I first saw this question here on the SDMB: John buys two pencils, a cheap pencil and an expensive pencil. There is no sales tax. John spends a total of $1.10 for the two pencils. The expensive pencil costs $1.00 more than the cheap pencil. How much did the cheap pencil cost? Almost everyone I ask gets this wrong. Engineers, accountants, and physicists all get it wrong. They all insist that the cheap pencil costs 10 cents. Even when I show them they are wrong, they still insist that the cheap pencil costs 10 cents.
What is it about this question that almost invariably invokes the wrong answer. There is no hidden trick or misleading information. What is going on here?
Look up Daniel Kahneman’s System 1 and System 2. System 1 is the intuitive, simple processing that gets us by thru most of our decisions. System 2 is much, more analytical.
The intuitive thing to do is to subtract $1.00 from $1.10 and get $0.10 since the starting data you have is “$1.10” and “minus $1.00”. At that point, most people say: “Right, found the answer.” without taking the extra steps of thinking that this means the expensive pencil costs $1.10 for a total of $1.20.
Why they insist that the cheap pencil costs 10 cents, maybe ego or having difficulty shifting their perspective. If you start with a conclusion, it can be difficult, emotionally or cognitively, to entertain other possibilities.
For the same reason you can write a paragraph with every word being written jumbled except the first and last letter, and people still understand it.
The brain is geared towards pattern recognition, and assumes small details are more likely to be noise errors than critical components, because they almost always are.
This question is kind of disingenuous though. It’s like asking, why are fish so rubbish at climbing trees or why do audiences seem surprised that nice lady wasn’t actually sawed in half?
The real answer is, because it was deliberately designed to be misleading, so that’s what it does.
You can train against this, but this leads to other non ideal conditions, which you could then further nitpick. So a second order answer would be, every potential ideal pragmatically conflicts with some other one. The best you can do is have some kind of balance between a specific element and its internal and external circumstances.
Like many “trick” questions, this one is structured to appear to have a simple, logical solution, but one that is misleading.
Just in case we have Dopers who haven’t figured it out or are mathematically challenged (imagine that!), this is the way a 9th-grade Algebra teacher would write the problem:
Let X = the cost of the cheap pencil
X + 1 = the cost of the expensive pencil (by definition)
The equation then is:
(Cheap pencil) + (expensive pencil) = 1.10
Substituting:
X + (X+1) = 1.10
Reducing:
X + X + 1 = 1.10
2X + 1 = 1.10
2X = 1.10 - 1
2X = .10
X = .10/2
X = .05
So the cost of the cheap pencil is 5 cents. Or a nickel, everywhere except Australia, where they don’t call a 5-cent piece a nickel.
Also, it’s not a conclusion that one ever faces in real life.
No one ever says to you, “Here’s a $1.10.” Go buy a couple of pencils, but just make sure one is exactly a dollar more that the other, OK?"
More likely it will be:A: “How much is a pencil?”
B: “The basic one is 5 cents, and the deluxe one costs a dollar more for $1.05.”
A: “Mmm. Okay, give me one of each.”
B" “That’ll be $1.10”
Yes, like many problems, it’s artificial and contrived and unrealistic. But that doesn’t make it unimportant. Someone who can’t solve it correctly is at risk for making a similar mistake on a non-contrived, real-world problem that is important.
There is an unfortunate tendency on the part of many people, when faced with a mathematical problem, to “solve” it by doing a calculation, without paying enough attention to whether it’s the appropriate calculation for that particular problem.
Wasted time, for one thing. Training against intuition and pattern recognition means more decisions need to be made as a result of conscious consideration, meaning that in some situations, opportunity has expired before action is taken.
Stupid example: 10 kids in a room and you call “Who wants ice cream?”. One kid hesitates to consider “What does this question really mean? What does it actually mean to want something? Is he asking whether we, in the room want ice cream, or whether people, in general want ice cream? Is he actually questioning the desirability of ice cream? Do people in general want ice cream? How could we measure that question without asking everyone? Does it depend on the flavour?”
Guess which kid is last in line.
Yes, by all means. My point is that this type of phrasing is so unnatural, one should know from the onset that it’s going to involve algebra. Only algebra problems are presented this way, and it’s clear immediately that the “obvious” answer is not correct. I don’t buy the premise of the OP that “everyone” gets it wrong. People who have some memory of middle school math should just reflexively start to set up the equation when they hear a situation described in such an unlikely way.
You do if you are consistent in using “20%” to mean 20% of the original figure
Travel uphill for an hour at 30mph and then downhill for an hour at 60mph and indeed your average speed is 45mph because you’ve covered 90 miles in 2 hours
In both cases your questions are set up as “gotchas” because the information is not complete.
But the standard for computing percentages is to use the amount before the given transaction as the denominator - not the amount from before some previous transaction.
Stock prices are a good example. If your stock reports a drop of 10% one day, then posts a gain of 10% the next day, then it hasn’t recovered.
