The length, the breadth and the height of a cuboid are in the ratio...

Question

The length, the breadth and the height of a cuboid are in the ratio 1:2:3. If the length, breadth and height are increased by 100%, 200% and 200% respectively , then the increase in the volume of the cuboid is

A) 5 times
B) 6 times
C) 12 times
D) 17 times
E) 20 times

A) 5 times
B) 6 times
C) 12 times
D) 17 times
E) 20 times

anurag356 · Answer

The question is simple however it was added to show that the final answer is not affected by the ratio of length,breadth, height i.e for any ratio the volume will increase by 17 times only for the corresponding percentage increments.

Consider the length as x
b=2x and h=3x

Now length increased by 100% i.e new l =2x
Now breadth increased by 200% i.e new b =6x
Now height increased by 200% i.e new h =9x

New volume =2x*6x*9x=\(108x^3\)
Old volume = \(6x^3\)

New volume/old volume=\(108x^3/6x^3=18\)

New volume= 18*old volume
= (1+17) old volume
= old volume+ 17 old volume

Therefore the old volume increased by 17 times.

Now consider l=b=h=x

Now length increased by 100% i.e new l =2x
Now breadth increased by 200% i.e new b =3x
Now height increased by 200% i.e new h =3x

New volume = \(18x^3\)
old volume = \(x^3\)

New volume / old volume = 18
Old volume increased by 17 times.

Therefore the quicker way to solve the problem is simply by considering only the percentage or say the observation is that the ratio is just a distraction.

Please give a kudos if you find the info useful :-D

EMPOWERgmatRichC · Answer

Hi anurag356,

This question is essentially about the Percentage Change Formula and a bit of Geometry. When you're dealing with a question that does NOT include any values to work with, it's often helpful to TEST VALUES.

Here, we're told that the dimensions of a rectangular solid (re: cuboid) - the length, width and height - are in the ratio of 1:2:3.

Let's TEST VALUES and say that...

Length = 1
Width = 2
Height = 3

Next, we're told that the length, width and height are increased 100%, 200% and 200% respectively. Thus, they would now be..

Length = 1+1 = 2
Width = 2 + 2(2) = 6
Height 3 + 2(3) = 9

We're asked for the increase in the VOLUME of the shape (and the answers hint that we should use the Percentage Change Formula).

Initial Volume = (L)(W)(H) = (1)(2)(3) = 6

New Volume = (L)(W)(H) = (2)(6)(9) = 108

Percentage Change = (New-Old)/Old = (108-6)/6 = 102/6 = 17

Thus, the new volume is 17 times greater than the old volume.

Final Answer:  D

GMAT assassins aren't born, they're made,
Rich

IanStewart · Answer

This question really doesn't make sense - where is it from? First it talks about 'cuboids', which you will absolutely never see on the GMAT. The word 'cuboid' is not a synonym for 'rectangular prism' (your normal rectangular box shape), and you don't get the volume of a general cuboid just by multiplying the lengths of the edges.

Second, the answer choices don't make sense. Filling in the blanks, the correct answer says "the increase in the volume is 17 times". 17 times what? It needs to say  something  after the word 'times'. The most natural way to interpret the question is to think it asks "the new volume is what times the old volume", and the correct answer to that question is '18'. But that's not what they mean - they mean "the  increase  in the volume is what times the old volume" which is a very strange way to interpret the question.

Anyway, if we assume, as I'm sure they meant, that we have a rectangular box, then the ratio of the length, height and width is irrelevant. When you increase something by 100%, you're multiplying by 2, and when you increase by 200%, you're multiplying by 3. So if our dimensions originally were a, b and c, our original volume is abc, and our new volume is (2a)(3b)(3c) = 18abc = 18 times the old volume.

GMATinsight · Answer

CONCEPT : Since there is no absolute value of any dimension of the Cuboid (Rectangular Solid/box) so in such case we can assume any number for the dimensions as per the ration of the dimensions given

Let, Length(L) = 1 Unit, Breadth(B) = 2 Units and Height(H) = 3 units of the Rectangular Solid

Then Volume = L x B x H = 1 x 2 x 3 = 6 Cube Units

New Length = L + 100% of L = L+L = 2L = 2*1 = 2
New Breadth = B + 200% of B = B+2B = 3B = 3*2 = 6
New Height = H + 200% of H = H+2H = 3H = 3*3 = 9

Then Volume = New Length x New Breadth x New Height = 2 x 6 x 9 = 108 Cube Units

New Volume = a * Old Volume

i.e. a = New Volume / Old Volume = 108/6 = 18 times

Since New Volume = 18 times of Old Volume
therefore Volume has increased by 18V-V = 17 times

i.e. Volume has increased by= 17 times

Answer: option  D

anurag356 · Answer

The only reason this question was added is that during my GMAT classes , everyone was considering the ratios part and coming to a solution. This takes more time and as proved by me and others as well considering ratios is NOT required. Hence just to let everyone know about this and SAVE time in the actual exam I added the question.

Sometimes seeing the bigger picture is a lot more important. After all, one of the key factors that makes GMAT difficult is its timing constraints.

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