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# What is the volume of a certain rectangular solid?

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Re: What is the volume of a certain rectangular solid?  [#permalink]

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08 Dec 2014, 07:32
kevin627 wrote:
deepbidwai wrote:
Information in (2) is redundant as from (1) we can know all required info.
(1) Two adjacent faces of the solid have areas 15 and 24, respectively.
So, one common side has to be 3. i.e. 5*3 and 8*3. From this info we can say area of the face is 8*5= 40.
Hence only (1) is sufficient.
Answer should be A and not C.

DeepakB

I also don't understand why A is not sufficient.

We need to determine the length of 3 sides.

If two adjacent faces share a side with respective areas 15 and 24, the only common prime they share is 3, which must be their shared side length.

If the area of one face is 15 and one side is 3, then the other side is 5.
If the area of one face is 24 and one side is 3, then the other side is 8.

Therefore, we have three side lengths: 5, 8, 3

Perhaps my mistake is assuming that the dimensions must be integers?

Have you read this: what-is-the-volume-of-a-certain-rectangular-solid-90748.html#p772095 ???
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Re: What is the volume of a certain rectangular solid?  [#permalink]

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08 Dec 2014, 08:06
Bunuel wrote:
kevin627 wrote:
deepbidwai wrote:
Information in (2) is redundant as from (1) we can know all required info.
(1) Two adjacent faces of the solid have areas 15 and 24, respectively.
So, one common side has to be 3. i.e. 5*3 and 8*3. From this info we can say area of the face is 8*5= 40.
Hence only (1) is sufficient.
Answer should be A and not C.

DeepakB

I also don't understand why A is not sufficient.

We need to determine the length of 3 sides.

If two adjacent faces share a side with respective areas 15 and 24, the only common prime they share is 3, which must be their shared side length.

If the area of one face is 15 and one side is 3, then the other side is 5.
If the area of one face is 24 and one side is 3, then the other side is 8.

Therefore, we have three side lengths: 5, 8, 3

Perhaps my mistake is assuming that the dimensions must be integers?

I did see your post before. After reading more carefully, I can clearly see why (1) is insufficient. Thanks
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What is the volume of a certain rectangular solid?  [#permalink]

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06 Jan 2015, 16:55
Where am I going wrong on this?

If a side has area of 15 can't we factor to say that the side must be some combo of factors and since they are adjacent they must share a factor for the like side...so if 15 was 3x5 then 24 must be 8x3. Through this we get 8x3x5 for area?

I know I am wrong this was my initial reaction the question and trying to avoid it. Is it chance for fractional sides that makes this solution incorrect?
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Re: What is the volume of a certain rectangular solid?  [#permalink]

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06 Jul 2015, 01:50
Bunuel wrote:
This question is from Official Guide and Official Answer is C.

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 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 --> $$l^2*h^2*d^2=(l*h*d)^2=15*24*40=24^2*5^2$$ --> $$V=l*h*d=24*5=120$$.

Hope it helps.

Where am I wrong-

a x b = 15
b x c = 24

a x b = 5 x 3
b x c = 3 x 2 x 2 x 2

I have broken into prime numbers.

3 is common, so b must be 3.

a = 5
c = 8

we can find the area now? where I am going wrong? according to me statement A is sufficient.
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Re: What is the volume of a certain rectangular solid?  [#permalink]

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06 Jul 2015, 07:45
honchos wrote:
Bunuel wrote:
This question is from Official Guide and Official Answer is C.

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 --> $$l^2*h^2*d^2=(l*h*d)^2=15*24*40=24^2*5^2$$ --> $$V=l*h*d=24*5=120$$.

Hope it helps.

Where am I wrong-

a x b = 15
b x c = 24

a x b = 5 x 3
b x c = 3 x 2 x 2 x 2

I have broken into prime numbers.

3 is common, so b must be 3.

a = 5
c = 8

we can find the area now? where I am going wrong? according to me statement A is sufficient.

Frankly, I don't know what to add after 3 pages of discussion...

In the very post you are quoting are TWO examples, which give TWO different answers.
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What is the volume of a certain rectangular solid?  [#permalink]

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09 Jul 2015, 13:17
Please correct me if I' wrong, it's just my observation:
I won't discuss why St1 and St2 alonne are not sufficient, as Bunuel have aleady shown some examples with yes/no outcome.
But St1+St2) I a solid rectangular all 3 different faces CAN NOT be adjacent on one side - so "1 feet" can not be can not be shae by all faces, hence --> let's find a common factors for all the areas --> 15=3*5 , 40=8*5, 24=3*8 as you see we have the numbe 3,5 and 8 just multiply them 3*5*8=120.

Here's is impotant to notice, if by St1) we haf some options 15*1 or 5*3 when both statements combined, we can eliminate 15*1, it can be only 5*3........... hope that was somehow clear...
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What is the volume of a certain rectangular solid?  [#permalink]

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28 Sep 2015, 17:48
Bunuel wrote:
This question is from Official Guide and Official Answer is C.

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 --> $$l^2*h^2*d^2=(l*h*d)^2=15*24*40=24^2*5^2$$ --> $$V=l*h*d=24*5=120$$.

Hope it helps.

Attachment:
800px-Cuboid.png

Sorry to bring this topic up again. I went through the discussion of this question but I'm still not satisfied with the explanation of statement 2.
Statement 2 seems a bit too ambiguous to me.
If we consider statement 1 to analyse statement 2, then the 2 opposite faces are the faces which were not included in statement 1.
If this is true, then solution above is perfect. Although it doesn't seem logical to me to consider statement 1 for analysing statement 2.
Let us not consider statement 1 at all and just focus on statement 2.
It says "each of the two opposite faces" has area 40.
There are three pairs of opposite faces. Red face and it's opposite face, blue face and it's opposite face and, yellow face and it's opposite face.
Question says each of the two opposite faces has area 40. So, each of the 6 faces must have area = 40.
Let l,b and h be the length, breadth and height respectively. Then, lb=40, bh = 40 and lh = 40.
Now,
$$lb*bh*lh = l^2b^2h^2 = 40*40*40$$
or $$l^2b^2h^2=64000$$
or $$lbh = 80\sqrt{10}$$
SUFFICIENT

Why is my solution wrong ?
Why do we have to consider constraints in statement 1 to analyse statement 2 individually ?
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Re: What is the volume of a certain rectangular solid?  [#permalink]

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28 Sep 2015, 19:25
kunal555 wrote:
Bunuel wrote:
This question is from Official Guide and Official Answer is C.

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 --> $$l^2*h^2*d^2=(l*h*d)^2=15*24*40=24^2*5^2$$ --> $$V=l*h*d=24*5=120$$.

Hope it helps.

Attachment:
800px-Cuboid.png

Sorry to bring this topic up again. I went through the discussion of this question but I'm still not satisfied with the explanation of statement 2.
Statement 2 seems a bit too ambiguous to me.
If we consider statement 1 to analyse statement 2, then the 2 opposite faces are the faces which were not included in statement 1.
If this is true, then solution above is perfect. Although it doesn't seem logical to me to consider statement 1 for analysing statement 2.
Let us not consider statement 1 at all and just focus on statement 2.
It says "each of the two opposite faces" has area 40.
There are three pairs of opposite faces. Red face and it's opposite face, blue face and it's opposite face and, yellow face and it's opposite face.
Question says each of the two opposite faces has area 40. So, each of the 6 faces must have area = 40.
Let l,b and h be the length, breadth and height respectively. Then, lb=40, bh = 40 and lh = 40.
Now,
$$lb*bh*lh = l^2b^2h^2 = 40*40*40$$
or $$l^2b^2h^2=64000$$
or $$lbh = 80\sqrt{10}$$
SUFFICIENT

Why is my solution wrong ?
Why do we have to consider constraints in statement 1 to analyse statement 2 individually ?

Good question but what you are assuming is that every pair of opposite sides have area 40. Statement 2 talks about just 1 pair of opposite sides. Bunuel covers this at what-is-the-volume-of-a-certain-rectangular-solid-90748.html#p772623

The statement couldve done better to remove this ambiguity but as it is an official question, it is not my place to doubt the wording of the question.

Hope this helps.
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Re: What is the volume of a certain rectangular solid?  [#permalink]

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22 Jan 2016, 09:59
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.

MATH REVOLUTION VIDEO SOLUTION:

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What is the volume of a certain rectangular solid?  [#permalink]

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Updated on: 16 Apr 2018, 13:00
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Top Contributor
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

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Re: What is the volume of a certain rectangular solid?  [#permalink]

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23 Dec 2016, 11:31
tingle15 wrote:
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.

Why not? A cube falls under the definition of ''rectangular''.
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Re: What is the volume of a certain rectangular solid?  [#permalink]

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17 May 2017, 23:55
Bunuel wrote:
This question is from Official Guide and Official Answer is C.

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 --> $$l^2*h^2*d^2=(l*h*d)^2=15*24*40=24^2*5^2$$ --> $$V=l*h*d=24*5=120$$.

Hope it helps.

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 ?

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Re: What is the volume of a certain rectangular solid?  [#permalink]

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18 May 2017, 00:09
Shruti0805 wrote:
Bunuel wrote:
This question is from Official Guide and Official Answer is C.

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 --> $$l^2*h^2*d^2=(l*h*d)^2=15*24*40=24^2*5^2$$ --> $$V=l*h*d=24*5=120$$.

Hope it helps.

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 ?

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]

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28 Jul 2017, 08:04
Total different faces of cubes is 3

Statmnt 1 area of 2 faces given

Statmnt 2 area of 1 face given

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Re: What is the volume of a certain rectangular solid?  [#permalink]

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30 Aug 2017, 06:56
Bunuel
The second statement is ambiguous as each of the two opposite faces means any two opposite faces and note that rectangular solid is also a cube solid.
this question is ambiguous and hence is not correct basically.
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Re: What is the volume of a certain rectangular solid?  [#permalink]

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30 Aug 2017, 07:02
parham wrote:
Bunuel
The second statement is ambiguous as each of the two opposite faces means any two opposite faces and note that rectangular solid is also a cube solid.
this question is ambiguous and hence is not correct basically.

I understand what you mean but don't agree. I tried to explain this question several times on previous 3 pages, so I've already said what I had to say. This is an Official Guide question though, so you are not agreeing with them...
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