Problem Solving

In the figure above, an edge, a face, and two vertices of a cube are indicated. If e, f, and v denote the number of edges, faces, and vertices, respectively, of a cube, which of the following is true?

(A) e + f = v + 8
(B) e + f = v + 6
(C) e + f = 20 -v
(D) e - f = v - 2
(E) e - f = 2v

OA is D

Please explain

Regards
Sachin

e = 12
f = 6
v = 8


{A} 12 + 6 = 8 + 8
NO

{B} 12 + 6 = 8 + 6
NO

{C} 12 + 6 = 20 - 8
NO

{D} 12 - 6 = 8 - 2
6 = 6
YES

{E} 12 - 6 = 2*8
NO

[spoiler]{D}[/spoiler]

sachin_yadav wrote:In the figure above, an edge, a face, and two vertices of a cube are indicated. If e, f, and v denote the number of edges, faces, and vertices, respectively, of a cube, which of the following is true?

(A) e + f = v + 8
(B) e + f = v + 6
(C) e + f = 20 -v
(D) e - f = v - 2
(E) e - f = 2v

OA is D

Please explain

Regards
Sachin

No. of Edges = (4 on the top face) + (4 on the bottom Face) + (4 Vertical edges attaching top face with Bottom one)

i.e. No. of Edges, e = 4+4+4 = 12

No. of Faces, f = 6

No. of Vertices, v = 8 (4 on top face and other 4 on the bottom face)

e+f = 12+6 = 18 (for First three options)
but v+8 = 8+8 = 16
and v+6 = 8+6 = 14
and 20-v = 20-8 = 12
Therefore Option A, B and C are ELIMINATED

e - f = 12 - 6 = 6 (for Option D and E)
Option (D): v - 2 = 8 - 2 = 6 BINGO!!!

Answer: Option D

In the figure above, an edge, a face, and two vertices of a cube are indicated. If e, f, and v denote the number of edges, faces, and vertices, respectively, of a cube, which of the following is true?

(A) e + f = v + 8
(B) e + f = v + 6
(C) e + f = 20 -v
(D) e - f = v - 2
(E) e - f = 2v

No. of edges = no. of lines = e = 12
No. of faces = f = 6
No. of vertices = no. of points where three lines meet = v = 8

A) e + f = v + 8
=> 12 + 6 = 8 + 8 (nah)

B) e + f = v + 6
=> 12 + 6 = 8 + 6 (no)

C) e + f = 20 - v
=> 12 + 6 = 20 - 8 (no way)

D) e - f = v - 2
=> 12 - 6 = 8 - 2 (Bingo)

E) e - f = 2v
=> 12 - 6 = 2*8 (Not)

Well, we already know what e, f and v are, so its a matter of seeing how they fit together.
e is 12 (4 edges on top, 4 going down the sides and 4 on the bottom)
f is 6 (there are 6 faces on a cube)
v is 8 (4 vertices on top, 4 on the bottom)

Now, we just have to plug these variables into each of the equations to see which one it us true.

(A) 12+6 does NOT equal 8+8
(B) 12+6 also does NOT equal 6 +8
(C) 12+6 also does NOT equal 20-8
(D)12-6 DOES equal 8-2 (There is our answer)

On test day, once you find the right answer and have double checked it, I would recommend moving on, since there is limited time.

Hope this helps!