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22 Feb 2022, 05:15
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D
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Difficulty:
95%
(hard)
Question Stats:
33% (02:57) correct
67%
(03:19)
wrong
based on 122
sessions
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A bicyclist travels 8 miles due west at a constant speed. Next, she rides x miles in a straight line in a direction somewhere between north and east, traveling at half the speed. She stops when she is due north of her starting point, at which time she is y miles from her original location. She then rides, at 1/3 of her original speed, due south for x/2 miles, at which point she ends her trip, more than x/3 miles from her starting point. If x and y are integers, how many total miles did she cycle?
(A) 15
(B) 24
(C) 32.5
(D) 33.5
(E) 40
Are You Up For the Challenge: 700 Level Questions
(A) 15
(B) 24
(C) 32.5
(D) 33.5
(E) 40
Are You Up For the Challenge: 700 Level Questions
07 Mar 2022, 00:09
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Bunuel
Attachment:
IMG_20220307_122553__01.jpg [ 767.64 KiB | Viewed 6790 times ]
Referring to the diagram above:
From Pythagoras theorem: x² = 8² + y² => y² = x² - 64
Also, since D is greater than x/3, we have:
y > x/2 + x/3
=> y > 5x/6
=> y² > 25x²/36
Substituting y² from the previous equation:
x² - 64 > 25x²/36
=> 11x²/36 > 64
=> x² > 64*36/11
=> x² > 209.45
=> x > 14.5
We need both x and y as integers
The Pythagorean triplet that satisfies x (hypotenuse), y and 8 is:
17² = 15² + 8²
So, x = 17 and y = 15
(Verify: D = y - x/2 = 15-8.5 = 6.5 > 17/3)
Total distance = 8 + 17 + 17/2 = 33.5 km
(I have used km in the solution - should be miles)
General Discussion
kungfury42
Joined: 07 Jan 2022
Last visit: 31 May 2023
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GMAT 1: 740 Q51 V38
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07 Mar 2022, 18:11
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sujoykrdatta
Hi sujoykrdatta, since in this question we've been given that both x and y are integers, can we work this problem backwards by considering known pythagorean triplets that contain 8?
Two such triplets that I can recall are (6, 8, 10) and (8, 15, 17). Of these the triplet (6, 8, 10) can be rejected as 6 !> (5/6)*10 a condition which we must satisfy ( y > 5x/6 )
The only remaining triplet (8, 15, 17) does satisfy this condition so we have no reasons to drop it. Fact check: 15 > (5/6)*17 Using this triplet we can arrive at our answer as 8+17+8.5 = 33.5 which matches with option D.
Is this a mere shot in the dark that went right? Can you please share your thoughts on the same? Thank you.
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