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RAWDuke
09-23-2002, 01:00 PM
I'm sure you've had this discussion before, but I couldn't find it in a search.

First, the puzzle:
You're driving a race car on a one mile oval track. You drive one lap at 30 MPH. How fast do you have to drive the second lap in order to average 60 MPH for BOTH laps?

I first saw this in Ask Marilyn several years ago. (Marilyn, BTW, along with Isaac Asimov and James Randi are my heroes!) She got plenty of letters from people, many angry, because they couldn't understand the answer.
When I present this to people I know, the immediate (and wrong) answer is "90 MPH." My older brother (PHD) is the only person I know who got it right.

I won't give the answer yet, but my main question is Why do people have such a problem figuring this one out?

Ethilrist
09-23-2002, 01:09 PM
Because we see 30 + 90 = 120 ; 120/2 = 60. If there's some kind of trick answer, people have problems with it because they hate trick answers. If it's a semantic trick, like, "You can't average anything over 30 mph for the first lap because we've already stated the speed" it's just gonna annoy people something fierce.

Kn*ckers
09-23-2002, 01:18 PM
Ethilrist 's summed it up. If it's a trick question, I'm gonna beat someone. I hate those.

On the other hand, if it's genuine math, I'm stumped. I thought it was 90.

Kn(dumber than she looks)ckers

dwc1970
09-23-2002, 01:24 PM
The only way that this could work using the simple and obvious answer is to have a car that could instantly accelerate straight to 90 mph after going 30 on the first lap, which would be impossible. One has to account for acceleration, which complicates the problem. One would have to know the rate of acceleration to truly get an accurate average, and unfortunately my math is not advanced enough to make such a calculation.

RAWDuke
09-23-2002, 01:27 PM
It's not a trick question, it is genuine math.
I know, your initial reaction is "30+09 = 120 and devided by 2 is 60" but you don't compute averages by comparing rates of speed, rather you calculate distance driven divided by time.

Lamar Mundane
09-23-2002, 01:27 PM
To average 60 mph (1 mile/min.), you must drive two laps (2 miles) in 2 minutes. Lap 1 at 30 mph (2 mins/mile) takes you two minutes. Therefore you must finish the second lap in 0 minutes. No problem.

Genseric
09-23-2002, 01:28 PM
It is because it takes you two minutes to drive one mile at 30 miles per hour, and no matter how much faster you drive the next lap, it's still going to take more time to go another mile? To average 60 MPH you have to cover two miles in two minutes. Just a wild guess.

Legomancer
09-23-2002, 01:28 PM
Okay, if you average 60mph for both laps, that means you have to take 2 minutes total to drive both laps. (60mph = 1 mile per minute. Each lap is a mile, so two laps = two miles = 2 minutes.)

In the first lap, we went 30mph. 30mph = 1 mile in 2 minutes.

OOPS! We ALREADY took the 2 minutes in the first lap! We can only hit that 60mph if we take 0 minutes to do it. Unless we can travel instantaneously, that ain't happening.

Even if we drive 90mph in the second lap, 90mph = 1.5 miles per minute. So it'll take 40 seconds to do the lap. You would have taken a total of 2 minutes, 40 seconds to do both laps. 2:40 = 160 seconds. Divide by 2 and that's 80 seconds per lap, which is over 60mph.

Beastal
09-23-2002, 01:28 PM
I'm guessing the answer isn't 60 mph because your second lap will be quicker than your first, hence you aren't travelling 90 mph for the same length of time that you are travelling at 30 mph.

Genseric
09-23-2002, 01:29 PM
Wow, I guess we ain't so dumb after all.

AndrewL
09-23-2002, 01:30 PM
Very tricky question. I got the wrong (90MPH) question at first, then had to sit down and work out the math to get the real answer. And it's not a trick question.

moe.ron
09-23-2002, 01:31 PM
I don't know the answer to this question, but my guess would be that you can't average 60 mph.

Ideally, at 30mph, it would take 2 minutes to get around the track. While at 60 mph it would only take one minute to get around the track. Therefore, if your average mph was 60, you would have already completed the 2nd lap, while if you were going 30 mph would just be completing the first. Do I get a cookie?

Why people would get this wrong is probably because the average of 30 and 90 is 60, but this fails to account for time. It's really just a question of how you approach problems, conventionally or creatively. The experienced problem solver knows not to jump to conclusions, while most others will look for easy answers.

RAWDuke
09-23-2002, 01:40 PM
ANDREWL, I like that. It's not a trick question, but it's a tricky question!

GENSERIC, not such a wild guess, and I guess I'm impressed too! Not such dumb peoples here!

I'm curious, though. If you mention this puzzle to your friends, how many instantly say "90?" And how many understand the explaination when you say they're wrong?

hawthorne
09-23-2002, 01:51 PM
If you were going to average 60 mph over two 1 mile laps, you'd do it in 2 minutes. The puzzle is therefore massive.

Genseric
09-23-2002, 01:52 PM
Almost everyone who thinks about it for ten seconds will say 90 MPH. (Unless they are incapable of any math at all.) It's only those who are knowledgeable in physics and math, or those who know that 90 is not the correct answer and give it more thought (and understand what the concept of Miles Per Hour, really means) who get it right.

hawthorne
09-23-2002, 01:53 PM
Polite, but slow. That's a first for me.

The Ace of Swords
09-23-2002, 01:53 PM
Put me down for "infinite." Averaging 60 MPH would mean you've already completed both laps at T=1, whereas in the puzzle you're just finishing your first lap.

If that's not a trick question, I don't know what is. Are you sure the puzzle's worded correctly?

-Ace

RAWDuke
09-23-2002, 02:04 PM
ACE0SPADES, either infinite or not possible are correct.

I guess "trick question" depends on your definition. I don't see it as a trick, because all the information you need is provided. "Tricky," yes, because the easy answer is incorrect, but that's true with most brain teasers.

Genseric
09-23-2002, 02:10 PM
If you take two more laps at 120, it will average 60 MPH, though. But that wasn't the question.

Scylla
09-23-2002, 02:18 PM
It is slightly angering because it's a trick question. It's only tricky because of the way it's phrased.

It's one of these "It sounds like it's asking one thing, but it's really asking another" sort of questions.

It has little to do with math, and everything to do with semantic analysis. The fact that people not to get it says to me that there is poor communication going on when the question is asked, and the failing is on the part of the asker.

It seems as if the question is asking for the average of two seperate units (laps around the track) while in fact it is asking for the average rate of a single unit.

roadrunner70
09-23-2002, 02:24 PM
...so what happened to the other dollar?

:::d&r:::

Genseric
09-23-2002, 02:35 PM
Well, it is a Brain Teaser. If they were clear and unambiguous, then they would just be ordinary questions. The point is to delude you into thinking you have the correct answer, when in fact, you don't, because you didn't read the question carefully enough.

Zebra
09-23-2002, 02:39 PM
What you have to catch is the 1 mile part. Why is that fact put into the question? Why is 1 mile important? What if it were five miles? If you only take in the 30 mph 1 lap and ? for lap 2 that will lead you to the wrong answer because you didn't take in the 1 mile per lap.

The Ace of Swords
09-23-2002, 03:11 PM
The distance is unimportant. Clearly, I cannot average, after going distance X at Y, go distance X again so as to average 2Y, since I'd already be done.

Try changing the distance to 30 Miles -- same answer.

RawDuke, I think you hurt your own question! The units you gave are impossible, but many people are stumped by the general puzzlers that involve calculating (seemingly) both time and distance simultaneously.

Try this, solvable puzzler. A lap is 50 miles, and Racer A goes 75 MPH full out. Racer B, conserving his Nitrous, goes 50 MPH for one lap. How fast does he have to go on the second lap to catch Racer A?

Ethilrist
09-23-2002, 03:28 PM
So this falls under the heading of "trick question," which people hate. Step back from the keyboard and prepare to receive a Wet Willie you will not soon forget.

RAWDuke
09-23-2002, 04:05 PM
ACE, I need to assume that both racers finish the second lap at the same time? If not, racer b could catch racer by flooring it and going, say, a million MPH and catch A in, ummmm, real quick!
Otherwise, at 75 MPH, A will finish 2 laps in 1 1/3 hours. Racer B would finish one lap in one hour, that leaves him 1/3 hour to travel 50 miles, which he could do at 150 MPH. Correct?

ETHILRIST, I would be happy to step back and receive that "Wet Willie" and might even enjoy it!
Would you first:
a. Explain just what a Wet Willie is?
b. Explain (and I'm not trying to be rude here, I'm truly curious) why do you think this is a "trick?" What is it about this teaser that is ambigous? I guess I don't see why people feel tricked.

The Ace of Swords
09-23-2002, 04:16 PM
Your answer is correct.

The original question is set up to "trick" one, IMO, not by the wording, but by the impossible answer, and the use of a track oval, instead of a straight line back and forth.

That said, I enjoy a good trick question, though "Infinite" feels a little unsatisfying.

-Ace

RAWDuke
09-23-2002, 05:30 PM
The distance is unimportant. Clearly, I cannot average, after going distance X at Y, go distance X again so as to average 2Y, since I'd already be done.

I read this at work, and it just didn't sink in. I just re-read it. I didn't get the "distance in unimportant" part. I guess the ability to visualize a one mile track helped me figure it out.

But you're right, ACE the distance doesn't matter. This is a very elegant explaination.

But I still have to disagree with the idea that this is a "trick" question.

To me, this is a "trick" question:
"Which weighs more, a pound of feathers or a pound of gold?"
To answer this question correctly, you need to know that gold is weighed in Troy measurement, in which there are 12 ounces to a pound. Which means that a pound of feathers weighs more.

Treviathan
09-23-2002, 06:14 PM
Funny, when I read the thread title, I knew it was either going to be the question about the three men who pay $30 to stay at the hotel, or this one.

Shade
09-23-2002, 07:44 PM
The thing to remember is than there are different sorts of averages. For most things the most common average is the mean (add them all up and divide by the number of things).

But average speed means "total distance travelled divided by total time taken" or "speed that if I had gone the whole way at, would have taken me the same time" because this is the only one that's any use.

The trick is that this is different to the normal average. Both are averages, but the one wanted here is worked out differently, as many people have pointed out.

Cabbage
09-23-2002, 08:48 PM
How about the speed of light? No time passes for you as you complete the second lap, so you manage to complete both laps in two minutes. In order to average faster than 60 mph, simply travel faster than the speed of light. ;)

Horseflesh
09-23-2002, 09:03 PM
So the smartass answer to this question is that the second lap is made in 0 seconds. There would be no way for an observer or the driver to discern whether or not the car had made the second lap.

The laws of physics don't apply here according to how the question is worded. As dwc1970 pointed out, you'd have to have a car that could instantly accelerate to 30MPH and travel exactly one mile for the first lap (not to mention we're not taking into account tire friction, air resistance, engine performance consistancy, and inprecise measurement equipment). Even if the required average were dropped to 40MPH for both laps, the second lap would need to be driven at 50MPH and the car would need to instantly accelerate from 30MPH to 50MPH exactly at the finish/start line. Even The Fast and the Furious didn't have cars like that.

Cabbage
09-23-2002, 09:32 PM
Actually, I don't see any trick in the way the problem is worded. It really is a legitimate question--there's no trick, just an answer that most don't expect (or even accept, many times). In fact, try this wording:

You drive the first lap averaging (so any time "wasted" for acceleration or whatever is taken into account here) 30 mph. How fast must you average for the second lap in order for the average speed for the entire trip to be 60 mph?

I guarantee most people will still answer 90 mph, though the answer is still "impossible" (minus any relativistic effects ;)).

monica
09-23-2002, 10:43 PM
Is the 30 constant in the first lap?

Zebra
09-26-2002, 10:31 AM
The distance is important in that you have to think about the distance the driver has to travel. Yes the answer is the same no matter how big the track but if a person only thinks

1 lap = 30
both laps = 60
so 2nd lap must = 90

is not taking into account the lap = 1 mile in the original question.

Because you are not traveling at laps per hour, you doing miles per hour.

RAWDuke
09-26-2002, 10:59 AM
So the smartass answer to this question is that the second lap is made in 0 seconds. There would be no way for an observer or the driver to discern whether or not the car had made the second lap.

Actually, this isn't smartass at all. It's correct.

How about the speed of light? No time passes for you as you complete the second lap, so you manage to complete both laps in two minutes.

Even at the speed of light, some time would elapse (the time it takes light to travel one mile)
You're talking a smidge under 60 MPH average, but it's still under.

Thudlow Boink
09-26-2002, 11:00 AM
If you're interested in why people have trouble with math problems like this one, may I recommend the wonderful book Mathsemantics by Edward MacNeal? It may be out of print, but if you can track down a copy, it's well worth reading.

kabbes
09-27-2002, 09:16 AM
We actually ask this question in our interviews for graduates. A remarkable number give the wrong answer. Personally I gave the right answer within about 2 secs. And I hadn't heard it before. It just seems obvious to me.

It's obvious because when you average a speed, you should do so weighted by the time you travelled at that speed. For instance if you travel 2 hours at 30mph and 1 hour at 60mph then your average speed is (2x30 + 1x60)/(2+1) {Note this is just another way of saying speed = distance/time}

In the question you can't average 30 + 90 to get 60 because you will have travelled one lap in 2 mins and one lap in 40 seconds. Anyone that has ever worked with weighted averages should immediately recognise that that doesn't work. In this instance the average of 30 and 90 is (30x2 + 90x0.6666)/2.6666

pan

Ethilrist
09-27-2002, 09:26 AM
Originally posted by RAWDuke
ETHILRIST, I would be happy to step back and receive that "Wet Willie" and might even enjoy it!
Would you first:
a. Explain just what a Wet Willie is?
b. Explain (and I'm not trying to be rude here, I'm truly curious) why do you think this is a "trick?" What is it about this teaser that is ambigous? I guess I don't see why people feel tricked.
a. To perform a Wet Willie, one first wets one's index finger to the first joint by inserting it into one's mouth; then, one approaches the Willie recipient and inserts it into the (hopefully unsuspecting) victim's ear and rotates the finger vigorously. This is rumored by some to be strangely pleasant, and by others to be monumentally unpleasant.
b. When presented with a word problem, the basic assumption is that there is a way to solve the problem. If, in fact, there is no answer, it's kind of irritating. Suppose, for example, I presented you with a popup ad saying, "Punch the monkey to win $1,000;" if you punched the monkey but didn't win the $1,000 because of some bizarre arbitrary reason, you would be justifiably annoyed.

Pábitel
09-27-2002, 10:40 AM
Such questions are not about logical thinking or being good at physics or anything else. They are about stating a simple question in the most obscure way possible in order to make a fool out of the person you target with your little practical joke.

How many people would miss it if it were asked more clearly?

"If you travel a mile at a rate of 1/2 mile per minute taking two minutes how fast will you have to travel for a second mile to cover a total distance of 2 miles in two minutes?"

It is the intentionally poorly asked question not anyone's answer to it that is in error.

Someone mentioned that they use this question for interviews for graduates. What?!?!

If I were met with a silly practical joke by someone I was interviewing with for any reason I would at least refuse to play and at most walk out. I cannot imagine why you would want to subject a person already in a stressful situation to a pointless amusement for you and you fellow interviewers. I honestly cannot imagine any useful information you could possible expect to garner by asking the question.

Playing a practical joke on someone by dropping rotten eggs on them is no worse than playing this kind of practical joke. There are about a dozen ways to write a joke question. Some have a "correct" answer. Some, like this, have no answer. When people hear one they immediately start thinking, "Which kind of joke is it?" not, "What is the right answer?" Even in this thread there was as much discussion about what kind of a trick question it was as there was discussion about what the "right" answer was.

Play with these if you want but don't fool yourself into thinking that you are a mental giant and others are idiots just because you got the joke first.

The Ace of Swords
09-27-2002, 11:47 AM
And divining the obscure is not any hallmark of intelligence?

All I know is....








ME MENTAL GIANT! ! HA! HA! RUN LITTLE PEOPLES FROM MY PUZZLE CRUSHING MIND!!


Oh, damn, I think I fooled myself.

Cabbage
09-27-2002, 12:23 PM
Even at the speed of light, some time would elapse (the time it takes light to travel one mile). You're talking a smidge under 60 MPH average, but it's still under.
Well, actually I was being facetious (since you can't travel the speed of light), and I am not a physicist, but it's my understanding that if you could travel the speed of light, no time would pass for you. So you would complete the second lap in literally zero time, and complete both laps in two minutes. It's also my understanding that if you could travel faster than the speed of light, time would flow backwards for you--in doing that, you could actually average over 60 mph.

Tiburon
09-27-2002, 12:30 PM
Hm. I read what Degrance wrote and I have to respectfully disagree that asking a question like that posed in the OP is as bad as dropping rotten eggs on someone. Part of this is because I do not see asking questions like this as "practical jokes."

I like these riddles because if I get the wrong answer, I don't assume it was the asker's intention to make a complete fucking asshole out of me. I think it is just a fun little puzzle that you need to asked in order to appreciate. Lemme take a poll, though -

Would you rather be asked more questions like that posed in the OP or have some rotten eggs dropped on you?

Tibs.

amarinth
09-27-2002, 12:34 PM
Originally posted by Degrance
Such questions are not about logical thinking or being good at physics or anything else. They are about stating a simple question in the most obscure way possible in order to make a fool out of the person you target with your little practical joke.

How many people would miss it if it were asked more clearly?

"If you travel a mile at a rate of 1/2 mile per minute taking two minutes how fast will you have to travel for a second mile to cover a total distance of 2 miles in two minutes?"
What kind of physics classes did you take?

Did the problems say "John traveled 3 miles in 2 hours, which averages to 1.5 miles/hour. What was his average velocity?"

Given the rate and the distance, I expect a person to be able to figure out the time. I should not have to explicitly state it because it is right there. The question is phrased as any general 2 part problem, rate, distance, find the missing variable, use it and and the second distance to find the second rate. Basic algebra/physics. There is nothing hidden, there's nothing tricky, and it is very simple math. I see nothing wrong with asking graduate students to do that. I see nothing wrong at all in expecting that potential hires be able to do very basic algebra (but then, I work in tech). The useful information gained is that they're able to work through a problem instead of jumping to a conclusion, and that they have that pesky multiplying thing down pat.

Not a trick question at all.

amarinth
09-27-2002, 01:01 PM
Originally posted by amarinth
Did the problems say "John traveled 3 miles in 2 hours, which averages to 1.5 miles/hour. What was his average velocity?"I need to amend this to say "John traveled 3 miles in a straight line..."

RAWDuke
09-27-2002, 02:44 PM
How many people would miss it if it were asked more clearly?
"If you travel a mile at a rate of 1/2 mile per minute taking two minutes how fast will you have to travel for a second mile to cover a total distance of 2 miles in two minutes?"



So.. we should make it easy enough so it's no longer a "brain teaser?"

This reminded me of the joke about "Math History"

Teaching Math in 1950: A logger sells a truckload
of lumber for $100. His cost of production is 4/5
of the price. What is his profit?

Teaching Math in 1960: A logger sells a truckload
of lumber for $100. His cost of production is 4/5
of the price, or $80. What is his profit?

Teaching Math in 1970: A logger exchanges a set
"L" of lumber for a set "M" of money. The cardinality
of set "M" is 100. Each element is worth one dollar.
Make 100 dots representing the elements of the set
"M". The set "C", the cost of production, contains
20 fewer points than set "M." Represent the set "C"
as a subset of set "M" and answer the following
question: What is the cardinality of the set "P"
for profits?

Teaching Math in 1980: A logger sells a truckload
of lumber for $100. Her cost of production is $80
and her profit is $20. Your assignment: Underline
the number 20.

Teaching Math in 1990: By cutting down beautiful
forest trees, the logger makes $20. What do you
think of this way of making a living? Topic for
class participation after answering the question:
How did the forest birds and squirrels feel as the
logger cut down the trees? There are no wrong
answers.

a. To perform a Wet Willie, one first wets one's index finger to the first joint by inserting it into one's mouth; then, one approaches the Willie recipient and inserts it into the (hopefully unsuspecting) victim's ear and rotates the finger vigorously. This is rumored by some to be strangely pleasant, and by others to be monumentally unpleasant.

Yaagghhhh!! Put me in the Monumentally Unpleasant catagory. Although....in the right situation....with the right person...maybe....Yaagghhhh!!!!!



b. When presented with a word problem, the basic assumption is that there is a way to solve the problem. If, in fact, there is no answer, it's kind of irritating.

I hope we can agree to disagree on this one. If you didn't have all the information you needed to solve the problem, I would agree that it would be a "trick" question. Such as if this were multiple choice, and "this is not possible" were not one of the choices, I would be very irritated.
However, everything you need to know is presented (except it's assumed you have a basic math and english knowledge)

ZooMetropolis
09-27-2002, 03:22 PM
Originally posted by Tiburon

Would you rather be asked more questions like that posed in the OP or have some rotten eggs dropped on you?
A: More questions........ yes, more questions.

Voyager
09-27-2002, 07:29 PM
Here's the general solution, showing that it doesn't matter how long the track is.

Say the distance around the track is d.

The time needed for the first lap is d/30 (units of furlongs per fortnight or whatever.)

The time needed for two laps at an average of 60 mph is

2d/60 = d/30, exactly the same amount of time as it took to go one lap, so you'd have to do it at infinite (> c) speed.

As for travelling at the speed of light - it might work in the reference frame of the traveler, but not in the reference frame of the observer.

Nice puzzle.

Johnny L.A.
09-27-2002, 09:14 PM
After a full day at work and a long commute, my brain is a bit frazzled. Let me get this straight: In order to average 60mph over two miles, you have to travel two miles in two minutes. (Assume instantaneous accelleration, since most of this sort of problem usually uses a "perfect system".) Since you have used two minutes to go one mile, you have no time left to go the remaining mile. If you drove the second mile at 600mph, then it would take you six seconds, right? So you will have made two circuits in 2.1 minutes. That averages to about 57mph.

Right?

Dr. Lao
09-28-2002, 12:00 AM
Originally posted by Johnny L.A.
Right? Yes, you are correct. And the question is not a trick. If you calculate average velocity the way you are supposed to (total distance travelled / total time) then it quickly becomes apparent that you cannot average 60 MPH over 2 miles if you travelled the first mile at 30 MPH. OK, maybe it is a little bit of a trick.

Pábitel
10-01-2002, 08:10 AM
Originally posted by Tiburon
Hm. I read what Degrance wrote and I have to respectfully disagree that asking a question like that posed in the OP is as bad as dropping rotten eggs on someone. Part of this is because I do not see asking questions like this as "practical jokes."

Just to be clear I intended this to be in regards to asking this type of question in an interview or other situation where one does not like to look bad, where some smug interviewer can sit and look down his/her nose at the object of their little amusement. In that case, yes, I would prefer the rotten eggs at least then the jerk couldn't rationalize his objectionable activities.

Pábitel
10-01-2002, 08:18 AM
Originally posted by amarinth
There is nothing hidden, there's nothing tricky, and it is very simple math. I see nothing wrong with asking graduate students to do that. I see nothing wrong at all in expecting that potential hires be able to do very basic algebra (but then, I work in tech). The useful information gained is that they're able to work through a problem instead of jumping to a conclusion, and that they have that pesky multiplying thing down pat.

Not a trick question at all.

The original question asks, "How fast do you have to drive the second lap in order to average 60 MPH for BOTH laps?" That is directly stating that there is a speed, a correct answer, at which this problem can be solved. When you build that assumption into the question then you are attempting to trick the person into thinking that there is a correct answer. There is not a correct answer. Therefore this is a trick question.

Got it?

Bryan Ekers
10-01-2002, 08:52 AM
Want to watch me drive the second lap?

Want to see it again?


(old joke, I know)

RAWDuke
10-01-2002, 09:07 AM
The original question asks, "How fast do you have to drive the second lap in order to average 60 MPH for BOTH laps?" That is directly stating that there is a speed, a correct answer, at which this problem can be solved. When you build that assumption into the question then you are attempting to trick the person into thinking that there is a correct answer. There is not a correct answer. Therefore this is a trick question.

But there IS a correct answer. "It's not possible" is a perfectly correct answer, and you are given everything you need in the original puzzle to reach this answer.
You are tricking yourself when you assume something that's not correct, and not implied.

Mr. Cynical
10-01-2002, 09:24 AM
This is a math question designed to throw you off. Were it not, the question would ask if it were POSSIBLE, rather than how to do it.

Deception + distraction = trickery.

Is that easy enough for you guys to figure out? Sheesh.

amarinth
10-02-2002, 02:57 AM
Originally posted by Degrance


The original question asks, "How fast do you have to drive the second lap in order to average 60 MPH for BOTH laps?" That is directly stating that there is a speed, a correct answer, at which this problem can be solved. When you build that assumption into the question then you are attempting to trick the person into thinking that there is a correct answer. There is not a correct answer. Therefore this is a trick question.

Got it?
No.

It doesn't require any outside information, it doesn't require that the solver do anything but set up the equations and solve the problem. You do the stated problem exactly the same way that you would do the problem if you were calculating for an average speed of 57mph for both loops. Would that have been a trick question because the answer is 600mph rather than 84mph?

The problem can be solved. And it can be solved the way that someone should expect to solve this kind of problem. This isn't tricky.

Gary T
10-02-2002, 12:56 PM
What makes this a trick question in some views is that it asks for something specific which cannot be provided. There exists no speed at which one can drive to correctly answer "How fast do you have to drive the second lap...".

By asking WHAT the speed would be, the question implies that there IS such a speed. The correct answer (it's impossible) is not a response to the question actually asked, but to the unasked question of whether there is ANY speed at which it can be done.

The Ace of Swords
10-02-2002, 01:46 PM
I agree, hence the second question I provided, which exactly the same percentage of people will get wrong, as the underlying math problem is what trips you up, though Problem A's answer is annoyingly unsatisfying.

Anyhow, are we moving on yet?

Here, I'll post another brain teaser in a new thread.

-Ace

psychobunny
10-02-2002, 09:13 PM
Originally posted by Horseflesh
Even if the required average were dropped to 40MPH for both laps, the second lap would need to be driven at 50MPH and the car would need to instantly accelerate from 30MPH to 50MPH exactly at the finish/start line. Even The Fast and the Furious didn't have cars like that.

Actually, to average 40mph you would have to drive the second lap at 60 mph.:D

ITzSmores
02-19-2017, 02:14 AM
You do the 1st lap in 2 minutes/30mph and you have to complete a total of 2 laps at this rate 2 laps would take 4 minutes and would be averaged at 45 mph to get an average of 60mph on the second lap you would have to do the second lap at 60mph cause 30+60/2 is equal to 60 aswell the laps are never said to be consecutive nor does it say he doesn't build up speed until he hits 30mph and then keeps a steady speed for his first lap and if the laps aren't consecutive then he could build up speed until he gets to 60mph after the first lap and then once he hits it steady speed for the second lap average those out and he drives an average of 60mph. Or you can take 45 break it into 3 get 15 and multiple that by the 4 minutes and get 60 for the second lap (btw you divide by 3 cause that's how long the 2 laps would take)

Mean Mr. Mustard
02-19-2017, 08:02 AM
You do the 1st lap in 2 minutes/30mph and you have to complete a total of 2 laps at this rate 2 laps would take 4 minutes and would be averaged at 45 mph to get an average of 60mph on the second lap you would have to do the second lap at 60mph cause 30+60/2 is equal to 60 aswell the laps are never said to be consecutive nor does it say he doesn't build up speed until he hits 30mph and then keeps a steady speed for his first lap and if the laps aren't consecutive then he could build up speed until he gets to 60mph after the first lap and then once he hits it steady speed for the second lap average those out and he drives an average of 60mph. Or you can take 45 break it into 3 get 15 and multiple that by the 4 minutes and get 60 for the second lap (btw you divide by 3 cause that's how long the 2 laps would take)

Whoa, take a breath Smores. The car finished its lap nearly 15 years ago and has long ago been sold for scrap.


mmm

Little Nemo
02-19-2017, 11:20 AM
You do the 1st lap in 2 minutes/30mph and you have to complete a total of 2 lapsYou're good so far.

at this rate 2 laps would take 4 minutes and would be averaged at 45 mphNo. Two laps are two miles. Two miles in four minutes is 30 mph.

to get an average of 60mph on the second lap you would have to do the second lap at 60mph cause 30+60/2 is equal to 60No. Thirty plus sixty is ninety. Ninety divided by two is forty-five.


aswell the laps are never said to be consecutive nor does it say he doesn't build up speed until he hits 30mph and then keeps a steady speed for his first lap and if the laps aren't consecutive then he could build up speed until he gets to 60mph after the first lap and then once he hits it steady speed for the second lap average those out and he drives an average of 60mph. Or you can take 45 break it into 3 get 15 and multiple that by the 4 minutes and get 60 for the second lap (btw you divide by 3 cause that's how long the 2 laps would take)I'm not sure what you're trying to say here but the numbers aren't going to work.

Rysto
02-19-2017, 12:20 PM
Zombie thread or not, the original problem exposed an interesting cognitive quirk for me. My initial impression of the answer was "it has to be more than 90mph" but I wasn't immediately sure how much, and I immediately dismissed trying to to the math in my head because the problem was presented in miles. If you took the same numbers but in kilometres (not converted to km, but 30km/h around a 1km track, etc) I would have come to the answer basically immediately. But as soon as I saw "miles" my brain just shut down and said "Nope, I don't know how those work."

SigMan
02-19-2017, 12:28 PM
The answer is the speed of light.

A lap is one mile. He drove 30 mph for one lap so he took two minutes.

two laps at 60 mph is also two minutes.

Do the math.

RTFirefly
02-19-2017, 12:47 PM
Whoa, take a breath Smores. The car finished its lap nearly 15 years ago and has long ago been sold for scrap.
The sun has gone down and the moon has come up
And long ago somebody left with the cup
But he's driving and striving and hugging the turns
And thinking of someone for whom he still burns

So, back in 2002, they were debating whether this was a trick question. I'd say it's a tricky question, but it's not a trick question.

At least IMHO, a trick question is a question constructed in a way to throw you off the track. This one isn't constructed that way; the words we have for these things are constructed that way.

You're looking at what look like whole numbers, but it's really about fractions - not because of any deliberate setup, but because 30 mph and 60 mph are really fractions, they just don't look like it. And people don't think of them as fractions. (Not to mention, most people would rather not have to deal with fractions any more than necessary.)

And to answer the long-ago OP's question, that's why people are thrown by this problem. They don't think of 30 mph as 30 miles/60 minutes. Even though it's implicitly there in 'mph,' they don't see it. You need to think about this problem in the right way to solve it (or rather, realize there's no solution), and the right way to think about it is encoded in those three little letters, mph. But most people will just think, 30 and what average to 60? Well, 90, duuuuh. :smack:

Johnny L.A.
02-19-2017, 12:49 PM
The sun has gone down and the moon has come up

When the sun goes down, and the moon comes up
I turn into a teenage goo goo muck

P-man
02-19-2017, 03:13 PM
As bad as I am at any math beyond the basic stuff, logic puzzles like this rarely stump me. That's why I reject the " higher math is necessary to learn to think logically" trope.

RTFirefly
02-19-2017, 05:01 PM
As bad as I am at any math beyond the basic stuff, logic puzzles like this rarely stump me. That's why I reject the " higher math is necessary to learn to think logically" trope.No doubt, higher math will help you think logically, if you've got an aptitude in that direction. (Not so much the material, but rather the fact that most of the problem sets in graduate-level math consist of doing proofs. You either learn to think logically, or you flunk.) But it's certainly not the only route there, by a long shot.

And the math in math-related brain teasers is usually just grade-school arithmetic, with occasional forays into first-year algebra from junior high. So it's not like you need heavy-duty math for the brain teasers. If I ever see one that requires calculus, it'll be the first time for me.

Ulf the Unwashed
02-19-2017, 07:12 PM
So, back in 2002, they were debating whether this was a trick question. I'd say it's a tricky question, but it's not a trick question.



Exactly correct. As a noted math educator once pointed out, "Trick questions are not okay. Tricky questions, on the other hand, are encouraged." This is absolutely the second, not the first, for all the reasons you give.

I'm intrigued, looking back over the thread at someone's comment that this question is "really" about semantics and not about math at all. Quite the contrary: it's a math question. It's a good bet that people who answer 90 mph don't really understand how time/distance ratios work. If I taught the appropriate grade level, I don't know if I would use the question as part of an assessment; but it could be very useful for probing kids' actual understanding.

Snowboarder Bo
02-19-2017, 07:21 PM
The sun has gone down and the moon has come up
And long ago somebody left with the cup
But he's driving and striving and hugging the turns
And thinking of someone for whom he still burns

He's going the distance
He's going for speed
She's all alone (all alone)
In her time of need

pulykamell
02-19-2017, 09:58 PM
The sun has gone down and the moon has come up
And long ago somebody left with the cup
But he's driving and striving and hugging the turns
And thinking of someone for whom he still burns

So, back in 2002, they were debating whether this was a trick question. I'd say it's a tricky question, but it's not a trick question.

I would agree. The same knowledge required to solve this question is required to solve a similar question with a more mundane (but still somewhat tricky) answer. Instead of how fast to average 60 mph over the trip, how fast would you have to go to average 45 mph? It requires knowing average speed = distance/time. We know it's a 2 mile trip. So to average 45 mph over the course of two one-mile laps, we would need to traverse 2 miles in 2/45 of an hour, or 2 2/3 minutes. Since the first mile was at 30 mph, that took 2 minutes, meaning we need to cover 1 mile in 2/3 minutes for the second lap. That works out to 90 mph.

If we followed the same principles (as has already been stated in this thread years ago), we would realize that an average speed of 60 mph means traversing 2 miles in 2 minutes. At 30 mph, our first mile will have taken 2 minutes, meaning the second mile must be instantaneous (or simply not possible.)

It's the same math to get the answer to the modified question as it is the OP's question. I don't really find that to be a trick question so much as a question that anyone who understands the principles involved to be able to get. And the benefit of asking the question in this way is that you don't actually need to do much math to figure it out (it's something you can easily figure in your head), whereas in my question, you actually do have to a little bit of work.

psychobunny
02-20-2017, 02:57 AM
I thought I already killed this thread once. The sad thing is that when I was reading this and came to the post about how to answer the question if the average was 40 mph I immediately wanted to correct the poster but reading on I saw that somebody already had. I was going to just quote their post but I thought it might be piling on. Today I came back to the thread and managed to notice that not only was the thread 15 years old, but the poster who insisted on correcting the post that I was going to quote was myself. Apparently, I am still insufferable.

pulykamell
02-20-2017, 10:11 AM
cause 30+60/2 is equal to 60

Funny enough, that is literally correct (that is, in following the order of operations 30+60/2 does equal 60), but the expression you want is (30+60)/2, which, as noted before, is 45. It almost looks to me as if you typed 30+60/2 into Google or a calculator app or something and accepted the answer that came out.

Just do a little sanity check: if you want the average of the numbers 30 and 60, it's going to be a number right in between both of them. 60 is not a number that looks to be midway between 30 and 60, does it? Even if you can't do this in your head, you should be able to tell that numbers like 15, 60, 90 simply cannot be the average because the answer must be between 30 and 60.

Similarly, how can two laps at 30 mph average to 45 mph? Think about it. Does that make sense to you? If I'm going 30 mph the whole time, I'm going 30 mph, no matter how far I go. The distance is immaterial to my average speed.

These little sanity checks should tip you off that something is wrong with the way you're approaching the question or setting up your equations/expressions. Don't just plug in numbers in a formula or calculator and accept whatever answer you get before thinking about whether the answer makes sense. I've been saved many times on math and science tests involving math by sanity checking my answers and realizing I've missed a decimal point somewhere, or I forgot to convert units, or I didn't take a reciprocal somewhere, etc. See if the answer makes sense in terms of the problem.

Emergency911
02-20-2017, 01:14 PM
I'm just glad that Mary didn't go to the store and buy 300 melons.....no one ever goes to the store and buys 300 melons! :)

Hari Seldon
02-20-2017, 01:24 PM
Here is a related question. Which would do more for reducing CO2: raising your gas mileage from 20 to 40 MPG or raising it from 40 to 100 MPG?

Raising it from 20 to 40 exactly halves your fuel consumption and CO2 generation. You can't do better than that, no matter how high you raise it unless you stop burning fuel. Going from 40 to 100 reduces it to a fifth of your original consumption but since 1/2 - 1/5 = 3/10, you have carried out a reduction only 30% of your original consumption by this.

Ludovic
02-20-2017, 04:07 PM
I disagree that it is not a trick question. A non-trick question supposes that there is a valid answer to the question. There is no answer to the question "how fast do you need to go to...".

SigMan
02-20-2017, 05:12 PM
There is no answer to the question "how fast do you need to go to...".

Why I said speed of light because of the impossibility. :cool:

Dr. Strangelove
02-20-2017, 05:46 PM
I disagree that it is not a trick question. A non-trick question supposes that there is a valid answer to the question. There is no answer to the question "how fast do you need to go to...".

Maybe. But that has nothing to do with what people get wrong about the question. It could be this instead:
You drive one lap at 30 mph. How fast do you have to drive a second lap in order to average 45 mph?

Most people will answer 60 mph. The actual answer is 90 mph. The flaw in people's thinking isn't that they thought there was an answer but it didn't exist; it was that you can't average rates with the arithmetic mean.

P-man
02-20-2017, 05:49 PM
No doubt, higher math will help you think logically, if you've got an aptitude in that direction. (Not so much the material, but rather the fact that most of the problem sets in graduate-level math consist of doing proofs. You either learn to think logically, or you flunk.) But it's certainly not the only route there, by a long shot.

And the math in math-related brain teasers is usually just grade-school arithmetic, with occasional forays into first-year algebra from junior high. So it's not like you need heavy-duty math for the brain teasers. If I ever see one that requires calculus, it'll be the first time for me.

For a lot of folks there's probably a tendency to make these problems harder than they should be. I agree that the math is pretty low level; maybe it's advantage when it comes to solving these brain teasers to not know too much.

Bill Door
02-20-2017, 05:56 PM
You can't do a distance weighted average, you have to do a time weighted average. You've driven two minutes at 30 mph, you need to drive two minutes at 90 mph to average 60 mph. You need to drive three miles at 90 mph, and the constraint of only driving one lap is what makes it impossible.

Dr. Strangelove
02-20-2017, 09:50 PM
Here is a related question. Which would do more for reducing CO2: raising your gas mileage from 20 to 40 MPG or raising it from 40 to 100 MPG?

One can make it even more extreme:

What saves more fuel over a 100 mile trip:
- Properly inflating your truck's tires and taking the excess junk from the bed, so that it gets 13 mpg instead of 12.
- Aggressively "hypermiling" your Prius so that you achieve 70 mpg instead of 50.

The first is better, as you might have guessed.

Ulf the Unwashed
02-20-2017, 09:55 PM
Okay, regarding "trick questions" vs. "tricky questions," this conversation (by an amazing coincidence--no reali!) took place tonight in the education methods class I teach at a local college.

We are discussing the teaching of subtraction, and one student raises his hand.

"When do you introduce tricky questions in subtraction?" he wants to know.

I'm not sure what he means--I'm guessing he's thinking about problems in which you have to regroup twice, or maybe multistep problems, so I ask him to give me an example. "Sure," he says. "Julia has fifteen dollars. She spends all but eight dollars. How much does she have left?"

"Okay," I tell him, "that actually is not a tricky question, it is a trick question," and I quote the math educator I quoted above ("tricky questions are fair, trick questions are not"). "That isn't a math question," I emphasize, "and so the answer is never; I would never use it in a math class."

"Wait," says another student, "I don't get it. What do you mean, a trick question or a tricky question? What makes it not a math question? It seems pretty straightforward to me."

Aha, I think, she doesn't get it. I repeat the question. "What would you say is the answer?" I ask.

"Seven," she says, and can't resist adding "of course."

Aha, she really DIDN'T get it. "What about you?" I ask the student next to her. "Do you agree with Lisa? Disagree?"

"Agree," she says with confidence. "Julia has seven dollars left."

It's a small class, and I ask everybody but the original question-poser, and they all agree: the answer is seven. "Okay," I say, "I disagree; I think the answer is something else. Listen very carefully as Gabe tells us the question again--"

And he does so, this time emphasizing the key phrase in the question, and one by one the students understand the issue. Not a single one of them is pleased to have been "caught" by the wording of the problem. It is clear they wish to throw their laptops at Gabe for asking the question to begin with...

So. In my book at least, a question like Gabe's is a trick question. People are likely to get it wrong because they are not paying close enough attention to semantics--not because they are misunderstanding something about math. When you realize you have gotten Gabe's question wrong, and why, you will have learned nothing about math, nothing about the way that numbers work; you will only have learned that you should have listened more carefully to the original question.

In contrast, a question like the one in the OP is a tricky question. People are likely to get it wrong because they do not completely grasp how ratios work--because they are misunderstanding something about math. When you realize you have gotten the OP's question wrong, and why, you will have a new understanding about math.

--You may prefer some different terms to describe the difference than trick vs. tricky, and that's your prerogative; but this is the difference, and it's an important one.

igor frankensteen
02-20-2017, 11:02 PM
This was really a psychology question, and or a mathematics principles awareness question, originally.

Everyone including the OP got caught up in the math or side issues about the math or each other, so no one ever answered the title question directly. Guess it doesn't matter now.

Anyway, even the OP was wrong, there is only one right answer: it's impossible. Even infinite speed takes time. Just as the speed of light, though very very fast, takes time, infinite speed also takes time.

It's another "brain teaser trick" in a way: speed includes time as a component. If time isn't present, then you aren't talking about speed, you're talking about something else.

psychobunny
02-20-2017, 11:07 PM
The problem is that people don't set up the equation properly. It's a fairly easy solution if you take the time to set it up right.

If x is the length of the track
And y is the speed of the second lap (in mph)
then:

2x/60=x/30 + x/y

and it is easy to solve for y. The problem is that people are setting it up as:

2x X 60= (x X 30) + (x X y)

JohnGalt
02-21-2017, 01:38 PM
I teach a college-level class using Excel, and in the first round we do a payroll spreadsheet. Everyone has a different pay rate, works different number of hours, so you can calculate what each person earns. Then you sum that to get the total paid.

But if you multiply the total hours worked by everyone by their average pay rate, you get a slightly different answer. To show them why, I use the OP's question.

Without fail, they will answer 60 MPH.

Then we get into the explanation why you can't average rates, you have to use the values that go into the rate. I then show them examples other rates, such as MPG or GPA. You can't take a 1-credit class one semester and get an 'A' for 4.0, then the next semester fail 20 credits for 0.0, and average them to get a GPA of 2.0. I hope they get it, but some students still refuse to see the answer.

ftg
02-21-2017, 04:29 PM
Then we get into the explanation why you can't average rates, you have to use the values that go into the rate.

In one of Al Franken's books he has a chapter on really bad political Math. Doing naive averages, adding percentages, etc. One table someone put out had a column of numbers that just made no sense at all. Completely absurd "calculation".

People make really important decisions on the basis of horrible Math. Sleep well.

JohnGalt
02-21-2017, 05:00 PM
In one of Al Franken's books he has a chapter on really bad political Math. Doing naive averages, adding percentages, etc. One table someone put out had a column of numbers that just made no sense at all. Completely absurd "calculation".

People make really important decisions on the basis of horrible Math. Sleep well.

Do you remember which book? I'll dig through my copies and perhaps use that in class as well!

Wallaby
02-21-2017, 05:57 PM
All keen cyclists know this.

If you can average, say, 30ks riding on the flat, why can't I average 30ks if I go up and down on a hilly course (but wind up at the same altitude)? Surely I make up the average speed when I go slow on the way up, by going faster on the way down?

Nope. As explained many times above - you have to go faster for the same period of time (not distance) - and of course, going down the hill is a lot shorter elapsed time than going up.

All cyclists puzzle over this phenomenon once or twice when looking at their ride logs, before working it out.

I can average 30ks solo on a dead-flat, windless course (Beach Rd in Melbourne. Yes, I was there the one day it wasn't windy:p:p). If I ride in the Dandenong Ranges, I average about 24.

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