Let the no of boxes holding 12 glasses each be x and those holding 16 be y.
Thus, \(\frac{(12*x+16*y)}{(x+y)}\) = 15 --> y = 3x. Again, we know that y = x+16 --> Thus, 2x = 16 and x = 8, y= 24.

Thus, The total no of glasses -->(x+y)*15 = 32*15 = 480.

10 second approach :
As the average no of glasses per box is 15, thus the ratio of the total # of glasses to 15 must give an integer --> The options denote the total # of glasses, thus, we can safely eliminate A,C,D. Now for option B, the ratio gives --> \(240/15\) = 16 which means the total no. of boxes including both kinds is 16 in number. However, as it is given that there are 16 more of the larger boxes, then the no of boxes holding 12 glasses must be zero, and in this case, the average no of glasses per box would be nothing but 16, which contradicts the given fact. Thus, the only correct option is E.

E.
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Hi All,

Most Test Takers would recognize the "system" of equations in this prompt and just do algebra to get to the solution (and that's fine). The wording of the prompt and the 'spread' of the answer choices actually provide an interesting 'brute force' shortcut that you can take advantage of to eliminate the 4 wrong answers....

We're told that there are 2 types of boxes: those that hold 12 glasses and those that hold 16 glasses. Since the AVERAGE number of boxes is 15, we know that there MUST be at least some of each. We're also told that that there are 16 MORE of the larger boxes.

This means, at the minimum, we have...
1 small box and 17 large boxes = 1(12) + 17(16) = 12 + 272 = 284 glasses at the MINIMUM

Since the question asks for the total number of glasses, we can now eliminate Answers A, B and C....

The difference in the number of boxes MUST be 16 though, so we could have....
2 small boxes and 18 large boxes
3 small boxes and 19 large boxes
etc.

With every additional small box + large box that we add, we add 12+16= 28 MORE glasses. Thus, we can just "add 28s" until we hit the correct answer....

284+28 = 312
312+28 = 340
340+28 = 368
368+28 = 396

At this point, we've 'gone past' Answer D, so the correct answer MUST be Answer E.....But here's the proof....

396+28 = 424
424+28 = 452
452+28 = 480

Final Answer:

GMAT assassins aren't born, they're made,
Rich
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Is this a possible question in GMAT? I spent almost 5 mins to do all the calculations :D

Found it bit confusing...

Let x be the total number of boxes containing 12 Glasses and y be the total number of boxes containing 16 Glasses.
Also, \(y = x + 16.\)
\(12x + 16y = 15 (x+y) ---> y = 3x\)

\(x = 8, y = 24.\)
So, total would be \(12*8 + 24*16 = 480.\)

Traditional Approach:

L (Large Box) and S (Small Box)

equation 1 ..................... L = S + 16


equation 2..................... 12S + 16L / S + L = 15

L = 3S

now susbtitute:

S + 16 = 3S
S = 8

S + 16 = 32
therefore 480

Fastest way:

Already given 16 larger boxes at least : 16 (boxes) *16 (glasses) = 256; hence answer has to be greater than 256.
average is 15, thus the answer needs to be divisible by 15, as the boxes cannot be in fraction, only answer left 480.
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The quickest way i've found for any average problem is to take the difference between each group and the average and establish a ratio.

12 glasses is 3 away from 15 glasses; and 16 glasses is 1 away from 15 glasses. So the resulting ratio is 3x (16 glasses) to 1x (15 glasses). Prompt tells us that the ratio of boxes is 3x = 1x + 16, so x = 8. Then the number of 16 glass boxes would would be 3*8 = 24.

Total glasses = 8*12 = 96 + 24*16 = 384 => 480.

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