# Distance problem

#### tuanl

A cyclist practising for a road race found that one complete CIRCUIT of the course took exactly two hours and fifteen minutes. The course at one part was very hilly. His speed on the level was at the rate of 16 km/hr, uphill was 12 km/hr and downhill 24 km/hr. What is the distance round the course?

PS: I can't see why the answer is 36 km. This problem has 3 variables and only 2 equations, so I think it's not solvable. But I'm afraid it's tricky since the road race is CIRCUIT. For your reference, I got this problem from this site: http://cygnus.et.put.poznan.pl/~piotrw/rozneciekawe/zagadki.htm

#### pruse

Whatever distance he goes uphill he must go downhill, so

$$x/16+y/12+y/24 = 9/4$$
$$3x+4y+2y = 108$$
$$3x+6y = 108$$
$$x+2y = 36$$

#### tuanl

Whatever distance he goes uphill he must go downhill, so

$$x/16+y/12+y/24 = 9/4$$
$$3x+4y+2y = 108$$
$$3x+6y = 108$$
$$x+2y = 36$$

Thanks for your solution, our puzzle guru:D Can't believe I missed this simple trick

#### pruse

Thanks for your solution, our puzzle guru:D Can't believe I missed this simple trick

Hehe. No problem.

#### physEcon

Whatever distance he goes uphill he must go downhill

Isn't that assuming that the uphill and downhill parts have the same slope?

#### pruse

Isn't that assuming that the uphill and downhill parts have the same slope?

The problem mentions that there are 'level' parts. In completing a full circuit, to go from level ground back to level, you must obviously go a total downhill distance equal to your total uphill distance.

#### lobomattu

I still don't get it. The vertical distance of the uphill and downhill must equal, but not necessarily the distance traveled.

In a circuit, do you have to make a U-turn and go back the way you came?? If so, this problem makes sense to me

#### LochWulf

@ lobomattu: The solution shows that the total distance around the circuit must be 36, in order for the speed and total time criteria to be satisfied. Yet the total amount of flat distance (x) and uphill-downhill pairs (2y) remain variables, and can be satisfied by any number of (x, y) pairs---subject to the obvious constraints.

For example, let the flat terrain x = 18 km and the hills comprise the other 18 km (y = 9). With the given speeds, you'll see the cyclist makes the run in 2h 15m. Ditto for 10 km of flat terrain and 26 km of hills (13 uphill and 13 downhill), or any other x, y combo satisfying x + 2y = 36.

#### pruse

No, lobomattu actually has a point... vertical distance may be the same for the uphill and downhill parts, but the road distance may not. I guess you're supposed to assume the hills are symmetric, which is kind of silly.

Let's just dismiss this as a bad problem :P

#### LochWulf

Roger that....I was lulled into an assumption trap, drawn by the carefree days of my youth in a world stylized with flat term structures and symmetrical hills.

#### pruse

Roger that....I was lulled into an assumption trap, drawn by the carefree days of my youth in a world stylized with flat term structures and symmetrical hills.

A poet; I like that. We need more of those around here!

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