A rectangular wooden dowel measures 4 inches by 1 inch by 1

hazelnut01 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

Source : Manhattan Prep
OA=B

Nice question.
We can answer it by calculating total surfaces areas.

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
= 3/8
= B

Cheers,
Brent

Hi hazelnut01,

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...

Final Answer: B

GMAT assassins aren't born, they're made,
Rich