# This week's Car Talk puzzler

Short version:

A guy goes jogging. He jogs for a while over level ground at 8mph. He then starts up a hill, slowing to 6mph. At the top he turns and heads back, running at 12mph down the hill. He then finishes the jog along the level ground, again at 8mph.

When he gets home he sees he has been running for exactly two hours. So how far did he run?

I know there is a fairly simple way to solve this maths problem but I’ve always sucked at maths and so I’d be interested to hear how this is solved. Anyone?

Smartass answer: Zero. He is still at home after the jog, so he didn’t run anywhere.

Reasonable answer: since he maintained the same speed over flat ground for the same distance, then the other parts of the run have to be the same as well. I’m saying 17 miles, but I know that’s wrong.

H.

Numbers after “x” are subscript.

8x1 + 6x2 + 12x3 + 8x4 = y
x1 + x2 + x3 + x4 = 2
x1 = x4
6x2 = 12x3

x2 = 2x3

Figure it out from there.

I could be wrong though.

12 MPH and 6 MPH over the same distance averages out to 8 MPH. (Think of running 4 miles up the hill–that’d take 20 minutes at 12, 40 minutes at 6, giving you 8 MPH).

So, after 2 hours you get 16 miles.

Houngan is right–the setup of the question pretty much tells you the answer; you know that it has to average out to 8 MPH, or the problem would be unsolvable.

Sixteen miles.

Forget the 8mph level ground part for now, and assume (for easy calculation) the path up the hill is 12 miles long. It’ll take you two hours at 6mph to get up, and 1 hour at 12mph to get down. That’s a total of three hours to cover 24 miles, ie, 8mph overall for the hill. The distance itself doesn’t matter, as the 2:1 proportion will be the same.

So – he’s been running an average of 8mph the entire time, which makes for sixteen miles over two hours.

Edit: dammit Gav.

Er my guess was 17 too; we know at the very least he went 16.

The assumption therefore is that he spent equal times jogging flat, then uphill, then downhill, then flat again (if you split everything into half-hour segments).

Something makes me think there’s a trick in doubling the speed running downhill and how it’d only take half the amount of time it would take to go uphill or something but my brain is not working right yet.

— Alan

50%, motherfuckers.

/obligatory riddle snark

I thought this as well, but I think it applies to percentages, like returns on stocks, rather than speed. I admire the elegance of Gav & Andrew’s thought processes, but I lack subtlety and simply brute forced my way through the algebra and got 16 miles.

Let x = distance on flat ground, and y = distance from base to top of the hill. Then the total distance one way is x + y, which makes the round trip distance 2*(x+y).

The amount of time it takes to get to the top of the hill is: (x/8) + (y/6)

The amount of time it takes to get back home is: (y/12) + (x/8)

The total time is 2 hours, so:
(x/8) + (y/6) + (y/12) + (x/8) = 2

Simplifying & combining terms, we get:
(x/4) + (y/4) = 2,

which is equivalent to
2*(x+y) = 16

So 16 miles is the round trip distance

i like how avg(6,12) = 8 to you guys

Assuming it’s not the smart-alec answer, I’d suggest considering the two extreme cases:

1. The flat level ground really goes the whole way, and the hill is is negligible. So it took 2 hours at 8mph, and it’s 16 miles.

2. The flat level ground is negligible, and the rest is the hill. So the distance covered is 6x + 12(2-x) where x is the time used to go up the hill and 2-x is the time to go down (x < 2). But we also know that 6x = 12(2-x) because it’s the same distance both ways, so x=2(2-x), or x=4-2x, or 3x=4 or x=3/4. At that rate we learn that the distance covered is 6(3/4) + 12(5/4) = 78/4 = 19 1/2.

Since the answer to 1 is different from the answer to 2, in fact there is no right answer because we don’ t know how big the hill was, but it was between 16 miles and 19 1/2 miles.

Of course I could be wrong, because I often am at things like this.

I bolded your error. If 3x = 4, then x = 4/3 (not 3/4).

Total distance covered is then 6*(4/3) + 12*(2-[4/3]), or 8 + 12*(2/3), or 8 + 8 = 16, which is the same answer as your first extreme case.

Excellent! But anyway, the real point is the extreme end-cases approach shows the answer…

OK, OK, that’s not the point. The point is obviously that there’s some simple insight to demonstrate that the answer is 16 regardless of the hill, but that I missed it pretty egregiously.

Edit, oh yeah, and that people pointed it out earlier, but I didn’t want to spoil the problem, so I ignored the interim postings.

The jogger takes off from the runway.

Tom of Car Talk has passed away. RIP.
http://www.npr.org/2014/11/03/357428287/tom-magliozzi-popular-co-host-of-nprs-car-talk-dies-at-77

Sad news. That show was great, even the hokey jokes were heartwarming more than groan-worthy.

Hopefully Ray won’t be like his brother.

That’s sad. I wasn’t aware that Tom had alzheimer’s, which might be why they retired 2 years ago.

Alas.

Damn. RIP.

What a bummer :( I listen to the reruns regularly and those two had a special chemistry between them. Poor Ray lost his brother and his best friend.