Last visit was: 17 Jun 2024, 00:03 It is currently 17 Jun 2024, 00:03
Toolkit
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

# What is the volume of a certain rectangular solid?

SORT BY:
Tags:
Show Tags
Hide Tags
Intern
Joined: 19 Feb 2009
Posts: 32
Own Kudos [?]: 582 [321]
Given Kudos: 8
Math Expert
Joined: 02 Sep 2009
Posts: 93697
Own Kudos [?]: 632367 [132]
Given Kudos: 82322
Manager
Joined: 10 Feb 2010
Posts: 92
Own Kudos [?]: 692 [41]
Given Kudos: 6
General Discussion
Manager
Joined: 16 Feb 2010
Posts: 122
Own Kudos [?]: 1499 [1]
Given Kudos: 16
Re: OG-12 DS # 122 [#permalink]
1
Kudos
@ Bunuel - ha! amazing! : )
Manager
Joined: 15 Apr 2010
Posts: 83
Own Kudos [?]: 343 [6]
Given Kudos: 3
Re: OG-12 DS # 122 [#permalink]
6
Kudos
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.
Senior Manager
Joined: 13 May 2013
Posts: 312
Own Kudos [?]: 568 [0]
Given Kudos: 134
Re: OG-12 DS # 122 [#permalink]
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.

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

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.
Math Expert
Joined: 02 Sep 2009
Posts: 93697
Own Kudos [?]: 632367 [1]
Given Kudos: 82322
Re: OG-12 DS # 122 [#permalink]
1
Bookmarks
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.

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

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.
Senior Manager
Joined: 13 May 2013
Posts: 312
Own Kudos [?]: 568 [0]
Given Kudos: 134
Re: OG-12 DS # 122 [#permalink]
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.

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

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.
Math Expert
Joined: 02 Sep 2009
Posts: 93697
Own Kudos [?]: 632367 [0]
Given Kudos: 82322
Re: OG-12 DS # 122 [#permalink]
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.
Manager
Joined: 19 Sep 2008
Status:Please do not forget to give kudos if you like my post
Posts: 69
Own Kudos [?]: 206 [5]
Given Kudos: 257
Location: United States (CA)
Re: What is the volume of a certain rectangular solid? [#permalink]
5
Kudos
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.
Intern
Joined: 24 May 2013
Posts: 16
Own Kudos [?]: 55 [0]
Given Kudos: 21
Concentration: Operations, General Management
WE:Engineering (Energy and Utilities)
Re: What is the volume of a certain rectangular solid? [#permalink]
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
Math Expert
Joined: 02 Sep 2009
Posts: 93697
Own Kudos [?]: 632367 [0]
Given Kudos: 82322
Re: What is the volume of a certain rectangular solid? [#permalink]
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.
GMAT Club Legend
Joined: 12 Sep 2015
Posts: 6814
Own Kudos [?]: 30567 [5]
Given Kudos: 799
What is the volume of a certain rectangular solid? [#permalink]
5
Kudos
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

RELATED VIDEO

Originally posted by BrentGMATPrepNow on 30 Jul 2016, 07:46.
Last edited by BrentGMATPrepNow on 14 Oct 2020, 06:00, edited 2 times in total.
Manager
Joined: 24 Dec 2016
Posts: 93
Own Kudos [?]: 45 [1]
Given Kudos: 145
Location: India
Concentration: Finance, General Management
WE:Information Technology (Computer Software)
Re: What is the volume of a certain rectangular solid? [#permalink]
1
Kudos
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 ?

Math Expert
Joined: 02 Sep 2009
Posts: 93697
Own Kudos [?]: 632367 [1]
Given Kudos: 82322
Re: What is the volume of a certain rectangular solid? [#permalink]
1
Bookmarks
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.
Manager
Joined: 30 Apr 2023
Status:A man of Focus, Commitment and Shear Will
Posts: 136
Own Kudos [?]: 58 [0]
Given Kudos: 72
Location: India
GMAT 1: 710 Q50 V36
WE:Investment Banking (Investment Banking)
Re: What is the volume of a certain rectangular solid? [#permalink]
[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.
@Benuel GMATNinja AndrewN avigutman
GMAT Club Verbal Expert
Joined: 13 Aug 2009
Status: GMAT/GRE/LSAT tutors
Posts: 6950
Own Kudos [?]: 64244 [2]
Given Kudos: 1803
Location: United States (CO)
GMAT 1: 780 Q51 V46
GMAT 2: 800 Q51 V51
GRE 1: Q170 V170

GRE 2: Q170 V170
Re: What is the volume of a certain rectangular solid? [#permalink]
2
Kudos
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!
Manager
Joined: 23 May 2020
Posts: 90
Own Kudos [?]: 10 [0]
Given Kudos: 1526
Re: What is the volume of a certain rectangular solid? [#permalink]
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?
Math Expert
Joined: 02 Sep 2009
Posts: 93697
Own Kudos [?]: 632367 [0]
Given Kudos: 82322
Re: What is the volume of a certain rectangular solid? [#permalink]
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.
Intern
Joined: 27 Feb 2022
Posts: 37
Own Kudos [?]: 3 [0]
Given Kudos: 33
Re: What is the volume of a certain rectangular solid? [#permalink]
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?
Re: What is the volume of a certain rectangular solid? [#permalink]
Moderator:
Math Expert
93698 posts