Dear members,
I came across this week's manhattan question and have no clue how to go about solving it. So can someone make me understand please?
Line m and line n intersect, forming 4 angles. Does any of these angles measure greater than 120°?
(1) The product of the measures, in degrees, of the four angles is less than 2^10 3^4 5^4.
(2) The product of the measures, in degrees, of the four angles is greater than 2^14 5^4.
I dont know what OA is.
This Week's Manhattan challenge problem
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For 2 intersecting lines we know opposite sides are equal.
Let the 2 angles be x & y.
We know x + y = 180.
From (1) we know;
x^2 * y^2 = 2^10 * 3^4 * 5^4
Taking root, we have
xy = 2^5 * 3^2 * 5^2 = 7200.
We know, x+y = 180.
y = 180 -x
x(180-x) < 7200.
Angles can take values from (0-60). If x < 60, then Y will be greater than 120 (or) vice-versa. Sufficient.
Sufficient.
From (2);
xy= 2^7 * 5^2 = 3200.
And, similarly, x(180-x) > 3200.
The angles can take values from 20-160. We can't say if it will be less than 120 or greater. Insufficient.
A IMO.
Let the 2 angles be x & y.
We know x + y = 180.
From (1) we know;
x^2 * y^2 = 2^10 * 3^4 * 5^4
Taking root, we have
xy = 2^5 * 3^2 * 5^2 = 7200.
We know, x+y = 180.
y = 180 -x
x(180-x) < 7200.
Angles can take values from (0-60). If x < 60, then Y will be greater than 120 (or) vice-versa. Sufficient.
Sufficient.
From (2);
xy= 2^7 * 5^2 = 3200.
And, similarly, x(180-x) > 3200.
The angles can take values from 20-160. We can't say if it will be less than 120 or greater. Insufficient.
A IMO.
Last edited by shankar.ashwin on Mon Oct 17, 2011 4:09 am, edited 1 time in total.
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I guess you got the inequalities wrong.
Statement 1 says its less than 7200 and Statement 2 says its greater.
Statement 1 says its less than 7200 and Statement 2 says its greater.
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Also, I got a bit confused there. So did it my way. A bit length at first. So please correct me if m wrong
Statement 1 says,
x(180-x)<7200
180x-x^2<7200
-x^2+180x-7200<0
x^2-180x+7200>0 (Multiplying both sides by -1)
(x-120) (x-60) > 0
x> 120 and x>60
so when x is greater than 120, y is less than 60 and when x is greater than 60, y is less than 120.
so one of them is greater, hence sufficient
Statement 2 says,
xy>3200
x(180-x)>3200
similarly as 1, x^2 - 180x + 3200 < 0
(x-160) (x-20) < 0
x< 160 and x< 20.
Here, I don't understand, is it sufficient. It says its less than 160, so when x is less than 160, y is greater than 20. Accordingly, when x is less than 20, y is greater than 160.
First of all, is this explanation correct. And even if it is, m sure there is a better explanation than i have given. Please GMAT gurus, explain this problem.
Statement 1 says,
x(180-x)<7200
180x-x^2<7200
-x^2+180x-7200<0
x^2-180x+7200>0 (Multiplying both sides by -1)
(x-120) (x-60) > 0
x> 120 and x>60
so when x is greater than 120, y is less than 60 and when x is greater than 60, y is less than 120.
so one of them is greater, hence sufficient
Statement 2 says,
xy>3200
x(180-x)>3200
similarly as 1, x^2 - 180x + 3200 < 0
(x-160) (x-20) < 0
x< 160 and x< 20.
Here, I don't understand, is it sufficient. It says its less than 160, so when x is less than 160, y is greater than 20. Accordingly, when x is less than 20, y is greater than 160.
First of all, is this explanation correct. And even if it is, m sure there is a better explanation than i have given. Please GMAT gurus, explain this problem.
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Thanks for pointing out the mistake enjoylife1788 Edited the post. I was wondering where I went wrong
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Your Welcomeshankar.ashwin wrote:Thanks for pointing out the mistake enjoylife1788 Edited the post. I was wondering where I went wrong
But i need your help to understand how you reached to those range of values. I know its too stupid. But I am just not getting it right now.
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enjoylife1788 wrote:Your solution is correct. Solving the quadratic equation gives you limits for the equation. You know roots of the equation are 20,160.shankar.ashwin wrote:Thanks for pointing out the mistake enjoylife1788 Edited the post. I was wondering where I went wrong
Roots are when the equations change sign usually. So test the inequality for 3 segments.
(0-20) (20-160) and (160-180)
You can see the condition x(180-x)>3200 holds true only for the range (20-160).
From this we cannot say if one angle would be greater than 120.
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shankar.ashwin wrote:Ok I get it now. Thank Youenjoylife1788 wrote:Your solution is correct. Solving the quadratic equation gives you limits for the equation. You know roots of the equation are 20,160.shankar.ashwin wrote:Thanks for pointing out the mistake enjoylife1788 Edited the post. I was wondering where I went wrong
Roots are when the equations change sign usually. So test the inequality for 3 segments.
(0-20) (20-160) and (160-180)
You can see the condition x(180-x)>3200 holds true only for the range (20-160).
From this we cannot say if one angle would be greater than 120.
So, The answer should be A.
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