So far I’ve been very impressed with the number of correct answers I’m getting to each week’s Brain Twister. So…I’ve decided to make the question even more twisty, more diabolical. That doesn’t mean it is impossible. Good luck!

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?

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the total miles ( lets call ‘T’ ) covered will be, T= 8+ x + (x/2) = 8 + (3/2)x

the question forms a right triangle with base =8 , hypotenuse = x, perpendicular side = y..
recollecting the pythagoras triplet ( 8,15,17)

by that triplet , hypotenuse ( x) = 17 & y= 15.
solving this in the equation for T = 8 + (3/2) x = 8 + (3/2) 17 = 33.5 , option (D)..

This approach might not work if some other combination of the pythagoras triplets with 8 had been used..

Second method:

T = 8 + (3/2)x .

evaluating each option..

A) by this option , T will be 15.
plugging this value into our equation for T and solving for x …

15 = 8 + (3/2) x
7= (3/2) x

=> x = 14/3 which cannot be an integer,, the question states that x,y are integers.. hence this option can be eliminated.

Solving other options in a similar way would show that only option D (33.5) satisfies this condition as given below..

8 + (3/2)x = 33.5
(3/2)x = 25.5
x = 25.5 *2/5
x = 17 which is an integer. solving for y ( using pythagoras theorem ) yields 15 which is also an integer.

This is how it was solved. Total distance traversed d=8+x+x/2 with x and y being integers and to satisfy Pythagorean theorem x^2 = 8^2 + y^2.
Substitute d with option answers and check which satisfies x and y being an integer and Pythagorean triples. The immediate check would be for 33.5 and 32.5 as they seem to be more appropriate answers # just saying.

The total miles cycled is 8+x+x/2 = 8+3x/2
and it is given that x is an integer.
so starting with choice a i.e.15 will give x as 14/3 not an integer so choice a is incorrect.
doing this for all the choices option d i.e. 33.5 gives the value of x=17, an integer.
So the correct ans is 33.5 i.e. option d.

From the question, it is evident enough that the bicyclist moves in a shape of a right angled triangle, with one of the sides as 8 (base), the other x (hypotenuse) and the last as y.

Now, since x and y are both integers, and the only triplet that comes to my mind with 8 in it is 8,15,17 (this was the most trickiest thing to figure out!) ; x=17 and y=15
(also satisfied is the constraint that 17/3 < {15- 17/2}.

Now, since the girl traveled (8 + x + x/2) miles, the total number of miles she traveled are 33.5 (option D).

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answer D – 33.5-8 =25.5

25.5= X+X/2 = 3X/2

51/3 = 17 = x .. y is 15 only option where x is integer.

Hmm…take a look at the question again. It is about the total distance she traveled

Hi,

Is the answer (D) ?

I used two methods.

First method :

the total miles ( lets call ‘T’ ) covered will be, T= 8+ x + (x/2) = 8 + (3/2)x

the question forms a right triangle with base =8 , hypotenuse = x, perpendicular side = y..

recollecting the pythagoras triplet ( 8,15,17)

by that triplet , hypotenuse ( x) = 17 & y= 15.

solving this in the equation for T = 8 + (3/2) x = 8 + (3/2) 17 = 33.5 , option (D)..

This approach might not work if some other combination of the pythagoras triplets with 8 had been used..

Second method:

T = 8 + (3/2)x .

evaluating each option..

A) by this option , T will be 15.

plugging this value into our equation for T and solving for x …

15 = 8 + (3/2) x

7= (3/2) x

=> x = 14/3 which cannot be an integer,, the question states that x,y are integers.. hence this option can be eliminated.

Solving other options in a similar way would show that only option D (33.5) satisfies this condition as given below..

8 + (3/2)x = 33.5

(3/2)x = 25.5

x = 25.5 *2/5

x = 17 which is an integer. solving for y ( using pythagoras theorem ) yields 15 which is also an integer.

Thank you for the very thorough and well-thought out explanation

Ans D

work your way back using options

Yes, you can do it that way, but I’m not sure how accurate that would be. It seems that picking between (C) and (D) would be hard

The answer is (D) 33.5.

No great formula to solve upside down. Solved it down to up i.e. from answer to question.

Hmm…how exactly did you do that? I’m curious how you would discern between (C) and (D).

Answer is 33.5.

This is how it was solved. Total distance traversed d=8+x+x/2 with x and y being integers and to satisfy Pythagorean theorem x^2 = 8^2 + y^2.

Substitute d with option answers and check which satisfies x and y being an integer and Pythagorean triples. The immediate check would be for 33.5 and 32.5 as they seem to be more appropriate answers # just saying.

Interesting approach. But how would you differentiate between (C) and (D)?

The total miles cycled is 8+x+x/2 = 8+3x/2

and it is given that x is an integer.

so starting with choice a i.e.15 will give x as 14/3 not an integer so choice a is incorrect.

doing this for all the choices option d i.e. 33.5 gives the value of x=17, an integer.

So the correct ans is 33.5 i.e. option d.

Wow, that is a brilliant way of working with the answers and coming up with an easy equation.Nice job!

Since x and y are integers, 8,y,x should form pythagorean triples. Hence the possible values are 8,6,10 or 8,15,17.

Now we apply the next condition y – x/2 > x/3. Applying this inequality gives us x=17 and y=15.

So total distance traveled = 8 + x + x/2 = 33.5

Answer is (D)

Yep, that’s a good, elegant way of doing it

My answer is D.

From the question, it is evident enough that the bicyclist moves in a shape of a right angled triangle, with one of the sides as 8 (base), the other x (hypotenuse) and the last as y.

Now, since x and y are both integers, and the only triplet that comes to my mind with 8 in it is 8,15,17 (this was the most trickiest thing to figure out!) ; x=17 and y=15

(also satisfied is the constraint that 17/3 < {15- 17/2}.

Now, since the girl traveled (8 + x + x/2) miles, the total number of miles she traveled are 33.5 (option D).

Hi Siddharth,

Yes, the 8:15:17 triangle part is tricky, but once you figure that out the rest of the problem falls into place.

Since her path is on a RAT.

The equations to solve are –

1. x^2 = y^2 + 64

Since x and y are integers

So, (x,y) can be (10,6), (17,15) and so on…

2. y – x/2 > x/3 (Assuming she has not crossed the point)

=> y – 5/6x > 0

Solving these two together.

We x = 17, y =15

So Answer is 17 + 17/2 + 8 = 33.5 Hence D

PS – If we assume that she has crossed the point, we get x > 6y, which will never satisfy equation 1.

Yep, you got it! Nice approach