The way I started thinking about the problem is, you’ve got $1.10 to spend in total. $1 is spent on the difference between the bat and the ball. That leaves $.1 which is split evenly between the bat and the ball.
So what I end up doing is, as Tordmor says below:
1.10 − 1 = .10
.10 / 2 = .05
This is essentially the explanation given by wedrifid but I wrote it before reading his and tried to format it more consistently with your comment below.
I’m a bit weird with these sorts of arithmetic questions, my thought process went something like this: “Ok, 10 cents seems close, but that puts the bat at 90 cents more than the ball.. oh it seems like 5 cents and 1.05 works.” The answer just sort of pops into my head, not even thinking about the division step. Of course, I could do the simple maneuvering to get the answer, but it isn’t what I naturally do. I think this has to do with how I did math in grade school: I would never learn the formulas (and on top of this I would often forget my calculator) so I would rather come up with some roundabout method for approximating various calculations (like getting the root of a number by guessing numbers that “seemed close” to the root, usually starting at 1⁄2 the number and adjusting). Probably not really the best way to do things since there are much cleaner solutions, but this bad habit of arithmetic has sort of stuck with me through my mathematics degree (though I of course have picked up the relevant formulas by now!); instead of using the straightforward formula, I do this mental jiggling around of values and it pops into my head.
I don’t know, maybe that isn’t weird at all, but in any event no one has mentioned doing it yet.
The ‘No’ part changes it from a clever shortcut to just being wrong. You having a more finely tuned mathematical intuition doesn’t invalidate the rudimentary simultaneous equation.
You could just go at it the other way. Guess-and-check is one of the basic math strategies taught in schools, and is easy to apply to these questions:
‘I need something to answer this… 0.10 sounds good, but let’s check it first. 1+0.1 means the bat costs 1.10, and 1.10+0.10 = 1.2 - what a minute, that’s not 1.10! I better try some other values—I guess this isn’t so obvious after all!’
It’d be interesting if getting the wrong answer first is the quickest method of getting the right answer.
I recently read something like this, though I can’t remember where. The experiment went roughly like so:
Subjects were divided into two groups. In one, each subject was given 15 seconds to memorize an answer to a question for several seconds, and their performance recalling the answer later was recorded. In the other, each was asked to guess the answer, and was then given 7 or 8 or so seconds to memorize the correct answer. The time difference was to account for the time during which the second group’s members thought about the question.
So each person was exposed to the question for 15 seconds, in the first group, they were exposed to the answer for those same 15 seconds, in the second group, for half that.
The second group was better at recalling the answers.
I found a set of five experiments similar to the one I described. Getting the wrong answer first appears to be a good method to get to the right answer.
Specifically, we evaluated the benefits of testing novel science instructional content before learning. Thus, the likelihood of failed tests was high, but we were able to extend our theory of testing to better understand whether trying and failing on test questions actually improved learners’ longer term retention of subsequently presented information...in the current study, the prequestions required participants to produce nouns or descriptive statements that they were unlikely to be able to answer on the basis of prior knowledge (e.g., “What is total colorblindness caused by brain damage called?”). This allowed us to isolate and examine the effects of unsuccessful retrieval attempts.
...
Previous research has demonstrated the memory benefits of successfully answering test questions. The five experiments reported herein provide evidence for the power of tests as learning events even when the tests are unsuccessful. Participants benefited from being tested before studying a passage—a pretesting effect—although they did not answer the test questions correctly on the initial test, as compared with being allowed additional study time. Furthermore, the benefits of pretests persisted after a 1-week delay.
...
The current research suggests that tests can be valuable learning events, even if learners cannot answer test
questions correctly, as long as the tested material has educational value and is followed by instruction that provides answers to the tested questions.
I know that yields the correct answer but how do I know I should divide the expression by two from the problem statement?
You could do a couple of steps forward in solving the equations but the intuitive explanation is something along the lines of: Assuming the ball costs nothing the total comes to $1. If the total is more than that then it means that both the ball and the bat have an increased price by an equal amount. So the difference from $0 of the ball is going to be half the difference between the total and $1.
I really don’t know how people can correctly answer this on the fly! I have to solve {bat + ball = 1.10, bat = 1 + ball} to get the correct answer.
The way I started thinking about the problem is, you’ve got $1.10 to spend in total. $1 is spent on the difference between the bat and the ball. That leaves $.1 which is split evenly between the bat and the ball.
So what I end up doing is, as Tordmor says below:
1.10 − 1 = .10
.10 / 2 = .05
This is essentially the explanation given by wedrifid but I wrote it before reading his and tried to format it more consistently with your comment below.
Why is it split evenly? (I’m just wondering what your thought process is)
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I’m a bit weird with these sorts of arithmetic questions, my thought process went something like this: “Ok, 10 cents seems close, but that puts the bat at 90 cents more than the ball.. oh it seems like 5 cents and 1.05 works.” The answer just sort of pops into my head, not even thinking about the division step. Of course, I could do the simple maneuvering to get the answer, but it isn’t what I naturally do. I think this has to do with how I did math in grade school: I would never learn the formulas (and on top of this I would often forget my calculator) so I would rather come up with some roundabout method for approximating various calculations (like getting the root of a number by guessing numbers that “seemed close” to the root, usually starting at 1⁄2 the number and adjusting). Probably not really the best way to do things since there are much cleaner solutions, but this bad habit of arithmetic has sort of stuck with me through my mathematics degree (though I of course have picked up the relevant formulas by now!); instead of using the straightforward formula, I do this mental jiggling around of values and it pops into my head.
I don’t know, maybe that isn’t weird at all, but in any event no one has mentioned doing it yet.
A friend tried this on three of us earlier tonight (having read the same report). One out of three (all agnostics) got it right on the fly.
No you have to solve (1.10 − 1) / 2
The ‘No’ part changes it from a clever shortcut to just being wrong. You having a more finely tuned mathematical intuition doesn’t invalidate the rudimentary simultaneous equation.
I know that yields the correct answer but how do I know I should divide the expression by two from the problem statement?
You could just go at it the other way. Guess-and-check is one of the basic math strategies taught in schools, and is easy to apply to these questions:
‘I need something to answer this… 0.10 sounds good, but let’s check it first. 1+0.1 means the bat costs 1.10, and 1.10+0.10 = 1.2 - what a minute, that’s not 1.10! I better try some other values—I guess this isn’t so obvious after all!’
Right. I know how to get to a right answer but didn’t understand Tordmor’s expression.
Starting with
bat + ball = 1.10
bat = 1 + ball
substitute one into the other to eliminate it
1 + ball + ball = 1.10
simplify
1 + 2 ball = 1.10
then solve for ball
ball = (1.10 − 1)/2
then compute
ball = 0.05
This is literally how I would solve this problem. So you can see why I’m surprised people can answer it correctly on the fly.
The way it went in my head:
Huh, that’s obvious, it’s 1. Oh wait, ‘more than.’ So it’s half the remaining .10.
(Although I would say it took less time than reading that sentence takes.)
It’d be interesting if getting the wrong answer first is the quickest method of getting the right answer.
I recently read something like this, though I can’t remember where. The experiment went roughly like so:
Subjects were divided into two groups. In one, each subject was given 15 seconds to memorize an answer to a question for several seconds, and their performance recalling the answer later was recorded. In the other, each was asked to guess the answer, and was then given 7 or 8 or so seconds to memorize the correct answer. The time difference was to account for the time during which the second group’s members thought about the question.
So each person was exposed to the question for 15 seconds, in the first group, they were exposed to the answer for those same 15 seconds, in the second group, for half that.
The second group was better at recalling the answers.
I found a set of five experiments similar to the one I described. Getting the wrong answer first appears to be a good method to get to the right answer.
I get it, thanks!
You could do a couple of steps forward in solving the equations but the intuitive explanation is something along the lines of: Assuming the ball costs nothing the total comes to $1. If the total is more than that then it means that both the ball and the bat have an increased price by an equal amount. So the difference from $0 of the ball is going to be half the difference between the total and $1.