Fraction of the painted area

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gmattesttaker2: Legendary Member; Posts: 641; Joined: Tue Feb 14, 2012 3:52 pmFollowed by: 8 members; Thanked: 11 times

by gmattesttaker2 » Sat Feb 22, 2014 12:20 am

Hello,

Can you please assist with this:

The rectangular solid above is made up of eight cubes of the same size, each of which has exactly one face painted blue. What is the greatest fraction of the total surface area of the solid that could be blue?

A) 1/6
B) 3/14
C) 1/4
D) 2/7
E) 1/3

OA: D

However, I am getting A

I tried to solve as follows:

The area of each face of the cube is e^2, where e is the edge of the cube. Hence, let the face that is painted blue be of area e^2.

Total painted area = 8 (e^2)
Total area of all the cubes = 8 (6e^2)

Hence, greatest fraction of the total surface area painted blue = (8 e^2) / (8 ( 6e^2)) = [spoiler]1/6[/spoiler]

Thanks,
Sri

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Source: Beat The GMAT — Problem Solving | Sat Feb 22, 2014 12:20 am

theCodeToGMAT: Legendary Member; Posts: 1555; Joined: Tue Aug 14, 2012 11:18 pmFollowed by: 34 members; Thanked: 448 times; GMAT Score:650

by theCodeToGMAT » Sat Feb 22, 2014 2:53 am

Let cube has side "1"

Total Surface Area = 2 (4*1 + 1*2 + 2*4) = 2 (14) = 28

Colored Area = 4*2 = 8

Fraction = 8/28 = 2/7

[spoiler]{D}[/spoiler]

R A H U L

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Sat Feb 22, 2014 3:36 am

gmattesttaker2 wrote:Hello,

Can you please assist with this:

The rectangular solid above is made up of eight cubes of the same size, each of which has exactly one face painted blue. What is the greatest fraction of the total surface area of the solid that could be blue?

A) 1/6
B) 3/14
C) 1/4
D) 2/7
E) 1/3

OA: D

Surface area:
Number of faces in the front = 8.
Number of faces in the back = 8.
Number of faces on top = 4.
Number of faces on the bottom = 4.
Number of faces on the left = 2.
Number of faces on the right = 2.
Total number of faces = 8+8+4+4+2+2 = 28.

Blue faces:
Since each of the 8 cubes has exactly 1 face painted blue, the greatest number of blue faces that could contribute to the surface area = 8.

Resulting fraction:
8/28 = 2/7.

The correct answer is D.

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by [email protected] » Sat Feb 22, 2014 6:37 pm

Hi Sri,

Both Rahul and Mitch have properly explained the "math" behind this question, so I won't rehash that here. Instead, I'll walk you through the mistake that you made in your calculation.

You are correct that with 8 "faces" painted blue, the AREA of the greatest number of exposed faces would = 8(e^2). However, the TOTAL surface area of the shape in the picture is NOT 8(6e^2).

Notice how some of the "faces" are touching one another (and thus are NOT exposed). Those "touching" faces have to be removed from the calculation.

From the drawing, you can see that...
each of the 4 "end" squares has 2 "touching faces" and
each of the 4 "middle" squares has 3 "touching faces.

4(2) + 4(3) = 20. That means you have to remove 20 of the 48 possible faces. This would give us 28 possible faces.

Your math would then be correct:

8(e^2)/28(e^2) = 8/28 = 2/7

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