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# A cuboid of dimensions 4 in. x 6 in. x 7 in. is painted on all six

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A cuboid of dimensions 4 in. x 6 in. x 7 in. is painted on all six  [#permalink]

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24 Jan 2018, 10:39
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Difficulty:

95% (hard)

Question Stats:

39% (02:45) correct 61% (03:02) wrong based on 26 sessions

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A cuboid of dimensions 4 in. X 6 in. X 7 in. is painted on all six faces with the same color. It is then completely cut into identical small cubes, each of side 1 in. What is the ratio of the number of cubes with no face painted to the number of cubes with exactly one face painted to those with exactly two faces painted?

A) 10:19:11
B) 11:19:10
C) 20:19:11
D) 20:38:11
E) 20:38:33

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A cuboid of dimensions 4 in. x 6 in. x 7 in. is painted on all six  [#permalink]

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24 Jan 2018, 11:53
2
Total number of cubes: 7 * 6 * 4 = 168

cubes with no paint:
(7-2) * (6-4) * (4-2) = 40

one side painted cubes:
if we take the surface of area 6*7

except the squares are along the edges, all the squares are single painted.
i.e. (6-2)*(7-2)
also there are two parallel 6*7 faces, so number of one side painted cubes = 2 * (6-2) * (7-2) = 40

if we take the two parallel surfaces of area 4*7, number of one side painted cubes = 2 * (4-2) * (7-2) = 20
if we take the two parallel surfaces of area 4*6, number of one side painted cubes = 2 * (4-2) * (6-2) = 16
total number of one sided painted cubes = 40 + 20 + 16 = 76

three sided painted cubes: 8 (one at each vertex)

two side painted cubes : total cubes - (no-sided) - 1_sided - 3_sided = 168 - 40 - 76 - 8 = 44

required ratio => no_sided: one_sided : two_sided = 40 : 76 : 44 = 10 : 19 : 11 => (A)

Thanks,
always welcome any fastest approach to solving !
cheers.
A cuboid of dimensions 4 in. x 6 in. x 7 in. is painted on all six   [#permalink] 24 Jan 2018, 11:53
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