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Geometry and Numbers Question

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Geometry and Numbers Question [#permalink] New post 26 Dec 2013, 19:15
I saw these 2 questions in one of the handouts from a GMAT prep center. Cannot find the source anymore and for some reason was not able to solve these quickly and get the right answer. Sorry, do not have the answer choices.

1. An inverted isosceles cone is filled with water up to a height such that the length between the water level and the lip of the cone is 'h'. diameter of the lip is 5 and the total height of the cone is 3 and the diameter of the water filled portion is 3. Find'h'

This is a DS question. I remember getting stumped on the first option itself
2. The product of the digits of a 3 digit number 'n' is 54. Find the number 'n'
(1) n > 950

Could somebody explain how n could be greater than 950?

There is possibility that the answer choices mentioned/questions could be wrong
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Re: Geometry and Numbers Question [#permalink] New post 26 Dec 2013, 19:23
My solution for the 1st question -

Considering a 2D image, Area of the the top trapezium + area of water filled triangle = area of the entire object (here an isosceles triangle)

[1/2(5+3)h] + [1/2(3)(3-h)] = [1/2(5)(3)]

=> 4h + 4.5 - 1.5h = 7.5 => 2.5h = 12 => h = 24/5
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Re: Geometry and Numbers Question [#permalink] New post 08 Jan 2014, 20:25
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Hi sapp,

your inverted cone is, from a 2D perspective, an isoceles triangle with a base of 5, the water fills a portion of this triangle to have a base of 3.

Based on triangle proportionality, the height of the water is therefore 3/5ths of the height of the total triangle (which is 3) so the h, which is really the remaining height, is the remaining 2/5th's of the height of the triangle, or

2/5*3 = 6/5

for your second question, even though it doesn't look possible, don't panic, break the problem down: the product of 3 digits is 54. in product questions it helps to break down the number into primes: 54 = 2 * 3 * 3 * 3

now it's a 3 digit number bigger than 950, so it's CERTAIN that the hundreds digit has to be 9, which is great cuz it takes care of two of the 3's in 54

now we have a 2 and a 3 left over, but it seems kinda tough, since 923 and 932 are both below 950, so we need to pull a trick out of our sleeve: 1

see even if the number was 92311111111 the product would still be 54, because when you're taking the product (the multiplication) the 1 is basically a neutral value, so use it!

in this case, although we can't make 923 or 932, if we put 2 and 3 together, we get a 6, and 961 fits the bill quite nicely since its big enough, and contains all those pesky primes we've been trying to put together!

(916 woulda worked too, but it's smaller than 950...)

hope that helps!
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Re: Geometry and Numbers Question [#permalink] New post 08 Jan 2014, 20:40
Hi ddb1981,

Ah yes! the 1, that's what was missing from my thought process!

Could you explain where I went wrong with the area calculation to find the value of h?

Thanks a ton!
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Re: Geometry and Numbers Question [#permalink] New post 08 Jan 2014, 20:56
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glad I could help!

for the 1st question, I didn't see your thoughts on how to solve, but if I had to guess, the part that most people choke on is the triangle proportionality, here's a link that hopefully you'll find helpful

http://hotmath.com/hotmath_help/topics/ ... eorem.html

let me know if this helps, (and lemme know if it doesn't!)
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Re: Geometry and Numbers Question [#permalink] New post 08 Jan 2014, 21:29
Check the second post in this thread.

sapp25 wrote:
My solution for the 1st question -

Considering a 2D image, Area of the the top trapezium + area of water filled triangle = area of the entire object (here an isosceles triangle)

[1/2(5+3)h] + [1/2(3)(3-h)] = [1/2(5)(3)]

=> 4h + 4.5 - 1.5h = 7.5 => 2.5h = 12 => h = 24/5


Thanks!
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Re: Geometry and Numbers Question [#permalink] New post 09 Jan 2014, 20:15
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got it! here we go:

you were trying to add the trapezoid to the mini triangle to make the full triangle! method's right but your math's off:

(5+3)/2*h + 3(3-h)/2 = 3*5/2

here's where things went awry for you: you took the 4.5 on the right and forgot to make it negative when flipping it to the right, it shoudn't be 12 it should be 7.5 - 4.5 = 3!

from there 2.5h = 3, so h = 3/2.5, or by multiplying top and bot by 2, you get 6/5

ps: because all the denominators were 2, you should ALWAYS, by reflex, multiply the entire equation by 2, which would get rid of all that pesky fractional stuff

how was that?
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Re: Geometry and Numbers Question   [#permalink] 09 Jan 2014, 20:15
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