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by oquiella » Wed Dec 23, 2015 3:49 pm
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 wo opposite faces of the solid has area 40.


Explain reasoning
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Source: — Data Sufficiency |

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by Brent@GMATPrepNow » Wed Dec 23, 2015 4:10 pm
oquiella 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.
Image


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

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 = C

Cheers,
Brent
Last edited by Brent@GMATPrepNow on Mon Apr 16, 2018 12:50 pm, edited 2 times in total.
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by Amrabdelnaby » Sat Jan 02, 2016 4:06 am
Ok. I had a very different approach that made me think that statement 1 is sufficient.

Please tell me what i did wrong here.

I thought of 2 adjacent sides have areas of 15 and 24 then the following must be true:

LW = 15 or 24

WH = the other number

LW x WH = 15 x 24

the prime factorization of 15 and 24 yields to 3,3,2,2,2,5

Hence since LWxWh is LHW^2 then W^2 must be 3x3

Hence the width is equal to 3

so the volume of the triangle will be equal to 5x2x2x2x3, regardless of the allocation of those numbers to the lengths of the other sides.

for example if L is 10 and W is 4 or L is 20 and W is 2 the multiplication of those numbers will yield to the same volume. and hence I concluded that statement one is sufficient.

Why am I wrong????

PLs advise
Brent@GMATPrepNow wrote:
oquiella 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.
Image


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

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 = C

Cheers,
Brent
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by [email protected] » Sat Jan 02, 2016 10:35 am
Hi amrabdelnaby,

This DS question asks us for the VOLUME of the rectangular solid, so we need to know the length, width and height of this shape. If those numbers can change, then the volume will almost certainly change.

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

Fact 1 gives us the areas of two adjacent faces, but we do NOT know the exact dimensions...

IF....
the dimensions are 1x15 and 1x24, then the volume = (1)(15)(24) = 360
the dimensions are 5x3 and 3x8, then the volume = (5)(3)(8) = 120
Fact 1 is INSUFFICIENT

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by Matt@VeritasPrep » Fri Jan 08, 2016 2:25 pm
Amrabdelnaby wrote:Ok. I had a very different approach that made me think that statement 1 is sufficient.

Please tell me what i did wrong here.
I like the idea, but you can't assume that the sides are integers, especially not factors of 15 and 24. (The test WANTS you to assume this -- it would be nice if it were true -- but it doesn't have to be.)

Algebra could save us here: we have

WH = 15
HL = 24

So we could have H = 9, W = (15/9), and L = (24/9), or some other ghastly combination.
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