# Average speed

#### Mark44

Joined Nov 26, 2007
628
A certain road goes up one side of a hill and back down the other side. Each of the two segments is one mile long. If I drive my car up the hill at a constant speed of 30 mph, how fast must I drive the downhill segment to average 60 mph for the entire trip?

#### studiot

Joined Nov 9, 2007
4,998
Last time I did this I was going so fast I met myself coming back.

#### MusicTech

Joined Apr 4, 2008
144
Well logically:

(30+x)/2=60

30+x=120

x=90

so 90 MPH, but I did do that in my head, so it is probably wrong, can we have a hint

#### jpanhalt

Joined Jan 18, 2008
8,118
The time for a 2-mile trip at 60 MPH is 2 minutes. When you get to the top of the hill at 30 mph, there is no time left. That's what studiot meant. John

#### thingmaker3

Joined May 16, 2005
5,084
The car has a "teleportation gear," doesn't it?

#### mrmount

Joined Dec 5, 2007
59
At infinite speed!

#### Mark44

Joined Nov 26, 2007
628
Most of the respondents figured this one out. It's not difficult, but it does seem to fly in the face of intuitive reasoning, which falters when you try to take the average of things that are already averages.
Mark

#### Dave

Joined Nov 17, 2003
6,970
I bet you some people still don't get it when you explain it to them because it seems counter intuitive.

Dave

#### jpanhalt

Joined Jan 18, 2008
8,118
It's not difficult, but it does seem to fly in the face of intuitive reasoning, which falters when you try to take the average of things that are already averages.
Mark
How very true, and how often that mistake is made - not for car speeds, but in business. I can't count the number of times I have dealt with managers (many with MBA's) who didn't understand the problem. My usual advice was, get rid of the ratios, go back to the original data, add it together and recalculate.

When I would try to describe the statistical problems of dealing with ratios, they would typically dismiss it as some arcane detail. That is, until they met a real-life problem caused by misuse of ratios (averages), like budgets.
John

#### MusicTech

Joined Apr 4, 2008
144
Ok, I see, so it would require at least 90 MPH for 2 minutes to go down the hill, but at 90 MPH you would only have 40 seconds to do that. Sorry I am so slow to do that. I have never seen this before, so anyway that's pretty cool. I really want to share this with my math class.

#### Mark44

Joined Nov 26, 2007
628
I think you have it, but I'm not sure you're explaining it very well, so let me take a stab at it. To average 90 mph for the entire 2 mile trip, you would have to cover that distance in 80 seconds. The uphill mile has already taken you 2 minutes, or 120 seconds, making it impossible for you to travel both miles in less time than you've already used.

#### MusicTech

Joined Apr 4, 2008
144
yep, that's what i was trying to say. thanks. I was never good at writing.