My common sense (which is sometimes not all that common or sensical) tells me that if she traveled the 240 mile total trip in 5 hours, her average speed for the trip is simply 240/5 = 48 miles per hour.

I assume the car is not keeping track of the time that she?s driving. It does, of course, know the speed at which she?s driving, and it?s keeping track of how far she?s driven. So, I assume the car says to itself, ?She went 120 miles at 40, and 120 miles at 60. They?re the same distance, so the average speed is (40+60)/2 = 50 mph.?

Is anyone familiar with cars that do this? I?m curious to try it with the GPS in my car. Reset it and drive 20 miles at 40 mph and then speed up to 60 for the next 20. I?m inclined to think that it works off the time though. My handheld Garmin units show a lot more information than the car models, and will actually tell me overall average, moving average, time moving, time stopped, overall time, and a whole mess of other information.

# Driving to grandma's puzzler

**jt1979**#1

I don’t know about the puzzler, but I know GPS units can do weird things with distance and time calculations. The last time I reset the trip computer in my GPS it said my maximum speed was 98 MPH, which is ridiculously impossible in my situation.

**jt1979**#3

Well, they?re only accurate to a certain number of feet, and the car models are less accurate than the expensive outdoor models. The car models (usually) lock onto the road, so you can?t tell that it?s not real accurate, but if you?re using an outdoor model you can see that the position jumps around slightly sometimes, especially when getting poor signal. I?ll bet that when you reset it your position jumped so quickly it registered a higher speed.

That?d be my guess.

**ZombieWoof**#4

A young gal drove 120 miles at 40 mph on the way to see her grandmother. On the way home, she drove 120 miles at 60 mph. She figures it should have taken her 4.8 hours round trip but it took her 5 hours. How come?

Is it possible that her math was wrong? 120 miles at 40 mph is 3 hours. 120 miles at 60 mph is two hours. Add them together and it’s 5 hours. It’s a weighted average. Note that instead of taking 4.8 hours the trip took 5 hours, so instead of averaging 50 miles an hour, she averaged 48 miles an hour.

@jt1979: It’s not the car doing the math, it’s the driver. You’re overthinking.

The sum of the averages is not the same as the average of the sums.

```
Two trips. One 100 miles @ 1 MPH and the other 1 mile at 100 mph
Trip one would take 100 hours. Trip two would take 1/100 hour
*Total miles: 1.01
*Total hours: 1.01
Total average 1 mph Miles (1.01) divided by Total hours (1.01)
```

**jt1979**#6

I just went back and READ the puzzler. I guess I wasn’t listening closely enough when I heard the puzzler. I thought her car’s computer was telling her that her average speed for the whole trip was 50 mph. That made sense to me, because I doubt the car is keeping track of the time, but figured it was likely that the car would figure average speed over distance traveled.

I guess she’s just an idiot. There’s the puzzler answer - “She’s a dummy.”

**jt1979**#7

The sum of the averages is not the same as the average of the sums.Two trips. One 100 miles @ 1 MPH and the other 1 mile at 100 mph

Trip one would take 100 hours. Trip two would take 1/100 hour

*Total miles: 1.01

*Total hours: 1.01Total average 1 mph Miles (1.01) divided by Total hours (1.01)

What in the world are you talking about? Explain to us all how that?s not 101 miles and 100.01 hours. That is not, by the way, an average speed of 1 mph.

**ZombieWoof**#8

Mr. Meehan:

```
With due deference to your age and your very fine Gran Torino http://en.wikipedia.org/wiki/Gran_torino,
```

a. what drove you to obfuscate my perfectly satisfactory answer to the Puzzler, and

b. how does a 101 mile trip end up with a length of 1.01 miles?

I feel sure that there are some punk kids trampling your lawn right now. Once you’ve shooed them off I hope you will enlighten me. Your post makes exactly zero sense.

**missileman**#9

The real answer is 240 miles divided by 5 hours. The mathmatical answer when you use averages is wrong because it’s an average and some averages are not pefect. The reason is that math is “not” a perfect science. You can’t divide a dollar 3 ways.

When I was a teenage, my cousin and I helped my uncle one summer loading logs. The trucks had to be loaded with the large end of the load to the rear of the truck. The load was then calculated by using the width and height of the logs at the rear. This was then multiplied by the length of the logs. The lumber yard and the loggers agreed on this. The reason was that logs are wider at the bottom than at the top. This method assured the loggers would get a fair price for their logs. I still don’t think there is a precise way to measure logs to this day. Some mathmatical formulas get close but close is never precisely accurate.

A 350 engine is not really a 350 engine. It’s close to a 350 but if you do the math it’s not precise.

**jt1979**#10

The real answer is 5 hours.

That can’t be the real answer, because the question was about what caused the discrepancy. We know that it actually took her 5 hours; that was part of the information given in the puzzler.

Some averages are not perfect - because you can?t divide a dollar perfectly into thirds??? I don?t understand the correlation. You can?t divide a dollar into perfect thirds because it is made up of 100 discrete units (pennies), and 3 is not a factor of 100. I think that?s one of the reasons that we use base 60 to measure time; it has more factors. You can perfectly divide an hour into halves, thirds, fourths, fifths, sixths, tenths, twelfths, fifteenths, twentieths, thirtieths, and sixtieths.

Your examples of loggers and a 350 engine not really being 350 don?t really have to do with math being imperfect; they have to do with people rounding things to make measuring them easier, or understanding them less confusing.

**missileman**#11

OK. I thought I had heard about this before and I found it online. It’s a trap. Check this out. I really want to hear the answer when it comes out.

http://www.beatthegmat.com/mba/2010/09/15/common-mistakes-on-average-rate-questions

**meanjoe75fan**#12

The “arithmatic mean” only works if your’re dealing in the terms of the denominator.

MPH=Miles per hour=Miles/Hours. Thus, the arithmatic mean works perfectly IF you’re working in terms of hours.

(ex: 1hr. @ 60MPH avg + 1hr. @ 40MPH avg = 2hr. @ 50 mph avg.)

While she drove identical MILES each leg, the return leg (because she drove it slower) contained more HOURS. Thus, the larger amt. of time drive the average away from “right down the middle” towards the slower, longer leg.

(This is why, BTW, some claim the EU l/km is better than mi/gal. On paper, slowing down to raise the MPG on your work truck from 10 to 11 MPG looks “not worth it” compared to the Prius driver who does the same to raise his MPG from 50 to 55. In reality, you saved 5X as much fuel over a given distance.)

**ZombieWoof**#13

No, missileman, the mathematical answer is right. You just don’t get that it’s a weighted average. Your example about dividing a dollar three ways is wrong. Example: 60 cents, 20 cents, 20 cents. There. I divided a dollar three ways. If you’re just saying that 1/3 is .3 repeating, well, that’s right, but it’s irrelevant.

Let’s see where else you went wrong. “I still don’t think there is a precise way to measure logs to this day.” Hmmm, how about some sort of device that would measure their weight? We could call it a “scale”.

**ZombieWoof**#14

I did not think the concept of weighted average could be further obfuscated, but you have managed it.

**missileman**#16

It’s a trap. I guess you did not read the article I listed. Until you become an instructor for Kaplan, I will listen to the expert. Also…wood is not sold by weight…it’s sold by board feet. You cannot divide a dollar 3 ways equally and it’s very relevant. It’s always a pleasure to spar with you so I await your next comment.

**jt1979**#17

From the Kaplan article:

What caused the discrepancy is that she made the mistake they just told people not to make.The most common of these is to introduce a problem along the following lines: a truck driver drives from town A to town B at a speed of 40 miles per hour and returns from town B to town A at a speed of 60 miles per hour; what is the average speed for the journey? Here the test makers have only included information on speed, but do NOT choose 50 miles per hour, which will likely be listed as a choice. Instead, pick a number for the distance, which we know is the same each way, calculate the times in either direction and then divided total distance by total time (in this case, the answer is 48).

No trap, no trick, just an incorrect application of math, Missileman. The Puzzler is identical to the situation they just outlined in the article you found.

**jt1979**#19

Am I the only one who thinks that people want to make this more confusing than it should be? Total distance traveled/total time spent traveling = average speed?

Now, I?ll admit, I never took the SAT. The closest I took to “SAT” was the LSAT. Here in Iowa we take the ACT. I only got a 33 on the math portion. I know it?s only a very small fraction of the score that those who took the SAT get. Still though, this seems pretty simple. I can see how there could be confusion in the KAPLAN example given, if there?s no distance given (and you don’t realize you need to solve further). As it is though, this seems pretty elementary.

**circuitsmith**#20

Let me change what Joseph wrote a little.

Round trip: One 100 miles @ 1 MPH going out and 100 miles at 100 mph coming back in.

Trip out would take 100 hours. Trip in would take 1 hour.

*Total miles: 200

*Total hours: 101

Total average speed = (200 miles) / (101 hours) = 1.98MPH

Average of the two trip speed averages = (1MPH + 100MPH) / 2 = 50.5MPH

As meanjoe says the total average is lower because most of the time is spent at the low speed.

And this comes form someone who got a 720 math on the SAT, in 1976 when the #s were less inflated.