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Someone explain this to me plz...

higs

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2. A bat and a ball cost $1.10 in total. The bat costs $1 more than the ball. How much does the ball cost?

Many people give the first response that comes to mind—10 cents. But if they thought a little harder, they would realize that this cannot be right: the bat would then have to cost $1.10, for a total of $1.20. IQ is no guarantee against this error. Kahneman and Frederick found that large numbers of highly select university students at the Massachusetts Institute of Technology, Princeton and Harvard were cognitive misers, just like the rest of us, when given this and similar problems


-----------

It's an extract from this article which has already been posted to the forum a few times:

http://www.scientificamerican.com/article/rational-and-irrational-thought-the-thinking-that-iq-tests-miss/

Embaressed to admit that Even once the answer has been given I don't get how it's the conclusion lol. I just dont see how the correct answer is not 10cents lol. I get the other questions.
If someone could just break it down for my stoopid brain to make the connection that would be cool plz. :kodama1:

Thanks x
 

Blarraun

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Unknowns (variables):
Bat
Ball

Data:
Bat + Ball = 1.1

Bat = Ball + 1

Therefore:

Ball +1 + Ball = 1.1 (You think of a bat as a ball plus 1 'dorar')

Therefore:

2 Ball + 1 = 1.1 (so in fact you can add these two balls and now you have only one unknown: the ball)

Then:

1.1 - 1 = 2 Ball => 2 Ball = 0.1 /2 => Ball = 0.05

(You subtract the 1 from the total cost and you are left with 0.1, but you still have 2 balls, so you divide by 2 to see the price of a single ball)

Answer:

The ball costs 0.05 and the bat costs 1.05, they cost 1.1 together.

The important thing is to consider that 1.1 is a price of both the bat and the ball and that the bat costs 1$ more than the ball, it doesn't mean that you can subtract 1 from the total price, because you were never given a precise price of the items, you were given a system of two equations with two unknowns which is perfectly solvable.

The illogic of subtracting 1 from the total price:

Ball + Bat = 1.1
1.1 - 1 = Ball (somehow people manage to lose a bat in this process, as if it was fine to equate the bat with 1$)

I suppose learning how to solve linear and algebraic systems (matrices) of equations isn't taught everywhere, or maybe people just don't think in these terms, it is the kind of question an elementary school kid shouldn't have a problem with.

If anyone feels they'd like to brush up on their elementary maths, or learn something advanced, you can do it for free here. I believe you can find lessons about linear equations such as the one in question as well.

Hey and don't be embarrassed if you are willing to learn. Every question is valuable.
 

onesteptwostep

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maths. it's teh magiek
 

Blarraun

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maths. it's teh magiek
No and I really hate people who make it seem maths is something beyond common man's grasp. Such people are only perpetuating the ignorance.

I assume you were joking, don't give me reasons to point out the stupidity of calling maths inaccessible.:twisteddevil:
 

onesteptwostep

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maths is heathen writing! your pluses and minuses and your derivatives and integrals will not get you into hea-ven! *slams fist into pulpit*
 

Tannhauser

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Pure algebra style: let x be the ball, y the bat.

x + y = 1.1
y = x + 1

Putting the second equation into the first yields 2x = 0.1 or equivalently x = 0.05
 

higs

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you INTP are smartness and kindness :o

So essentially the mistake people make in the question is assuming that the bat = exactly 1 dollar more than ball when that is not actually a given. Thanks Blarraun and tanhauser. I find you are both excellent additions to the forum. I am a lurker mostly but I read it a lot :phear:

I still admit that the steps you have laid out somewhat fly over my head a bit, but I will use my khan Academy account to make sure I fully understand at some point in the future. I never bothered to listen in most classes unfortunately and have many holes in my cognitive abilities where I did not autonomously teach myself. Particularly maths and formal logic which always just made me go "ugh" BOREDOM without a second thought from the earliest age. Not sure why. some form of ADD or just laziness plagues me no doubt. Luckily my verbal intelligence is okay and I like reading and wondering.
 

Minuend

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If the ball costs 0.10, then the bat costs 1.10 (one dollar more).

0.10 + 1.10 = 1.20 <---- wrong total

That's why it's the wrong answer. People mistakenly add 0.10 + 1.00 in their head, but that would mean that bat only costs 90 cents more than the ball.

The right answer is 0.05 (ball) + 1 = 1.05 (bat)

0.05 (ball) + 1.05 (bat) = 1.10 <--- correct total
 

Lot

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If the ball costs 0.10, then the bat costs 1.10 (one dollar more).

0.10 + 1.10 = 1.20 <---- wrong total

That's why it's the wrong answer. People mistakenly add 0.10 + 1.00 in their head, but that would mean that bat only costs 90 cents more than the ball.

The right answer is 0.05 (ball) + 1 = 1.05 (bat)

0.05 (ball) + 1.05 (bat) = 1.10 <--- correct total

This made so much more sense now.
 

Tannhauser

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you INTP are smartness and kindness :o

So essentially the mistake people make in the question is assuming that the bat = exactly 1 dollar more than ball when that is not actually a given. Thanks Blarraun and tanhauser. I find you are both excellent additions to the forum. I am a lurker mostly but I read it a lot :phear:

I still admit that the steps you have laid out somewhat fly over my head a bit, but I will use my khan Academy account to make sure I fully understand at some point in the future. I never bothered to listen in most classes unfortunately and have many holes in my cognitive abilities where I did not autonomously teach myself. Particularly maths and formal logic which always just made me go "ugh" BOREDOM without a second thought from the earliest age. Not sure why. some form of ADD or just laziness plagues me no doubt. Luckily my verbal intelligence is okay and I like reading and wondering.
I have to say it is hard to see exactly why we have a tendency to think the ball is 0.1. Probably some mental picture that divides the total of 1.1 into 1 and 0.1 and just assumes this is the answer. Very interesting problem.

Maybe it is helpful to realise that, given only the condition that the bat is 1 dollar more than the ball, you have infinitely many solutions:

0.01+1.01
0.02+1.02
0.03+1.03
...

and so on.. where only one of them has the total of 1.1, namely 0.05+1.05.
 

The Gopher

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If you want to know why people get this wrong then I can help you!

Ultra simple version is we tend to use a more instinct style of thinking with these kind of questions compared to actually manually looking for the answer. This question is designed to trick that instant response. In your case once you had that response you saw no reason why another answer would make sense and didn't manually work out the maths you just kept doing the instinct type thing.

There is nothing wrong with that, if we manually thought through everything in detail we would be ISTJ's. :D

(This post is for people who haven't read the article and wanted an MBTI related joke)
 

onesteptwostep

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Er for those who are more pictoral-algebraic (will be doing the work part of Tann's post):

x = ball
y = bat

x + y = 1.10
y = (x + 1)

Then we do our work.. insert the second equation into the first (the y into the 1st formula).

x + (x + 1) = 1.10 then
2x + 1 = 1.10
2x = 0.10
x = 0.1/2
x = 0.05
ball = 0.05

Needless to say this type of question will never be brought up to you in real life ever.
 

higs

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I'm going to run any math problems I have by this forum from now on.
 

Blarraun

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You can solve it the way I and Tann did it, but you can also do this:
For X = ball and Y = bat

X + Y = 1.1
Y = X +1

So what you can do is rearranging the terms so that they line up one under another, like this:

X + Y = 1.1
(-X) + Y = 1

All I did was moving X from the right side of the equation to the left, so it naturally gives X the opposite sign ( now it's a negative X or minus X). Then to line up Y under the first equation, I put the -X in front, with the addition sign, relying on the commutativity of addition in mathematics, meaning that I can change the order of operands and doing so doesn't affect the result.

After they are lined up, you can treat these two equation as 3 vertical ones and you see X is above -X, well imagine you subtract all three horizontally and you get something like this:

X - (-X) = 2X
Y - Y = 0
1.1 - 1 = 0.1

So if you calculate what happens, X - (-X) means, two minus signs become a plus and you get 2X

Y is gone and there is 0.1 left.

So if you now put the elements back to the original equation, you will see:

2X + 0 = 0.1

At this point you can divide by two, 0 can be removed since it's a zero, you can have as many as you like and they don't affect the result.

So you get the familiar X = 0.05

This method is a more graphical way, relying on the vertical column of this matrix (system of variables). The alternate method becomes much more useful as the number of equations and individual terms increases, say someone asked about ball, bat, glove and a few other items and laid out their relations to each other.


Needless to say this type of question will never be brought up to you in real life ever.
It already has in this thread and your reply is a silly one.
Math is much more than practical down to earth thinking, it's a framework, it allows exploring new horizons. I don't care why you dislike maths, but you come to every math thread I see and try to diminish it in some way or another.
I'm going to run any math problems I have by this forum from now on.
Good to hear this, it's supposed to be a forum of thinkers, not just whiny philosophers.
*Eyes Onestep* "Who made you that pulpit, I bet it was some engineer."
 

Hadoblado

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Getting it wrong is natural, I got it wrong with my most immediate instinct, but then it's sort of like I try to ram the answer into a hole too small for it in my head, and I know it's wrong.

I then quickly articulated a proof in my head for why my initial answer was wrong:

If the ball costs 0.10, then the bat costs 1.10 (one dollar more).

0.10 + 1.10 = 1.20 <---- wrong total
~ Minuend

^this one

Then I sort of intuit that because there's a 10c difference and I'm changing both totals, I need to apply half to each, and arrive at the correct 5c/1.05c split. I then feed it through that same 'hole' in my mind and find that it fits snugly, try to wiggle it for good measure, and then walk away assuming I'm probably right.

I'm not really a maths guy, but I tend to do well at puzzles because I second-guess myself a lot. This took no more than ten seconds, but in that time I tried a solution, instinctively found it wrong, proved it wrong, intuited a path to the next answer, arrived at that answer, tested it, then tested it again. When I see people not able to arrive at these answers, it seems to me that they're not actually looking for ways that they're wrong. It's not a failure of processing power or anything like that, it's more like they're pushing so hard in a different direction they never get back on track.

Conversely, I do poorly at anything that requires committing to a single line of logic at an intuitive level - this is usually a requirement for tasks with high time pressure. I often find myself outperformed at a gross holistic level where second-guessing is maladaptive.
 

Cheeseumpuffs

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maths is heathen writing! your pluses and minuses and your derivatives and integrals will not get you into hea-ven! *slams fist into pulpit*
*starts grumbling about classical theorems of integration and how much he hates them*

I'm going to run any math problems I have by this forum from now on.
Ooh, please do :D
 

Urakro

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I got a lot of enthusiasm from reading the OP, but it turns out everything I wanted to say is already mentioned.

A similar situation that I ran into as a real life example. Say after I spent a total of $100 on a stereo system. Somebody asks me how much the selling price was before taxes. I lost the receipt ( :facepalm: ) , and all I know is that taxes were an additional 15% on top of the selling price.

My quick-thinking circuits pounced on an answer of $85. It's the wrong answer, but to my amusement, I wasn't the only person who quickly came to that conclusion. After working out the precise answer, the guess is close but off by ~$2.
 

Reluctantly

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2. A bat and a ball cost $1.10 in total. The bat costs $1 more than the ball. How much does the ball cost?

Many people give the first response that comes to mind—10 cents. But if they thought a little harder, they would realize that this cannot be right: the bat would then have to cost $1.10, for a total of $1.20. IQ is no guarantee against this error. Kahneman and Frederick found that large numbers of highly select university students at the Massachusetts Institute of Technology, Princeton and Harvard were cognitive misers, just like the rest of us, when given this and similar problems


-----------

It's an extract from this article which has already been posted to the forum a few times:

http://www.scientificamerican.com/article/rational-and-irrational-thought-the-thinking-that-iq-tests-miss/

Embaressed to admit that Even once the answer has been given I don't get how it's the conclusion lol. I just dont see how the correct answer is not 10cents lol. I get the other questions.
If someone could just break it down for my stoopid brain to make the connection that would be cool plz. :kodama1:

Thanks x
huh. I thought 10 cents at first as well. Kind of a mindfuck when I realize how stupid that was. heh
 

Yellow

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No and I really hate people who make it seem maths is something beyond common man's grasp. Such people are only perpetuating the ignorance.:twisteddevil:
Preach!

Seriously though, this was one of my biggest hurdles as a math teacher.
 

cheese

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I went a totally different, equally useless 'intuitive' way first, and read it as the bat being 10c more expensive than the ball. Why? Because I saw '10c' a couple of times. My answer was thus 50c and 60c, making sure to add up to $1.10. When I actually read the question properly, I got the right answer - but not before being a total conclusion-jumping poophead. So you're not alone in your struggles higs.

These guys also struggle with numbers.
https://www.youtube.com/watch?v=yHxHQ00uvcc
 

Sabreena

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Then I sort of intuit that because there's a 10c difference and I'm changing both totals, I need to apply half to each, and arrive at the correct 5c/1.05c split. I then feed it through that same 'hole' in my mind and find that it fits snugly, try to wiggle it for good measure, and then walk away assuming I'm probably right.

I'm not really a maths guy, but I tend to do well at puzzles because I second-guess myself a lot. This took no more than ten seconds, but in that time I tried a solution, instinctively found it wrong, proved it wrong, intuited a path to the next answer, arrived at that answer, tested it, then tested it again. When I see people not able to arrive at these answers, it seems to me that they're not actually looking for ways that they're wrong. It's not a failure of processing power or anything like that, it's more like they're pushing so hard in a different direction they never get back on track.

Conversely, I do poorly at anything that requires committing to a single line of logic at an intuitive level - this is usually a requirement for tasks with high time pressure. I often find myself outperformed at a gross holistic level where second-guessing is maladaptive.
Yeah, I sort of intuited it too. I was wondering how you all came up with formulas on the spot, pluggin g in numbers and moving thiem around and shit. It amounts to the same thing, I guess. Your intutiion taps into the part of your brain that knows how numbers fit together, and feeds you the answer without necessarily revealing the process used to develop the answer (I have no idea how to express this in proper psychological terms.)

This is one reason why I'm complete shit at tutoring/teaching. You have to explain things in broken-down logic in the formats that people understand. And for me, constructing formulas out of thin air doesn't come naturally. I just forget that other people don't run through the same intutive process that I do.

I have a similar problem with time pressure if the task is something external, something that involves responding right away, like coming up with arguments on the spot. Or normal social interaction for that matter.
 
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