defective item each, the remaining 4 shipments must account for 10 defective bottles.

We can distribute the 10 defective bottles across the 4 shipments in such a way that none of them has 4 or more defective bottles: for example, 2, 2, 3, 3. However, we can also distribute the 10 defective bottles such that one shipment does have at least 4 defective bottles: for example, 2, 2, 2, 4. Since we have two different outcomes, the information provided is not sufficient.

Answer: A.

Sanjay9822 · Answer

Given:
 
 The factory produced 12 shipments of bottles. 
 Each shipment contains 15 bottles. 
 The average number of defective bottles per shipment is 1.5.  
This means the total defective bottles across all shipments is:
12 * 1.5 = 18 defective bottles.
We need to determine if  any shipment contained at least 4 defective bottles .
 
Statement (1):
 
 One-third of the shipments had no defective items. 
 One-third of the shipments had exactly 1 defective item each.  
 Explanation: 
One-third of 12 shipments = 4 shipments.
 
 4 shipments had 0 defective bottles. 
 4 shipments had 1 defective bottle each.  
Total defective bottles in these 8 shipments = (0 * 4) + (1 * 4) = 0 + 4 = 4 defective bottles.
This leaves:
18 - 4 = 14 defective bottles for the remaining 4 shipments.
If 14 defective bottles are distributed among 4 shipments, the average is:
14 / 4 = 3.5.
Since defective bottles must be whole numbers, at least one shipment must have 4 or more defective bottles.
 Statement (1) is sufficient. 
 
Statement (2):
 
 Two-thirds of the shipments had at most 1 defective item each.  
 Explanation: 
Two-thirds of 12 shipments = 8 shipments.
 
 These 8 shipments had at most 1 defective bottle each. 
 The maximum defective bottles among these 8 shipments is: 1 * 8 = 8 defective bottles.  
This leaves:
18 - 8 = 10 defective bottles for the remaining 4 shipments.
If 10 defective bottles are distributed among 4 shipments, the average is:
10 / 4 = 2.5.
Since defective bottles must be whole numbers, at least one shipment must have 4 or more defective bottles.
 Statement (2) is sufficient. 
 
Final Answer:
Each statement alone is sufficient.
Answer: D

twinkle2311 · Answer

Given :
Total bottles = 12 shipments × 15 bottles = 180 bottles. 
Average defective bottles per shipment = 1.5. 
Total defective bottles = 12 × 1.5 = 18 defective bottles.

Statement A: 
One-third (4 shipments) had no defective items. 
One-third (4 shipments) had exactly 1 defective item each. 
Remaining 4 shipments must account for the rest: 18 - (4 × 1) = 14 defective bottles. 
Average defects in these 4 shipments = 14/4 = 3.5. 
Since a defected bottle would be a whole number, at least one shipment will have 4 defects. 
 Statement A is Sufficient

Statement B:  
Two-thirds (8 shipments) had at most 1 defective item each. 
These 8 shipments can have a maximum of 8 defects. 
Remaining 4 shipments must account for 18 - 8 = 10 defects. 
Average defects in these 4 shipments = 10/4 = 2.5. 
Not enough information to guarantee 4 defects in any shipment. 
 Statement B is Insufficient

Answer: A

Krunaal · Answer

Avg. no. of defective bottles = 1.5 / shipment
Total no. of shipments = 12

Total no. of defective bottles = 1.5 * 12 = 18

Statement 1

4 shipments have 0 defective bottles, and 4 shipments have exactly 1 defective bottle, the remaining defective bottles = 18 - 4 = 14 which are in 4 remaining shipments.  If each of the 4 shipment had 3 bottles, then also 2 defective bottles remain out hence there is atleast one shipment that has atleast 4 defective bottles.   SUFFICIENT

Statement 2

If 8 shipments each had 1 defective bottle, the remaining defective bottles = 10.  The 4 remaining shipments can have 3, 3, 2, 2 defective bottles each or 4, 2, 2, 2 defective bottles . We're not sure whether any shipment contain at least 4 defective bottles.  INSUFFICIENT

Answer A.

hr1212 · Answer

Total shipments = 12

Defective pieces = 12*1.5 = 18

Statement 1 => 4 shipments - 0 defective pieces, 4 shipments - 4 defective pieces and 4 shipments - (18-4=14) defective pieces

If we optimally try to distribute these 14 pieces among 4 places, we get (3,3,4,4). So one of this shipment has to have atleast 4 defective pieces.
This statement is sufficient.

Statement 2 => 8 shipments - atmost 8 defective pieces, 4 shipments - (18-8=10) defective pieces.

If we divide 10 into 4 places, we get (2,2,3,3). So it's not required that any shipment will or will not have atleast 4 defective pieces.
This statement is not sufficient.

Answer: A

aviraj1703 · Answer

Total shipments = 12
Bottles in each shipment = 15
Total defective bottles = 12 * 1.5 = 18

(1)  One-third of the shipments had no defective items, and one-third of the shipments had exactly 1 defective item each.
4 shipments had no defective items
Next 4 had exactly 1 defective item so we are left with 14 defective items and 4 shipments
Now, in next 4 shipments to divide 14 bottles in each scenario at least 1 shipment has to have 4 defective items.
 Statement 1 alone is sufficient.

(2)  Two-thirds of the shipments had at most 1 defective item each.
8 shipments had at most 1 defective items 
That means 0 or 1 so based on that there can be or can not be at least 1 shipment with 4 defective items.
 Statement 2 alone is insufficient.

Answer: A

missionmba2025 · Answer

Total defective bottles = 1.5 * 12 = 28

(1) One-third of the shipments had no defective items, and one-third of the shipments had exactly 1 defective item each.

4 shipments didn't have any defective bottles
4 shipments had only 1 defect bottle

So, the rest 24 defective bottles must be in the last 4 shipments. Hence, we can conclude that at least one shipment had 4 defective bottle.

The statement is sufficient. Eliminate B, C, and E.

(2) Two-thirds of the shipments had at most 1 defective item each.

Best case scenario (to reduce the number of defective bottles on the last one third shipment) - each of the shipment has 1 defective bottle.

Remaining = 20 defective bottles in 4 shipment. Hence, we can conclude that at least one shipment had 4 defective bottles.

This statement is also sufficient.

Option D.

HarshaBujji · Answer

Total bottles = 12*15 =180. 
So defective bottles is 18. in 12 shipments.

We need to find that  did any shipment contain at least 4 defective bottles ? 
 Stmt (1) One-third of the shipments had no defective items, and one-third of the shipments had exactly 1 defective item each.
 
 4 shipments with no defect. 
other 4 shipments 4 defects.

The remaining 4 shipments should have 18-4=14 defects.

So a minimum 1 shipment contains more than 1. Avg is 3....+

Hence Stmt 1 is self-sufficient. So AD can be the answer.

Stmt ( 2) Two-thirds of the shipments had at most 1 defective item each.

So 8 have max one, So 8 max defects.

The remaining 4 shipments should have 10.

2,2,3,3 or 4,2,2,2. So this is insufficient. 
 
Hence IMO A

AviNFC · Answer

1. No defect=4
 1 defect each=4 
 total 4 defects in 8 shipments. 
 remaining 14 in 4 shipments.
 3,3,4,4 or 3,3,4,5 ..so minimum 4 is there in atleast 1 shipment. SUFFICIENT

2. let all 8 have 1 defect. so total 8 gone. remaining 10 in 4 shipments. 
2,2,2,4 or 2,2,3,3 .. .NOT SUFFICIENT

Answer A

OmerKor · Answer

Hi everyone

12 shipments, 15 bottles each. 1.5 per shipment if defective. 12*1.5 = 18 defective bottles in total. (1) 4*0=0 4*1=4 We left with 18-4=14 In order to get 14 bottles in 4 shipments: 2 of the shipments must contain at least 4 defective bottles. Sufficient. (2) 8*0=0 / 8*1=8 We left with 18-8=10 Two cases: 1) 8*0=0 and 4+4+5+5 2) 8*1=8 and 2+2+3+3 One has at least 4 and the other is not. Insufficient Answer A

bellsprout24 · Answer

Answer is  A. Statement 1 alone is sufficient

1) 1.5 = (4*0 + 4*1 + 4X)/12 --> 18 = 4 + 4X --> 14 = 4X --> X = 3.5 which means at least 1 of the 4 remaining shipments must have 4 defects in order for the overall average to be 1.5 since the remaining 4 shipments must have a total of 14 defects. Statement 1 is sufficient.
2) If all 8 shipments have 1 defect, then 1.5 = (8*1 + 4X)/12 --> 18 = 8 + 4X --> 10 = 4X --> X = 2.5 which means the remaining shipments may or may not contain 4 defects. 10 could be comprised of 2, 2, 3, 3; it could also be 2, 2, 2, 4. If all 8 shipments have 0 defects, then 1.5 = (8*0 + 4X)/12 --> 18 = 4X --> X = 4.5 which means the remaining shipments would definitely have at least 4 defects. However we do not have a definite answer since both are possible. Statement 2 is insufficient.

andreagonzalez2k · Answer

12 shipments x 15 bottles/shipment

defective bottles/12=1.5
defective bottles=18

¿any shipment with defective bottles>=4?

(1)
1/3 * 12 = 4 -> defective bottles=0
1/3 * 12 = 4 -> defective bottles=1

The remaining one-third of shipments (4) have x defective bottles each.

4*0+4*1+4*x=18
4x=14
x=3.5

As the number of defective bottles must be a non negative integer, there is no way to answer NO to the question.

SUFFICIENT

(2)
We can be as in (1) and the answer would be YES.

But, if 2/3*12=8 shipments have 1 defective item each:

8*1+4*x=18
4x=10
x=2.5

If the other third have (2,2,3,3) defective items each, the answer would be NO.

INSUFFICIENT

IMO A

Navya442001 · Answer

Statement (1):
 One-third of the shipments had no defective items, and one-third of the shipments had exactly 1 defective item each. 
 
 One-third of the shipments is: (1/3) × 12=4 shipments. 
 So, 4 shipments had no defective items, and 4 shipments had exactly 1 defective item each. The total number of defective bottles from these 8 shipments is: (0×4)+(1×4)=4 defective bottles. 
 Therefore, the remaining 4 shipments must account for the remaining defective bottles: 18−4=14 
 The average number of defective bottles for these 4 shipments is: 14/4=3.5. 
 Since the average for these 4 shipments is 3.5, at least one of them must contain more than 3 defective bottles (i.e., at least 4 defective bottles).  
 Statement (1) is sufficient.

Statement (2):
 Two-thirds of the shipments had at most 1 defective item each. 
 
 Two-thirds of the shipments is: (2/3) × 12=8 shipments  
 These 8 shipments have at most 1 defective item each. The maximum number of defective bottles in these 8 shipments is: 8×1=8 
 The remaining 4 shipments must account for the remaining defective bottles: 18−8=10 
 The average number of defective bottles for these 4 shipments is: 10/4=2.5 
 With an average of 2.5, it is possible that none of the shipments have 4 defective bottles. For instance, the defective counts could be distributed as 2,2,3,3, none of which is at least 4. Alternatively, it is possible that one shipment has 4 defective bottles. Hence, the information is insufficient.  
 Statement (2) is not sufficient.

Shruuuu · Answer

Total shipments = 12 
Each shipment contains = 15 bottles
Total bottles that need to be defective = Defective bottles / Total shipments = 1.5
DB / 12 = 1.5
DB = 18

i) So out of 12 shipments, 4 has 0 defective bottles
Out of other 8 shipments, 4 has 1 defective bottle each i.e. 4 defective bottles in total
In the remaining shipments, 14 defective bottles are there - 
If each shipment has 3 bottles defective then 4 * 3 = 12 defective bottles 
If each shipment has 4 bottles defective then 4 * 4 = 16 defective bottles

Since, 14 bottles are defective this implies, a few of the shipments will have > 3 defective bottles

SUFFICIENT

ii)2/3 of the shipment has atmost 1 defective item each; this creates 2 scenarios -

a) 6 shipments have 0 defective bottles 
2 shipments have 1 defective bottle each

So the remaining should have, 16 defective bottles i.e. few will have 4 or greater than 4. Hence, we get a yes

b) 2 shipments have 0 defective bottles
6 shipments have 1 defective bottle each

So the remaining should have, 12 defective bottles i.e. we can have 3 defective bottles in each shipment. Hence, we get a no

INSUFFICIENT

IMO A

pintukr · Answer

A factory produced 12 shipments of bottles, with each shipment containing 15 bottles. If the average number of defective bottles per shipment was 1.5, did any shipment contain at least 4 defective bottles?

(1) One-third of the shipments had no defective items, and one-third of the shipments had exactly 1 defective item each.

(2) Two-thirds of the shipments had at most 1 defective item each.

12 shipments (with average 1.5 bottles defective per shipment)
so, total defective bottles = 18

(1) One-third of the shipments had no defective items, and one-third of the shipments had exactly 1 defective item each.

4 shipments @ 0 defective/shipment => 0 defective bottles 
 4 shipments  1  defective/shipment => 4 defective bottles

so, 14 defective bottles in balance 4 shipments

(3,3,3,5) 
 (4,4,4,2)...In all options, there will be atleast one shipment with >= 4 defective bottles

YES is the answer; hence,

SUFFICIENT

(2) Two-thirds of the shipments had at most 1 defective item each.

If 8 shipments (2/3 of 12) has 1 defective each, then defective bottles = 8 
 balance 4 shipments has = 10 defective bottles

(3,3,3,1)... NO is the answer 
 (4,4,1,1)... YES is the answer

INSUFFICIENT

(A) is the CORRECT answer

prashantthawrani · Answer

Total shipments: 12  
  Bottles per shipment: 15  
  Average defective bottles per shipment: 1.5, so total of 18 defective bottles.
   
 (1) One-third of the shipments had no defective items, and one-third of the shipments had exactly 1 defective item each.

Out of 12 shipments, 4 had no defective items        , and 4 had 1 defective bottle each    4 defective.bot   ],    
   so 18-0-4=14 defective bottles must be present in remaining 4 shipments.   
   14/4=3.5 defective bottles per shipment   
   As no. of bottles can't be a fractional number it must be (3,     4     ,3,     4      ; 3,3,3,     5      ; etc) in remaining 4 shipments.

Sufficient. 
 
(2) Two-thirds of the shipments had at most 1 defective item each.

Out of 12 shipments, 8 had almost 1 defective bottle each      8 defective.bot     ],   
  S   o 18-8=10 defective bottles must be present in remaining 4 shipments.   
   10/4=2.5 defective bottles per shipment   
   As no. of bottles can't be a fractional number it must be (2,3    ,2,3     ; 2,2,2, 4      ; etc) in remaining 4 shipments.

Not sufficient.

Ans. A

Mardee · Answer

12 shipments, each containing 15 bottles
Average number of defective bottles per shipment is 1.5 => Total of 12*1.5=18 defective bottles across all shipments

1 =>
One-third of the shipments had no defective items = 0.33*12 = 4
Another one-third (4) had exactly 1 defective item each
Total defective items = 4*0 + 4*1 = 4

Total defective bottles among 4 shipments = 18-4 = 14
So, Distribution among 4 shipments defective bottles = 14/4 = 3.5, since it should be integer
Then, atleast one of the shipment has 4 defective bottles

2=>
Two-thirds of the shipments had at most 1 defective item each = 0.66*12 = 8

8 shipments had at most 1 defective item each, 
=> Maximum contribution from these shipments = 8*1 = 8 bottles
=> Minimum contribution from these shipments = 8*0 = 0 bottles

Thus,
Defective bottle between 4 shipment for maximum contribution = 18-8 = 10
Defective bottle between 4 shipment for minimum contribution = 18-0 = 18

=> Average distribution between 4 shipment for maximum contribution = 10/4 = 2.5 
Since it should be integer, no shipment would have 4 defective bottles
=> Average distribution between 4 shipment for minimum contribution = 18/4 = 4.5
Since it should be integer, at least one shipment would have 4 or more defective bottles

Since multiple possibilities possible, statement 2 is not enough

So 
 A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient

Rishm · Answer

I agree with A however it says in second condition that two thirds had at most 1 defective item each, meaning there could be 1 defective item max in each and they can also have 0 defective items, because of the word at MOST and not at least. It maybe that it has 1 defective item in 1 shipment and 7 no defects or maybe in 2/6 ratio. We get 4.1 or 4.25 as average for other defectives the remaining 4 shipments.

miag · Answer

Hi, This is good! In addition, I would advise just finding a case for a yes or a no to solidify your reasoning to reject Option B) (statement II alone is sufficient) Like you rightly said, minimum defectives needed in the other 4 shipments (leftover 1/3rd) would be when each of the other 8 have 1 defective so that leaves us with 10 defective across 4 shipments. There could be a case like 3,3,2,3 - which gives us the answer as No. Another case could be 4,2,2,2 - which gives us Yes. Hope this helps!

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