## Yoak: A Rather Odd Car Trip

Here’s a puzzle that sounds a little like those, “A train leaves…” questions we were all prepared for but rarely saw on the SAT, but with a twist.

You are going to take a drive from City A to City B and back, but in a rather unusual car. When travelling uphill, the car always moves at exactly 56 miles per hour. On level ground, it travels at 63 miles per hour and finally when travelling downhill it travels at 72 miles per hour. Assume that it transitions from one speed to another instantaneously and all of those other “mathematically perfect” qualities that make questions like this answerable.

You find that travelling from City A to City B takes exactly 4 hours of travel time. On the return trip, driving time sums to 4 hours and 40 minutes.

How far apart are Cities A and B?

## czarandy said,

April 20, 2009 at 8:02 pm

[spoiler]

It seems there is not a unique solution?

e.g., one choice is that they are 273 miles apart. Of this 105 is level ground, and 168 is downhill on the way there (uphill on the way back).

So on the way there it takes 105/63 + 168/72 = 4 hours

On the way back it takes 105/63 + 168/56 = 4 hours 40 min

But they could also be 271 miles apart: 101 miles level, 170 downhill on the way there, and 2 miles uphill on the way there.

etc.

[/spoiler]

## jyoak said,

April 21, 2009 at 9:10 am

czarandy, about your solution…

[spoiler]

101+170+2 is also 273, not 271. You are correct that the amount uphill, downhill and level aren’t fixed, but the 273 total length is fixed. That’s what made this problem interesting to me. You’re drawn to try to compute two equations in three unknowns which you can’t do. You also have to be quite particular about what speed you select for the relative conditions or you may not have a unique solution. I’ll be posting all about this soon in another comment, but if you want to take this further, take a crack at describing what speeds will and will not have a unique answer.

[/spoiler]

## nklein said,

April 21, 2009 at 11:29 am

That was a fun problem. I spent most of it grudgingly going through the motions expecting to find a range of answers. Alas, they collapsed nicely in the end. Bonus.

## pjrogers said,

April 21, 2009 at 3:50 pm

[spoiler]I also got distance=273 miles

I converted the miles per hour into minutes per hour and set up three equations:

x+y+z=d

9x+8y+7z=2016

7x+8y+9z=2352

doing some linear algebra showed that the matrix would simplify down to the equation 9d-2016=2352-7d. Solving for d gives 273.[/spoiler]

Fun!

## jyoak said,

May 4, 2009 at 3:26 pm

I’ve posted some more of my own thoughts as a followup in a later post here: http://mathfactor.uark.edu/2009/05/04/yoak-followup-to-a-rather-odd-car-trip/ .