Working out number of corners on a dodecahedron

Question

A dodecahedron has 12 pentagon faces, at each corner 3 faces touch. How many corners does it have?

What's the best way to figure this out?

mikemcgarry · Answer

Dear umairk , I'm happy to help. :-)

I love the Platonic Solids, but that's really a level of difficulty beyond the GMAT Quant section. Here's one way to think about it. Each pentagon has 5 vertices. There are 12 of them, so that's 60 vertices. But that obviously over-counts the vertices of the dodecahedron, because each vertex is getting counted three times, once for each pentagon meeting at it. So divide by 3 --- 60/3 = 20, which is the number of vertices on the dodecahedron . If you're curious, you can find out much more about the Platonic Solids here: https://en.wikipedia.org/wiki/Platonic_solid but, again, that's beyond what the GMAT will test. Mike :-)

umairk · Answer

Hi great! Thanks so much for that, it makes sense now and saves time from drawing out the whole thing.

It's actually for a management consulting aptitutude test, hence my bafflement!

umairk · Answer

A company's current revenue is £1million a year, 10% is profit. If over the next 3 years revenues increase by 10% a year and costs increase by £120000 a year, what will be the profit margin?

6.1 
3.2
4.6
-2.1
5.6

I worked it out as Revenues at year 3 are 1.331 million, Costs are 1.26 million and so profit is 0.071 million giving a profit margin of 0.071/1.331 *100= 5.33

Any advice would be appreciated, thanks!

umairk · Answer

How many ways are there to place a domino on an 8x8 chess board so that it covers exactly 2 squares (not counting rotations)

I cant think of how to do this without drawing out a chess board and manually counting...any help please!

umairk · Answer

ABC is a right angled triangle, AB= 4, BC= 5. Whats the area?

Poss answers= More info needed, 12, 6, 20, 10

Now, in such a scenario is it valid to say more info is needed as you are not sure which way they have labelled the triangle or is it just a straight forward question?

mikemcgarry · Answer

Dear umairk , I'm happy to help with this. :-)

I agree with all of your calculations. Currently revenue = £1M profit = £100K cost = £900K If revenue increases by 10%, that's £1M(1.1)^3 = 1.331M, which you correctly calculated. If costs increase by £120K a year, that's 900K + 3*120K = 1.26M which you correctly calculated. Difference in revenue and cost should be profit, which is 1.331M - 1.26M = 0.071M = 71K which you correctly calculated. In my understanding, profit margin would be (profit)/(revenue) x 100 = 0.071/1.331 *100= 5.33%, as you calculated and profit percentage would be (profit)/(cost) x 100 = 0.071/1.26*100 = 5.635% --- this looks suspiciously like answer (E) . These two terms are, I believe, how the concepts are discussed in American economics. I don't know if the exact terms are used the same way around the world. It may be that whoever wrote the question meant the thing we Americans call "profit percentage", but that person knew the same idea as "profit margin." The GMAT tends to steer clear of all ambiguities related to terminology. Rather than use a technical term that may have one meaning in one place and another meaning in another, they are scrupulously careful to say things such as " what are profits in the last year as a percentage of costs in that year " --- so that even a person unacquainted with the terminology could find the answer. The GMAT definitely expects folks to have a handle on the basic ideas of cost, profit, and revenue, as you do, but I have never seen them use some of these other terms, such as "profit margin" or "profit percentage". They will always be very explicit about that: they are not interested in testing the precise meaning of more obscure economic terms. Does all this make sense? Mike :-)

mikemcgarry · Answer

Dear umairk , This is a great counting problem, and I am happy to help. :-)

First of all, here's a post on difficult counting problems that might help you. https://magoosh.com/gmat/2013/difficult- ... -problems/ Think about it this way. The domino can be placed on the board either vertically or horizontally. Let's look at the horizontal arrangements first. In a single row of eight squares, the left side of the domino can be located in seven of those eight squares. The left side can't be in the eighth, rightmost square, because then the right side of the domino would be hanging off the board. So in a single row, there are seven possible locations. Well, there are eight rows, so that's a total of 7*8 = 56 possible horizontal arrangements. Now, the number of vertical arrangements, by symmetry, must equal the number of horizontal arrangements --- 56 for each, for a total of 112 arrangements. Does all this make sense? Mike :-)

mikemcgarry · Answer

Dear umairk , I'm happy to help. :-)

Yes, you are right --- I believe the answer has to be "more info needed", because we don't know where the right angle is. It could be at B, in which case the area would be 10, or it could be at A, in which case AC = 3 and area = 6. Without either a diagram or a statement about where the right angle is, we cannot give a definitive answer. Mike :-)

umairk · Answer

Hi Mike

Yes that makes perfect sense. It seems so easy when you put it that way!

Many thanks
Umair

umairk · Answer

Hi

Yeah your right in terms of profit margin having the same meaning. It's interesting in that 5.3 isn't in the answer choices but as that is the closest it will have to do.

Again, thanks!

umairk · Answer

Great thanks, at least I got one right!

Working out number of corners on a dodecahedron

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