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?
Answer and Explanation
To solve this week’s Brain Twister, it is a good idea to draw out the cyclist’s journey. First, she goes 8 miles left (which is consistent with west). Next, draw a diagonal line in a roughly northeastern direction until that line is directly above the starting point. Next, draw a line straight down. At this point, you should pick up on the fact that you are drawing a triangle, but stopping slightly short, since the trip doesn’t quite take her back to her original starting point.
The next piece of information that is useful is the fact that x and y are integers. Right now, we have a leg that is ‘8 miles’. This could very well be a triangle that is based on the 3:4:5 dimensions. That would mean that the hypotenuse is 10, making the complete length of the final leg ‘6’. Notice, however, that ‘x’ would equal ‘10’, the hypotenuse of the triangle. if you plug in 10 for x, then she drives 10/2 = 5 miles south, at which point she is only 1 mile from her starting point. The question says that she ends up more than x/3, or 3.33 miles, from her starting point. Therefore, the answer could not be.
At this point, we have to think what are some other possible triangles in which all three sides are integers and one of those sides is ‘8’. If you don’t know, you might have to use the Pythagorean Theorem, working backwards. We have x^2 + 8^2 = y^2. Rearranging the equation, we get y^2 – x ^2 = 64, or (y-x)(y+x) = 64. These are integers we are dealing with, so we need to work with factors of 64, 16 and 4, and 32 and 2. Working with 16 and 4, as in (y-x) = 4, and (y+x) = 16, gives us y=10, x=6, which we already know is not the answer. Working with 32 and 2, we get (y-x) = 2, and (y+x) = 32. Solving for the variables, we get x = 17 and y = 15. At this point, make sure to remember the problem, which—oh so trickily!—made the hypotenuse equal to x. So therefore, x = 17, and y = 15. If the final leg of the triangle is 15, then she travels 17/2 = 8.5 miles. Therefore, she stops 6.5 miles from her starting point. Is that more than x/3 miles? Plugging in x, which equals 17, we get 5.67, which means she is more than x/3 miles from her starting point. Therefore, the triangle has to be a 8:15:17 (if you’ve never seen that triangle before, it’s a good one to remember, as it may come up on very difficult questions).
Now before you happily add up 8, 15, and 17 to get 40, remember that the question is asking for the length of her total trip, and she stopped before she reached her starting point. Add up 8+17 =25, then find how far she cycled south (y/3 = 8.5), and we get 33.5 (D).
Throughout the entire explanation, you might have been wondering why I never accounted for her differing speeds. The thing is it doesn’t really matter how fast she pedaled at any given moment, since we are only concerned with the total distance she covered.