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What is the volume of a certain rectangular solid?

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What is the volume of a certain rectangular solid? [#permalink] New post   Updated on: 14 Dec 2012, 05:52
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What is the volume of a certain rectangular solid?

(1) Two adjacent faces of the solid have areas 15 and 24, respectively.
(2) Each of two opposite faces of the solid has area 40.
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C

Originally posted by amod243 on 24 Feb 2010, 06:45.
Last edited by Bunuel on 14 Dec 2012, 05:52, edited 1 time in total.
Renamed the topic and edited the question.
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What is the volume of a certain rectangular solid? [#permalink] New post   31 Aug 2010, 06:26
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This question is from the Official Guide and the Official Answer is C.

About the rectangular solid:



In a rectangular solid, all angles are right angles, and opposite faces are equal, so a rectangular solid can have a maximum of 3 different areas for its faces; on the diagram: yellow, green, and red faces can have different areas. I say at max because, for example, a rectangular solid can be a cube and in this case, it'll have all faces equal. Also, it's possible to have only 2 different areas for the faces, for example, when the base is square and the height does not equal the side of this square.

The volume of a rectangular solid is Volume = Length * Height * Depth.

BACK TO THE ORIGINAL QUESTION:

What is the volume of a certain rectangular solid?

(1) Two adjacent faces of the solid have areas 15 and 24, respectively.

Let the two adjacent faces be the blue and yellow faces on the diagram --> \(blue = dh = 15\) and \(yellow = lh = 24\) --> we have 2 equations with 3 unknowns; not sufficient to calculate the value of each or the product of the unknowns (\(V = lhd\)).

To elaborate more:
If \(blue = dh = 151 = 15\) and \(yellow = lh = 241 = 24\) then \(V = lhd = 24115 = 360\);
If \(blue = dh = 53 = 15\) and \(yellow = lh = 83 = 24\) then \(V = lhd = 835 = 120\).

Two different answer, hence not sufficient.

(2) Each of two opposite faces of the solid has area 40.

The above just gives the areas of two opposite faces, so clearly insufficient.

(1)+(2) From (1): \(blue = dh = 15\), \(yellow = lh = 24\), and from (2) each of two opposite faces of the solid has an area of 40, so it must be the red one: \(red = dl = 40\) --> here we have 3 distinct linear equations with 3 unknowns; hence we can find the values of each and thus can calculate \(V = lhd\). Sufficient.

To show how it can be done: multiply these 3 equations --> \(l^2h^2d^2 = (lhd)^2 = 15*24*40 = 24^25^2\) --> \(V = lhd = 24*5 = 120\).

Answer: C.

Hope it helps.

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Re: OG-12 DS # 122 [#permalink] New post   24 Feb 2010, 08:27
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1) Two adjacent faces of the slid have areas 15 & 24, respectively.

2) Each of the opposite faces of the solid has area 40.

Let the length of the sides be a,b,c. Areas of each of the sides will be ab,bc,ca and the volume is abc

1. ab=15 ac=24 From these we cannot get individual values of a,b,c or the volume. So INSUFFICIENT

2. bc=40. From this we cannot get the volume. -- INSUFFICIENT

Combining we have:

ab=15
ac=24
bc=40

multiplying the LHS and RHS, you get:

(ab)(ac)(bc)=15*24*40
(a^2)(b^2)(c^2)=14400
or abc=120=Volume

So, C is the answer
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Re: OG-12 DS # 122 [#permalink] New post   31 Aug 2010, 14:40
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@ Bunuel - ha! amazing! : )
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Re: OG-12 DS # 122 [#permalink] New post   06 Oct 2010, 07:35
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Yes, the second statement is confusing but the catch is in the question itself. The question states a rectangular solid. All the faces cannot have an area of 40 if the solid is rectangular.
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Re: OG-12 DS # 122 [#permalink] New post   Updated on: 10 Dec 2013, 09:41
Hi Bunuel,

I understand everything you have done right up until the point where you solve 1+2) Wouldn't the equation be (15*40*24)^2?

Bunuel wrote:
This question is from Official Guide and Official Answer is C.

About rectangular solid:
Attachment:
800px-Cuboid.png

In a rectangular solid, all angles are right angles, and opposite faces are equal, so rectangular solid can have maximum 3 different areas of its faces, on the diagram: yellow, green and red faces can have different areas. I say at max, as for example rectangular solid can be a cube and in this case it'll have all faces equal, also it's possible to have only 2 different areas of the faces, for example when the base is square and the height does not equals to the side of this square.

Volume of rectangular solid is Volume=Length*Height*Depth.

BACK TO THE ORIGINAL QUESTION:

What is the volume of a certain rectangular solid?

(1) Two adjacent faces of the solid have areas 15 and 24, respectively --> let the two adjacent faces be blue and yellow faces on the diagram --> \(blue=d*h=15\) and \(yellow=l*h=24\) --> we have 2 equations with 3 unknowns, not sufficient to calculate the value of each or the product of the unknowns (\(V=l*h*d\)).

To elaborate more:
If \(blue=d*h=15*1=15\) and \(yellow=l*h=24*1=24\) then \(V=l*h*d=24*1*15=360\);
If \(blue=d*h=5*3=15\) and \(yellow=l*h=8*3=24\) then \(V=l*h*d=8*3*5=90\).

Two different answer, hence not sufficient.

(2) Each of two opposite faces of the solid has area 40 --> just gives the are of two opposite faces, so clearly insufficient.

(1)+(2) From (1): \(blue=d*h=15\), \(yellow=l*h=24\) and from (2) each of two opposite faces of the solid has area 40, so it must be the red one: \(red=d*l=40\) --> here we have 3 distinct linear equations with 3 unknowns hence we can find the values of each and thus can calculate \(V=l*h*d\). Sufficient.

To show how it can be done: multiply these 3 equations --> \(l^2*h^2*d^2=(l*h*d)^2=15*24*40=24^2*5^2\) --> \(V=l*h*d=24*5=120\).

Answer: C.

Hope it helps.

Originally posted by WholeLottaLove on 10 Dec 2013, 09:35.
Last edited by WholeLottaLove on 10 Dec 2013, 09:41, edited 1 time in total.
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Re: OG-12 DS # 122 [#permalink] New post   10 Dec 2013, 09:41
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WholeLottaLove wrote:
Hi Bunuel,

I understand everything you have done right up until the point where you solve 1+2) Could you elaborate a bit?

Bunuel wrote:
This question is from Official Guide and Official Answer is C.

About rectangular solid:
Attachment:
800px-Cuboid.png

In a rectangular solid, all angles are right angles, and opposite faces are equal, so rectangular solid can have maximum 3 different areas of its faces, on the diagram: yellow, green and red faces can have different areas. I say at max, as for example rectangular solid can be a cube and in this case it'll have all faces equal, also it's possible to have only 2 different areas of the faces, for example when the base is square and the height does not equals to the side of this square.

Volume of rectangular solid is Volume=Length*Height*Depth.

BACK TO THE ORIGINAL QUESTION:

What is the volume of a certain rectangular solid?

(1) Two adjacent faces of the solid have areas 15 and 24, respectively --> let the two adjacent faces be blue and yellow faces on the diagram --> \(blue=d*h=15\) and \(yellow=l*h=24\) --> we have 2 equations with 3 unknowns, not sufficient to calculate the value of each or the product of the unknowns (\(V=l*h*d\)).

To elaborate more:
If \(blue=d*h=15*1=15\) and \(yellow=l*h=24*1=24\) then \(V=l*h*d=24*1*15=360\);
If \(blue=d*h=5*3=15\) and \(yellow=l*h=8*3=24\) then \(V=l*h*d=8*3*5=90\).

Two different answer, hence not sufficient.

(2) Each of two opposite faces of the solid has area 40 --> just gives the are of two opposite faces, so clearly insufficient.

(1)+(2) From (1): \(blue=d*h=15\), \(yellow=l*h=24\) and from (2) each of two opposite faces of the solid has area 40, so it must be the red one: \(red=d*l=40\) --> here we have 3 distinct linear equations with 3 unknowns hence we can find the values of each and thus can calculate \(V=l*h*d\). Sufficient.

To show how it can be done: multiply these 3 equations --> \(l^2*h^2*d^2=(l*h*d)^2=15*24*40=24^2*5^2\) --> \(V=l*h*d=24*5=120\).

Answer: C.

Hope it helps.


When we combine the statements we have:
\(blue=d*h=15\).

\(yellow=l*h=24\).

\(red=d*l=40\).

Multiply these 3 equations --> \(l^2*h^2*d^2=(l*h*d)^2=15*24*40=24^2*5^2\) --> \(V=l*h*d=24*5=120\).

Hope it's clear.
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Re: OG-12 DS # 122 [#permalink] New post   10 Dec 2013, 09:45
Hello,

I tried to edit my question in time to better specify my question. I see that in 1 and 2 there are two equations with three unknowns whereas here, we have equations for length, depth, height. You multiply 15*40*24 but according to your formula, don't you multiply them together then square that result? Then how do you go from 24^2*5^2 to 24*5? I see that you take the root but why is that done and how in the context of this problem?

Thanks!

Bunuel wrote:
WholeLottaLove wrote:
Hi Bunuel,

I understand everything you have done right up until the point where you solve 1+2) Could you elaborate a bit?

Bunuel wrote:
This question is from Official Guide and Official Answer is C.

About rectangular solid:
Attachment:
800px-Cuboid.png

In a rectangular solid, all angles are right angles, and opposite faces are equal, so rectangular solid can have maximum 3 different areas of its faces, on the diagram: yellow, green and red faces can have different areas. I say at max, as for example rectangular solid can be a cube and in this case it'll have all faces equal, also it's possible to have only 2 different areas of the faces, for example when the base is square and the height does not equals to the side of this square.

Volume of rectangular solid is Volume=Length*Height*Depth.

BACK TO THE ORIGINAL QUESTION:

What is the volume of a certain rectangular solid?

(1) Two adjacent faces of the solid have areas 15 and 24, respectively --> let the two adjacent faces be blue and yellow faces on the diagram --> \(blue=d*h=15\) and \(yellow=l*h=24\) --> we have 2 equations with 3 unknowns, not sufficient to calculate the value of each or the product of the unknowns (\(V=l*h*d\)).

To elaborate more:
If \(blue=d*h=15*1=15\) and \(yellow=l*h=24*1=24\) then \(V=l*h*d=24*1*15=360\);
If \(blue=d*h=5*3=15\) and \(yellow=l*h=8*3=24\) then \(V=l*h*d=8*3*5=90\).

Two different answer, hence not sufficient.

(2) Each of two opposite faces of the solid has area 40 --> just gives the are of two opposite faces, so clearly insufficient.

(1)+(2) From (1): \(blue=d*h=15\), \(yellow=l*h=24\) and from (2) each of two opposite faces of the solid has area 40, so it must be the red one: \(red=d*l=40\) --> here we have 3 distinct linear equations with 3 unknowns hence we can find the values of each and thus can calculate \(V=l*h*d\). Sufficient.

To show how it can be done: multiply these 3 equations --> \(l^2*h^2*d^2=(l*h*d)^2=15*24*40=24^2*5^2\) --> \(V=l*h*d=24*5=120\).

Answer: C.

Hope it helps.


When we combine the statements we have:
\(blue=d*h=15\)
\(yellow=l*h=24\)
\(red=d*l=40\)

Multiply these 3 equations --> \(l^2*h^2*d^2=(l*h*d)^2=15*24*40=24^2*5^2\) --> \(V=l*h*d=24*5=120\).

Hope it's clear.
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Re: OG-12 DS # 122 [#permalink] New post   10 Dec 2013, 09:52
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WholeLottaLove wrote:
Hello,

I tried to edit my question in time to better specify my question. I see that in 1 and 2 there are two equations with three unknowns whereas here, we have equations for length, depth, height. You multiply 15*40*24 but according to your formula, don't you multiply them together then square that result? Then how do you go from 24^2*5^2 to 24*5? I see that you take the root but why is that done and how in the context of this problem?

Thanks!

Bunuel wrote:
WholeLottaLove wrote:
Hi Bunuel,

I understand everything you have done right up until the point where you solve 1+2) Could you elaborate a bit?



When we combine the statements we have:
\(blue=d*h=15\)
\(yellow=l*h=24\)
\(red=d*l=40\)

Multiply these 3 equations --> \(l^2*h^2*d^2=(l*h*d)^2=15*24*40=24^2*5^2\) --> \(V=l*h*d=24*5=120\).

Hope it's clear.


\(Volume=l*h*d\).

Now, if you multiply the 3 equations we have we get \(l^2*h^2*d^2*=(lhd)^2=15*24*40=24^2*5^2\) --> \(Volume=l*h*d\), thus the volume is the square root of \((lhd)^2=24^2*5^2\), so 24*5=120.

Hope it's clear.
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Re: What is the volume of a certain rectangular solid? [#permalink] New post   01 Feb 2014, 19:49
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This is more like a verbal question on meaning. Trick here is statement2 which makes this question hard to understand and easy to get it wrong.

(2) Each of two opposite faces of the solid has area 40.
===>someone can interpret this as pick any 2 opposite sides the area is 40. or exactly 2 opposite sides have areas of 40 each. but if you look at statement one then the latter makes more sense, thats the trick. Since OG question do not contradict on Statement1 and 2, this is an example of how you can verify the statement more carefully just by knowing what happened in 1. But remember while evaluating 2 do not bring that concept in 1. that seems hard to do in this question.

so for me this question is dubious, although answer should be C.
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Re: What is the volume of a certain rectangular solid? [#permalink] New post   04 Jun 2014, 03:00
Hi bunuel,
Though your explanation is correct but st#2 is confusing
It says:
Each of two opposite faces of the solid has area 40.

"Each" has area 40. It means from your fig, Red will have area =40 , but Blue face and (opposite to it face) both can also have area 40..

Thats how it is confusing. Can you clarify
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Re: What is the volume of a certain rectangular solid? [#permalink] New post   04 Jun 2014, 03:12
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nidhi12 wrote:
Hi bunuel,
Though your explanation is correct but st#2 is confusing
It says:
Each of two opposite faces of the solid has area 40.

"Each" has area 40. It means from your fig, Red will have area =40 , but Blue face and (opposite to it face) both can also have area 40..

Thats how it is confusing. Can you clarify


Does it matter which two opposite faces have the area 40?

"Each of two opposite faces of the solid has area 40" means that one pair of opposite faces has area 40.
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What is the volume of a certain rectangular solid? [#permalink] New post   Updated on: 14 Oct 2020, 06:00
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amod243 wrote:
What is the volume of a certain rectangular solid?

(1) Two adjacent faces of the solid have areas 15 and 24, respectively.
(2) Each of two opposite faces of the solid has area 40.


Target question: What is the volume of a certain rectangular solid?

Aside: A rectangular solid is a box

Statement 1: Two adjacent faces of the solid have areas 15 and 24, respectively.
There are several different rectangular solids that meet this condition. Here are two:
Case a: the dimensions are 1x15x24, in which case the volume is 360
Case b: the dimensions are 3x5x8, in which case the volume is 120
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: Each of two opposite faces of the solid has area 40.
So, there are two opposite faces that each have area 40.
Definitely NOT SUFFICIENT

Statements 1 and 2 combined:
So, we know the area of each face (noted in blue on the diagram below).
Let's let x equal the length of one side.



Since the area of each face = (length)(width), we can express the other two dimensions in terms of x.


From here, we'll focus on the face that has area 40.
This face has dimensions (15/x) by (24/x)
Since the area is 40, we know that (15/x)(24/x) = 40
Expand: 360/(x^2) = 40
Simplify: 360 = 40x^2
Simplify: 9 = x^2
Solve: x = 3 or -3
Since the side lengths must be positive, we can be certain that x = 3

When we plug x=3 into the other two dimensions, we get 15/3 and 24/3
So, the 3 dimensions are 3, 5, and 8, which means the volume of the rectangular solid must be 120.
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer =
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C


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Originally posted by BrentGMATPrepNow on 30 Jul 2016, 07:46.
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Re: What is the volume of a certain rectangular solid? [#permalink] New post   17 May 2017, 23:55
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Bunuel wrote:
This question is from Official Guide and Official Answer is C.

About rectangular solid:



In a rectangular solid, all angles are right angles, and opposite faces are equal, so rectangular solid can have maximum 3 different areas of its faces, on the diagram: yellow, green and red faces can have different areas. I say at max, as for example rectangular solid can be a cube and in this case it'll have all faces equal, also it's possible to have only 2 different areas of the faces, for example when the base is square and the height does not equals to the side of this square.

Volume of rectangular solid is Volume=Length*Height*Depth.

BACK TO THE ORIGINAL QUESTION:

What is the volume of a certain rectangular solid?

(1) Two adjacent faces of the solid have areas 15 and 24, respectively --> let the two adjacent faces be blue and yellow faces on the diagram --> \(blue=d*h=15\) and \(yellow=l*h=24\) --> we have 2 equations with 3 unknowns, not sufficient to calculate the value of each or the product of the unknowns (\(V=l*h*d\)).

To elaborate more:
If \(blue=d*h=15*1=15\) and \(yellow=l*h=24*1=24\) then \(V=l*h*d=24*1*15=360\);
If \(blue=d*h=5*3=15\) and \(yellow=l*h=8*3=24\) then \(V=l*h*d=8*3*5=90\).

Two different answer, hence not sufficient.

(2) Each of two opposite faces of the solid has area 40 --> just gives the areas of two opposite faces, so clearly insufficient.

(1)+(2) From (1): \(blue=d*h=15\), \(yellow=l*h=24\) and from (2) each of two opposite faces of the solid has area 40, so it must be the red one: \(red=d*l=40\) --> here we have 3 distinct linear equations with 3 unknowns hence we can find the values of each and thus can calculate \(V=l*h*d\). Sufficient.

To show how it can be done: multiply these 3 equations --> \(l^2*h^2*d^2=(l*h*d)^2=15*24*40=24^2*5^2\) --> \(V=l*h*d=24*5=120\).

Answer: C.

Hope it helps.

Show Spoiler
Attachment:
800px-Cuboid.png



Hi Bunuel,

Although I understand your solution, I'm still a little confused with the 2nd statement. On the GMAT, doesn't each mean All ? I mean, if a random statement is worded as : Each of the students got 5 dollars, doesn't it mean all the students got 5 dollars ? Also, isn't square just a special kind of a rectangle ?

Please help. Thanks in advance!
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Re: What is the volume of a certain rectangular solid? [#permalink] New post   18 May 2017, 00:09
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Shruti0805 wrote:
Bunuel wrote:
This question is from Official Guide and Official Answer is C.

About rectangular solid:



In a rectangular solid, all angles are right angles, and opposite faces are equal, so rectangular solid can have maximum 3 different areas of its faces, on the diagram: yellow, green and red faces can have different areas. I say at max, as for example rectangular solid can be a cube and in this case it'll have all faces equal, also it's possible to have only 2 different areas of the faces, for example when the base is square and the height does not equals to the side of this square.

Volume of rectangular solid is Volume=Length*Height*Depth.

BACK TO THE ORIGINAL QUESTION:

What is the volume of a certain rectangular solid?

(1) Two adjacent faces of the solid have areas 15 and 24, respectively --> let the two adjacent faces be blue and yellow faces on the diagram --> \(blue=d*h=15\) and \(yellow=l*h=24\) --> we have 2 equations with 3 unknowns, not sufficient to calculate the value of each or the product of the unknowns (\(V=l*h*d\)).

To elaborate more:
If \(blue=d*h=15*1=15\) and \(yellow=l*h=24*1=24\) then \(V=l*h*d=24*1*15=360\);
If \(blue=d*h=5*3=15\) and \(yellow=l*h=8*3=24\) then \(V=l*h*d=8*3*5=90\).

Two different answer, hence not sufficient.

(2) Each of two opposite faces of the solid has area 40 --> just gives the areas of two opposite faces, so clearly insufficient.

(1)+(2) From (1): \(blue=d*h=15\), \(yellow=l*h=24\) and from (2) each of two opposite faces of the solid has area 40, so it must be the red one: \(red=d*l=40\) --> here we have 3 distinct linear equations with 3 unknowns hence we can find the values of each and thus can calculate \(V=l*h*d\). Sufficient.

To show how it can be done: multiply these 3 equations --> \(l^2*h^2*d^2=(l*h*d)^2=15*24*40=24^2*5^2\) --> \(V=l*h*d=24*5=120\).

Answer: C.

Hope it helps.

Show Spoiler
Attachment:
800px-Cuboid.png



Hi Bunuel,

Although I understand your solution, I'm still a little confused with the 2nd statement. On the GMAT, doesn't each mean All ? I mean, if a random statement is worded as : Each of the students got 5 dollars, doesn't it mean all the students got 5 dollars ? Also, isn't square just a special kind of a rectangle ?

Please help. Thanks in advance!


Pay attention to the highlighted part:
"(2) Each of TWO opposite faces of the solid has area 40" means that one pair of opposite faces (two opposite faces) has an area 40.

As for your other question: yes, a square is a special type of a rectangle.

I suggest you to go through the previous pages of discussion where you can find several different ways of solving the question as well as answers to many questions and doubts.

Hope it helps.
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Re: What is the volume of a certain rectangular solid? [#permalink] New post   29 Jun 2023, 10:05
[quote="Bunuel"]This question is from Official Guide and Official Answer is C.

Thank you in advance. I have 2 basic questions:

1. Why cant the sides be a fraction as well? say 30*0.5; 48*0.5 (or) 10*1.5; 16*1.5? The question doesn't mention they are whole numbers.

2. Also, in which conditions can assume the numbers used are whole numbers and not fractions. I have seen in DS we dont assume the format of numbers (can be rational/ irrational)?

Tried searching for answer on #2 more widely but to no avail. Hoping to get some help on the above.

Thank you.
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Re: What is the volume of a certain rectangular solid? [#permalink] New post   05 Jul 2023, 11:55
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Tukkebaaz wrote:

Thank you in advance. I have 2 basic questions:

Why cant the sides be a fraction as well? say 30_0.5; 48_0.5 (or) 10_1.5; 16_1.5? The question doesn't mention they are whole numbers.

Also, in which conditions can assume the numbers used are whole numbers and not fractions. I have seen in DS we dont assume the format of numbers (can be rational/ irrational)?

Tried searching for answer on #2 more widely but to no avail. Hoping to get some help on the above.

Thank you.

You can't ever assume whole numbers unless that condition is specifically given. In this case, you don't need to.

With both statements together, we're essentially given the area of each face (15, 15, 24, 24, 40, 40). That gives us three equations:

    xy = 15
    yz = 24
    xz = 40

Solving the above gives x = 5, y = 3, and z = 8. There aren't any fractions here, but that's just the way the math works out -- we don't need to make any assumptions about whole numbers.

I hope that helps!
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Re: What is the volume of a certain rectangular solid? [#permalink] New post   19 Aug 2023, 08:45
Hi Bunuel

I am dubious about the quality of this question.
A rectangle can be a square but a square cannot be a rectangle. From statement 2 one can infer the cuboid to be a cube?
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Re: What is the volume of a certain rectangular solid? [#permalink] New post   19 Aug 2023, 09:05
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Vegita wrote:
Hi Bunuel

I am dubious about the quality of this question.
A rectangle can be a square but a square cannot be a rectangle. From statement 2 one can infer the cuboid to be a cube?


This is an official question, which means it's the best of the best.

The read part is not correct: a square IS a rectangle. All squares are rectangles but not vice-versa.
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Re: What is the volume of a certain rectangular solid? [#permalink] New post   24 Aug 2023, 09:47
amod243 wrote:
What is the volume of a certain rectangular solid?

(1) Two adjacent faces of the solid have areas 15 and 24, respectively.
(2) Each of two opposite faces of the solid has area 40.


The second statement seems confusing, is it not telling that each face surface area is 40 m2 and as square is a special case of rectangle. Solid is a cube and statement 2 alone sufficient to answer?
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Re: What is the volume of a certain rectangular solid? [#permalink]
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