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# A rectangular wooden dowel measures 4 inches by 1 inch by 1 inch. If

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Math Expert
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A rectangular wooden dowel measures 4 inches by 1 inch by 1 inch. If [#permalink]

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17 Jun 2015, 04:55
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A rectangular wooden dowel measures 4 inches by 1 inch by 1 inch. If the dowel is painted on all surfaces and then cut into 1/2 inch cubes, what fraction of the resulting cube faces are painted?

(A) 1/3
(B) 3/8
(C) 7/16
(D) 1/2
(E) 9/16

Kudos for a correct solution.
[Reveal] Spoiler: OA

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Re: A rectangular wooden dowel measures 4 inches by 1 inch by 1 inch. If [#permalink]

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17 Jun 2015, 05:21
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Bunuel wrote:
A rectangular wooden dowel measures 4 inches by 1 inch by 1 inch. If the dowel is painted on all surfaces and then cut into 1/2 inch cubes, what fraction of the resulting cube faces are painted?

(A) 1/3
(B) 3/8
(C) 7/16
(D) 1/2
(E) 9/16

Kudos for a correct solution.

Number of pieces in which First Side of Box of Dimension 4 will be cut in cubes of Dimension (1/2)inch = 4/(1/2) = 8
Number of pieces in which Second Side of Box of Dimension 1 will be cut in cubes of Dimension (1/2)inch = 1/(1/2) = 2
Number of pieces in which Third Side of Box of Dimension 1 will be cut in cubes of Dimension (1/2)inch = 1/(1/2) = 2

i.e. Total Number of smaller cubes of Dimension (1/2 each) = 8 x 2 x 2 = 32

METHOD-1
Total Cubes painted on 3 faces = Cubes available on the vertices of Box = No. of Vertics of Box = 8
Total Cubes painted on 2 faces = Cubes available on the edges of Box x 4 = 6 x 4 = 24
Total Cubes painted on 1 faces = Cubes available on the Faces but not on the edges of Vertices
Total Cubes painted on 0 faces = Cubes NOT available on anyone of the Faces

i.e. Total Painted faces = (8*3) + (24*2) + 0 + 0 = 72
Total faces = No. of Cubes x Faces on each cube = 32x6 = 192

Fraction = 72/192 = 3/8

[Reveal] Spoiler:
B

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Re: A rectangular wooden dowel measures 4 inches by 1 inch by 1 inch. If [#permalink]

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17 Jun 2015, 05:22
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Bunuel wrote:
A rectangular wooden dowel measures 4 inches by 1 inch by 1 inch. If the dowel is painted on all surfaces and then cut into 1/2 inch cubes, what fraction of the resulting cube faces are painted?

(A) 1/3
(B) 3/8
(C) 7/16
(D) 1/2
(E) 9/16

Kudos for a correct solution.

Number of pieces in which First Side of Box of Dimension 4 will be cut in cubes of Dimension (1/2)inch = 4/(1/2) = 8
Number of pieces in which Second Side of Box of Dimension 1 will be cut in cubes of Dimension (1/2)inch = 1/(1/2) = 2
Number of pieces in which Third Side of Box of Dimension 1 will be cut in cubes of Dimension (1/2)inch = 1/(1/2) = 2

i.e. Total Number of smaller cubes of Dimension (1/2 each) = 8 x 2 x 2 = 32

METHOD-2
Total painted Area = Total Surface Area of Bigger Box = 2 (lb+bh+lh) = 2(4x1 + 1x1 + 4x1) = 18

Total Surface Area of all the smaller cubes = No. of Smaller Cubes x Surface Area of Each Cube

Please Note: Surface Area of Each Cube = $$6a^2$$ where $$a$$ is the each dimension of Cube

i.e. Total Surface Area of all the smaller cubes = $$32 * [6x(1/2)^2]$$ = 48

Painted faces as fraction of total faces = Painted Area / Total area of smaller Cubes = 18/48 = 3/8

[Reveal] Spoiler:
B

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Posts: 5538
Re: A rectangular wooden dowel measures 4 inches by 1 inch by 1 inch. If [#permalink]

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17 Jun 2015, 05:45
Bunuel wrote:
A rectangular wooden dowel measures 4 inches by 1 inch by 1 inch. If the dowel is painted on all surfaces and then cut into 1/2 inch cubes, what fraction of the resulting cube faces are painted?

(A) 1/3
(B) 3/8
(C) 7/16
(D) 1/2
(E) 9/16

Kudos for a correct solution.

Hi,
we have to find the painted faces and total faces..

1)painted faces..
we have four faces of size 4*1 sq inches...when cut at 1/2 inches , it will give us 8*2 external (painted) faces.. total faces 8*2*4=64faces
remaining two faces of size 1*1sq inches...when cut at 1/2 inches, it will give us 2*2 external (painted) faces.. total faces 2*2*2=8faces
total 64+8=72 faces

2) total number of faces: number of cubes=4*1*1/(0.5*0.5*0.5)=8*2*2=32..
each cube has 6 faces.. so total faces=32*6=192

fraction of faces painted=painted faces/total faces=72/192=3/8
ans B
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Re: A rectangular wooden dowel measures 4 inches by 1 inch by 1 inch. If [#permalink]

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17 Jun 2015, 06:53
1
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Bunuel wrote:
A rectangular wooden dowel measures 4 inches by 1 inch by 1 inch. If the dowel is painted on all surfaces and then cut into 1/2 inch cubes, what fraction of the resulting cube faces are painted?

(A) 1/3
(B) 3/8
(C) 7/16
(D) 1/2
(E) 9/16

Kudos for a correct solution.

Solution -
Total area of the Wooden dowel(Painted) = 2*(4*1+1*4+1*1) = 18 inch^2
No of cubes after cut into 1/2 inch = (4*1*1)/(1/2) = 32
Area of 1/2 inch cube = 6*(1/2)*(1/2) = 3/2 inch^2

Total area of all the 1/2 inch cubes = 32*(3/2) = 48 inch^2*

Fraction of the resulting cube faces are painted = 18/48 = 3/8. ANS B.

Thanks

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Re: A rectangular wooden dowel measures 4 inches by 1 inch by 1 inch. If [#permalink]

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18 Jun 2015, 23:45
Bunuel wrote:
A rectangular wooden dowel measures 4 inches by 1 inch by 1 inch. If the dowel is painted on all surfaces and then cut into 1/2 inch cubes, what fraction of the resulting cube faces are painted?

(A) 1/3
(B) 3/8
(C) 7/16
(D) 1/2
(E) 9/16

Kudos for a correct solution.

The surface area of the wooden dowel is the area of each of the 6 faces. There is are four 4x1 faces and two 1x1 faces, for a total of 4*4+2=18 square inches that are painted.
If it is cut into 1/2 inch cubes, then that means that it will be cut into 8x2x2 1/2 inch cubes. The surface area of each of these cubes is 6*1/2*1/2=6/4=1.5 square inches. There are a total of 8x2x2=32 cubes, so the total surface area is 32*1.5=32+16=48 square inches. 18/48 square inches are painted. 18/48=3/8 so answer is B.
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Re: A rectangular wooden dowel measures 4 inches by 1 inch by 1 inch. If [#permalink]

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19 Jun 2015, 00:14
Hello,

I am a little bit confused here. Are we looking for the fraction of the resulting, painted surface areas, or for the fraction of the number of resulting faces that are painted?

So, the ratio of "total inches of resulting painted surface areas"/"total inches of surface areas" or for the ratio of "total number of resulting faces that are painted"/"total number of resulting faces"?
Math Expert
Joined: 02 Aug 2009
Posts: 5538
Re: A rectangular wooden dowel measures 4 inches by 1 inch by 1 inch. If [#permalink]

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19 Jun 2015, 00:36
pacifist85 wrote:
Hello,

I am a little bit confused here. Are we looking for the fraction of the resulting, painted surface areas, or for the fraction of the number of resulting faces that are painted?

So, the ratio of "total inches of resulting painted surface areas"/"total inches of surface areas" or for the ratio of "total number of resulting faces that are painted"/"total number of resulting faces"?

Hi,
we are looking for the faces and that is mentioned in the Q..
A rectangular wooden dowel measures 4 inches by 1 inch by 1 inch. If the dowel is painted on all surfaces and then cut into 1/2 inch cubes, what fraction of the resulting cube faces are painted
so the coloured portion in your query is the ratio you have to find
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Re: A rectangular wooden dowel measures 4 inches by 1 inch by 1 inch. If [#permalink]

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22 Jun 2015, 06:18
Expert's post
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Bunuel wrote:
A rectangular wooden dowel measures 4 inches by 1 inch by 1 inch. If the dowel is painted on all surfaces and then cut into 1/2 inch cubes, what fraction of the resulting cube faces are painted?

(A) 1/3
(B) 3/8
(C) 7/16
(D) 1/2
(E) 9/16

Kudos for a correct solution.

MANHATTAN GMAT OFFICIAL SOLUTION:

If you draw a picture, this problem becomes a matter of counting:

Total Cubes = (4 inches × 2 cubes per inch) × (1 × 2) × (1 × 2) = 32 cubes

Total Cube Faces = 32 cubes × 6 faces per cube = 192 faces total

We now consider the faces that were painted on the front and back of the dowel, the top and bottom of the dowel, and the ends of the dowel. In the diagram above, we can see 16 faces on the front, 16 faces on the top, and 4 faces on the end shown. Of course, there are other sides: the back, the bottom, and the other end.

The fraction of faces that are painted = 72/192 = 24(3)/24(8) = 3/8.

Notice that there is no shortcut to solving this kind of problem, so don't waste time looking for one—just draw the diagram and count.

 ! Even if you can easily picture 3-D shapes and objects in your head, it is still better to draw a picture on your scrapboard.Wrong answer choices are often those you might get by losing track of your progress as you process the object in your mind.

This kind of process can also help you with questions that deal with the relative size of different objects.

[Reveal] Spoiler:
Attachment:

2015-06-22_1814.png [ 38.37 KiB | Viewed 2614 times ]

Attachment:

2015-06-22_1815.png [ 29.26 KiB | Viewed 2598 times ]

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A rectangular wooden dowel measures 4 inches by 1 inch by 1 inch. If [#permalink]

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17 Aug 2017, 14:53
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Hey Experts- Bunuel mikemcgarry - Can you please validate my approach below. I used the volume method to find the number of cubes that would be contained in the cuboid instead of the counting technique. Is it alright to use this method? Are there any caveats to consider. For example- the volume approach often does not take into account the exact cuts and specifics of a figure. This approach might be working in this question, but does it work well with other scenarios as well?

BEFORE the cuts
There are 6 faces to the dowel, and each face is a rectangle.
4 of the faces have dimensions 4 by 1. So, the area of each rectangle = (4)(1) = 4
So, the total area of those 4 rectangles = (4)(4) = 16
The remaining 2 faces have dimensions 1 by 1. So, the area of each rectangle (square) = (1)(1) = 1
So, the total area of those 2 rectangles = (2)(1) = 2

So, BEFORE the cuts, the TOTAL surface area of the dowel = 16 + 2 = 18
In other words, there are 18 square inches of paint.

AFTER the cuts
Each cube has dimensions 1/2 by 1/2 by 1/2
So, the VOLUME of each cube = (1/2)(1/2)(1/2) = 1/8
BEFORE the cut, the VOLUME of the dowel = (4)(1)(1) = 4
So, the NUMBER of cubes = 4/(1/8) = 32
So, after the cuts, there are 32 mini cubes

Each individual mini-cube has 6 sides, and each side is a 1/2 by 1/2 square.
So, the area of ONE square = (1/2)(1/2) = 1/4
So, the total surface area of one mini cube = (6)(1/4) = 6/4 = 3/2
So, the TOTAL surface are of all 32 mini cubes = (32)(3/2) = 48 square inches

So, the 32 mini cubes have a TOTAL surface are of 48 square inches, and 18 square inches are painted.

So, the correct answer is 18/48
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Re: A rectangular wooden dowel measures 4 inches by 1 inch by 1 inch. If [#permalink]

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17 Aug 2017, 15:43
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Expert's post
bkpolymers1617 wrote:
Hey Experts- Bunuel mikemcgarry - Can you please validate my approach below. I used the volume method to find the number of cubes that would be contained in the cuboid instead of the counting technique. Is it alright to use this method? Are there any caveats to consider. For example- the volume approach often does not take into account the exact cuts and specifics of a figure. This approach might be working in this question, but does it work well with other scenarios as well?

BEFORE the cuts
There are 6 faces to the dowel, and each face is a rectangle.
4 of the faces have dimensions 4 by 1. So, the area of each rectangle = (4)(1) = 4
So, the total area of those 4 rectangles = (4)(4) = 16
The remaining 2 faces have dimensions 1 by 1. So, the area of each rectangle (square) = (1)(1) = 1
So, the total area of those 2 rectangles = (2)(1) = 2

So, BEFORE the cuts, the TOTAL surface area of the dowel = 16 + 2 = 18
In other words, there are 18 square inches of paint.

AFTER the cuts
Each cube has dimensions 1/2 by 1/2 by 1/2
So, the VOLUME of each cube = (1/2)(1/2)(1/2) = 1/8
BEFORE the cut, the VOLUME of the dowel = (4)(1)(1) = 4
So, the NUMBER of cubes = 4/(1/8) = 32
So, after the cuts, there are 32 mini cubes

Each individual mini-cube has 6 sides, and each side is a 1/2 by 1/2 square.
So, the area of ONE square = (1/2)(1/2) = 1/4
So, the total surface area of one mini cube = (6)(1/4) = 6/4 = 3/2
So, the TOTAL surface are of all 32 mini cubes = (32)(3/2) = 48 square inches

So, the 32 mini cubes have a TOTAL surface are of 48 square inches, and 18 square inches are painted.

So, the correct answer is 18/48

Dear bkpolymers1617,

I'm happy to respond. :-)

What you did is fine. I would think that this approach might take slightly longer than the counting method, but whatever makes the most intuitive sense to you is often the easier and faster method. This is a relatively straightforward problem. As you move into harder problems, and you are wondering about the relative strengths of two different approaches, experiment: do them both, and see which one is faster. Ideally, you will find that you are confident in more than one way to solve a number of problems.

Does this make sense?

Mike :-)
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Re: A rectangular wooden dowel measures 4 inches by 1 inch by 1 inch. If [#permalink]

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12 Jan 2018, 12:07
Hi All,

This question can be approached in a couple of different ways (most of which involve lots of math steps). Based on the 'spread' of the answer choices though, you can do a little bit of math and use a bit of logic to get to the correct answer.

To start, it would help to physically draw the rectangular solid that is described (including the "cut lines"). Since a (1 in.) x (1 in.) x (1 in.) cube will contain eight 1/2 in. mini-cubes, the (4 in.) x (1 in.) x (1in.) dowel will end up being cut into 4(8) = 32 mini-cubes.

When the outside of the dowel is painted, you should recognize that each of the smaller cubes will have paint on either 3 faces (the 8 'corner' pieces) or 2 faces (all of the non-corner pieces - 24 in total). Put a different way - the 8 corner pieces have HALF of their faces painted and all of the other pieces have a THIRD of their faces painted. This then can be looked at as a Weighted Average. The AVERAGE fractional number of faces painted must be closer to 1/3 than to 1/2. There's only one answer that fits...

[Reveal] Spoiler:
B

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Re: A rectangular wooden dowel measures 4 inches by 1 inch by 1 inch. If   [#permalink] 12 Jan 2018, 12:07
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