One can make it even more extreme:
What saves more fuel over a 100 mile trip:
The first is better, as you might have guessed.
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.
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.
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)
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.
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!
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.