# A New Twist on a Classic Puzzle

“A bat and a ball cost $1.10 in total. The bat costs $1.00 more than the ball. How much does the ball cost?”

Take a minute to think about it … Do you have the answer? Many people respond by saying that the ball must cost 10 cents. Is this the answer that you came up with? Although this response intuitively springs to mind, it is incorrect. If the ball cost 10 cents and the bat costs $1.00 more than the ball, then the bat would cost $1.10 for a grand total of $1.20. The correct answer to this problem is that the ball costs 5 cents and the bat costs — at a dollar more — $1.05 for a grand total of $1.10.

So why do so many people answer incorrectly? The answer is that people often substitute difficult problems with simpler ones in order to quickly solve them. In this case, people seem to unconsciously substitute the “more than” statement in the problem (the bat costs $1.00 *more than* the ball) with an absolute statement (the bat costs $1.00). This makes the math easier to work with; if a ball and bat together cost $1.10 and the bat costs $1.00, then the ball must cost 10 cents.

Time and again research using the bat-and-ball problem has shown that that this intuitive process leads people astray. But are intuitions always detrimental to problem solving? In a 2014 *Journal of Cognitive Psychology* article, Université de Toulouse researcher Bastien Trémolière and Université Paris-Descartes researcher Wim De Neys sought to answer this question.

Trémolière and De Neys point out that the intuitively generated response to the bat-and-ball problem (that the ball costs 10 cents) is neither highly believable nor highly unbelievable. It is not unreasonable to think — especially for someone who isn’t an expert in baseball — that such a ball could cost 10 cents. They wondered how a person might respond if a similar problem cued an intuitive — but unbelievable — response. What would happen if the intuitive response contradicted other intuitions such as past knowledge about the cost of an item?

To find out, the researchers had participants answer a classic or a modified bat-and-ball-type problem. In the classic problem, participants were asked the following question:

“A Rolls-Royce and a Ferrari together cost $190,000. The Rolls-Royce costs $100,000 more than the Ferrari. How much does the Ferrari cost?”

In the modified version of the problem, participants were asked the following question:

“A Ferrari and a Ford together cost $190,000. The Ferrari costs $100,000 more than the Ford. How much does the Ford cost?”

As in the original bat-and-ball problem, people often will try to make the problem seem easier by unconsciously removing the “more than” wording in the problem, leading them to read the problem as saying either “The Rolls Royce costs $100,000” or “the Ferrari costs $100,000.”

The intuitive but incorrect answer is that the less expensive car (either the Ferrari or the Ford, depending on the problem) costs $90,000; however, in the modified version of the problem this answer (that the Ford costs $90,000) conflicts with people’s prior knowledge about Ford cars: The idea of a Ford being that expensive is not believable. This conflict is not present in the classic problem, as the thought of a Ferrari costing $90,000 would seem reasonable to most people.

The researchers found that significantly more people correctly answered the modified version of the problem than the classic version of the problem. The authors posited that when intuitive answers conflict with other intuitions, such as those based on past knowledge, people are more likely to engage in more deliberate and reflective reasoning leading to a higher likelihood that they will answer the problem correctly.

**Reference**

Trémolière, B., & De Neys, W. (2014). When intuitions are helpful: Prior beliefs can support reasoning in the bat-and-ball problem. *Journal of Cognitive Psychology*,* 26*, 486–490.

## Comments

ShathaNovember 9, 2016The thing is, why does the ball have to be $.05? It could have been .04 0r.03 and the bat would still cost more than $1.

TonyNovember 17, 2016I hear your pain. I feel as though psychologists and psychiatrists get together every now and then to prove how stoopid I am. However, after more than a little head scratching I’ve gained an understanding of this puzzle. It can be expressed as two facts and a question A=100+B and A+B=110, so B=? If B=2 then the solution would be 100+2+2 and A+B would be 104. If B=6 then the solution would be 100+6+6 and A+B would be 112. But as be KNOW A+B=110 the only number for B on it’s own is 5. My worry is; as I don’t spend 20 minutes studying every maths problem I’m faced with each day how many of these am I instinctively getting wrong?!

Dillon wittyDecember 8, 2016This is exactly what bothers me and resulted in me wanting to look up the question online. On the quiz the other 2 questions were definitive. This one technically could have more than one answer so this is where phycologists actually mess up when trying to give us a trick question. The ball at .4 and the bat at 1.06 doesn’t break the rule either.

TristanDecember 12, 2016No, the ball at 0.4 and the bat at 1.06 does break the ‘rule’. That would mean the bat would cost 1.02 dollars more than the ball. It’s a stupid question they need to state ‘exactly one dollar more’. Nothing clever about this is been designed to catch people out.

It’s not been resolved efficiently either.

The difference needs to equal 1.00 dollar. If the ball is 0.10c and the bat is 1.00 than the difference would only be 0.90c instead of 1.00. Hence you need to split the difference equally in order to get the 1 dollar difference.

DemI-gODMarch 16, 2017As you put it in formulation with your example that would give us this numbers:

Let a be the amount of ball asked

b be the amount of bat given

x be the total amount of the items

given in your example lets get the formula

if a = 1.00 + b

if b = 0.04

lets compute if it is correct

x = a + b

x = (1.00 + b) + 0.04

x = (1.00 + 0.04) + 0.04

x = 1.04 + 0.04

x = 1.08 which is not equal to 1.10

if we substitute:

a = 1.00 + b

b = 0.05

x = ?

x = (1.00 + b) + 0.05

x = (1.00 + 0.05) + 0.05

x = 1.05 + 0.05

x = 1.10 which is equal to 1.10 which is correct

Richard LabrieAugust 1, 2018Sorry, my friend. But, it does break the rule. If the ball costs $.04 cents and the bat costs $1.00 more than the ball, the bat costs $1.04. That equals $1.08, not $1.10 as the question states the two items must equal.

KRystalJanuary 25, 2017No, that would’t work, because together they have to cost $1.10.

If the ball cost $0.04 then the bat would cost $1.04, and $1.04 + $0.04 would only be $1.08, which is incorrect.

The ball costing $0.05 is the only answer that it correct.

🙂

tUNG nGUYENApril 27, 2018I hadn’t understood the post until I’ve read the comment of KRYSTAL on January 25, 2017.

My Math skill is good enough to solve this problem if the question didn’t confuse me with the “unclear” language usage.

If I rewrite the question, I think most of the people can solve it.

“A bat and a ball cost $1.10 in total. The bat costs $1.00 more than the ball. How much does the ball cost?”

// We have:

total_cost = 1.1

(bat + ball) == total_cost

(bat – ball) > 1

// Now, calculate the ball

RaamMarch 6, 2017because, the bat and the ball has to cost $1.10

Dale KOHLERApril 12, 2017The confusing part is understanding the language. The bat cost a dollar more than the ball. That means the price of the ball plus a dollar more. If the ball was less than 5 cents then the total would have to be less than $1.10.

TjenkinsMay 2, 2017Because if the ball is 4 cents..now the bat is $1.01 more and 3 cents the bat is 1.02 more…the bat IS a $1.00 more it said. Not just over a $1.00 more.

AlexOctober 12, 2017Except it says costs 1$ more… if you look at that as an absolute, then theres no other answer than 5 cents for the ball. The bat costs absolutely no more or less than 1$ than the ball. 1.05 for the bat and .05 for the ball is the only option.

Shad VickDecember 21, 2017Absolutely. More can be defined several wasy – as in this case you could say more = “additional”. So the the bat costs an additional $1 than the ball. $1 additional or more would make the bat worth $0.10. It’s a trick question not giving enough proper words to single out one answer by folks who are trying to deceive you.

IMranMay 12, 2018Coz it bat costs exactly $1 more. Not less not more

John GoeckermannMay 14, 2018You are ignoring the fact that the TOTAL fof bat and ball is $1.10, which means your $0.04 or $0.07 does not work. It does not say it ONLY has to cost one dollar more.

Robert OnumaAugust 18, 2018they have to add up to 1.10 so if the bat cost 1.04 or 1.03, that would mean that the the ball costs either 0.4 or 0.3 which when added would give you an answer less than 1.10

Diana BloomSeptember 23, 2018Yes!

RosemaryNovember 15, 2016I simply don’t understand why life has to be so complicated.

This goes to confirm why I would rather stick to linguistics!

Marlo eugeneFebruary 1, 2017Linguistics makes all the difference. The conceptual emphasis seems to lie within the word MORE.

X + Y = $1.10. If X = $1 MORE then that leaves $0.10 TO WORK WITH rather than automatically assign to Y

So you divide the remainder equally (assuming negative values are disqualified) and get 0.05.

Or with the Rolls Royce/Ferrari: RR = $145K | F = $45K

leigh HurshMarch 9, 2018Finally a way to say the problem…I have 90,000 to work with! How is that said in the problem…just the way you did? I am beginning to think this has trasnfer value for me for other problems as it should. thank you.

christian o’connorApril 12, 2017I don’t understand this…it has nothing to do with whether or not I believe the baseball only costs .10 or the bat $1. Here’s my logic: if I take a .10 baseball to the checkout and say I want to buy it, it costs me .10. When the cashier says, you can get the bat too, it’s $1 more than the baseball, I have to pay $1 for the bat. My total is $1.10, so how is the baseball not .10?

SApril 17, 2017The bat is a dollar MORE than the ball if you have a $.10 ball the bat be $1 more in total cost would be $1.10 thus the bat which is $1.10 + the ball which is $.10 would be $1.20.(I don’t have a cents sign on my phone, that’s why I use the dollar sign.)

SApril 17, 2017The bat is a dollar MORE than the ball if you have a $.10 ball and the bat is $1 more in total cost the bat would be $1.10 thus the bat which is $1.10 + the ball which is $.10 would be $1.20.(I don’t have a cents sign on my phone, that’s why I use the dollar sign.)

Caraid HolgateSeptember 20, 2017If I’m correct and allow for modification of the bat price to not be $1 then I suppose the bat can be $1.05 and the ball can be 5 cents it also $1 more?

I must admit I found the question not loose enough for that to be the case but this could be the answer? Maybe?

RhysOctober 12, 2017Bascally when they say ‘more than’ they mean that it also contains the price of the ball with the 1 dollar.

If someone says the ball is 0.05 and the bat is 1 dollar more than the ball, that means the bat is 1 dollar + the ball so the bat is 1.05.

an object is $1 more than x.

Is the same as saying:

an object = $1 + x.

Hope I helped someone lol.

Tamara reedNovember 7, 2017Although I agree that the correct answer to the question is that the bat is 1.05 and the ball .05, I completely agree with your logic. Everyone argues that the $1.00 answer is based on intuition, but I think it is a problem with linguistics. That is exactly what the clerk would say, but he’s incorrect. What he should be saying is the total is a dollar more than the ball. The clerk’s response (with different items) is so etched into the way we think it’s ridiculously difficult to get past. It’s like saying sunrise, which would more accurately be something like “earth spin.”

WayneDecember 11, 2017Well said, how do people get 1.20 ??? The ball in .10, the bat is 1 more, = 1.10 !!!

I don’t understand this…it has nothing to do with whether or not I believe the baseball only costs .10 or the bat $1. Here’s my logic: if I take a .10 baseball to the checkout and say I want to buy it, it costs me .10. When the cashier says, you can get the bat too, it’s $1 more than the baseball, I have to pay $1 for the bat. My total is $1.10, so how is the baseball not .10?

Wayne

Too Little Too LateFebruary 7, 2018Well Wayne, it nailed me too. The problem doesnt seem to stem from math but how we interpereted the question.

When we heard the bat is a dollar more than the ball, we broke the problem down into a very simple and instantly answered one. Ball is 10 cents, bat is a dollar more, bat is a $1.10

Unfortunately we forgot the first part of the story, that the bat AND ball total is $1.10.

So if, like we said, the bat is $1.10, and you add the cost of the ball, $.10, we end up with $1.20 for both, instead of $1.10 for both.

in order to make the cost come out right with the total, the ball has to be $.05 then at a dollar more than that the bat would be $1.05, for a total of $1.10

MARC LEWISDecember 5, 2018no your mistaking one thing for another. if the cashier said the ball is ten cents and if you want to buy the bat also its a dollar more to the transaction. this is not the same as her saying the price of the bat is exactly a dollar more than the ball. Because you thinking you get to establish the balls price in advance. set at 10 cents. The ball is bought last and the cashier has had to say that ball and bat are a set the bat is a dollar more than the balls price . if the total wasn’t already fixed then the ball could cost 80 cents and the bat would have to cost 1.80 cents but in our case the ball costs 5 cents and the bat costs 1.05 because the bat is a dollar more than the cost of the ball.

BloopJune 28, 2017So we’re supposed to assume that there’s no taxes or rounding and the bat is exactly $1 more than the ball to total exactly $1.10, right?

kenn kirbySeptember 5, 2017“A bat and a ball cost $1.10 in total. The bat costs $1.00 more than the ball. How much does the ball cost?”

Here’s the reason there are at least two answers to this question. One is the correct answer based on an linear analysis or response, the other is the correct answer based on a variable answer or response.

The first is linear. If the starting point, which is not clearly defined, is .10, then the answer is that the bat costs 1.00.

The second is variable. If the starting point is not determined, and the bat must be 1.00 more than the ball, then the answer is that the bat is 1.05.

However, if the bat must not be precisely 1.00 more than the ball, but merely 1.00 or more than 1.00 over the cost of the ball, then the answer is that the bat is 1.05 or greater than the ball, but no greater than 1.09, assuming that the measurement system is in terms of pennies being the lowest variable possible, and not half-pennies or another denomination.

These type of questions are confounding not because they are truly enigmas, but because the question is not clearly defined, which is why so many students are disadvantaged by poor test makers.

John wrightJanuary 14, 2018Why is this so tough to understand for some…

1.10 is the total…

A bat is 1.00 MORE than a ball totaling 1.10

Bat+1+ball=1.10

-1.00. =-1.00

Bat+Ball=0.10

+.5 +.5. =0.10 so ball =.5

Hello…!!!

Tina porterSeptember 15, 2017A dollar itself is still that amount regardless of the ball’s price. Ex: bat= 1.00 + ball = 1.10 total. But, *more* (in terms of this equation) seems to factor the ball’s price as an add-on. A dollar more than 5 cents. Ex: 1.00 + .5 = bat. 1.05 + .5 = 1.10 total.

Umar khaledJanuary 1, 2018please solve the Rolls-Royce & Ferrari problem…

Dan BeigelJanuary 20, 2018Yes, I too would like to see the “correct” answer to the Ferrari -Rolls-Royce question.

It is posed yet again differently than the bat+ball

question.

douglasJanuary 16, 2018If the ball is .05, then the wording “the bat cost one dollar more than the ball” would be the “cost of the ball” “plus” a “dollar.” .

Tried hardJanuary 25, 2018It is about how the mind works not about how smart you are

NormFebruary 5, 2018Aliens are looking down on earth and saying “they rule the planet”?

Mind boggling how simple math can trick people. All of the “ya buts” are coming from people who got it wrong and are trying to justify it. Yikes!!!

Now get back to watching Springer.

Will FowlesMarch 15, 2018The original question is from a three question cognitive reasoning test. I forget exact figures but when the three questions were put to a group of MIT students, something like less than third got all three right. The construction, language and amount would have all been very carefully designed to make it as easy to quickly get wrong as possible. That was the whole point.

BMarkMarch 15, 2018If you’re still confuse about this question, trust me you’re not dumb. As rightly stated by others, this is mostly a language problem rather than math. And I believe it’s clearly composed to prove a point rather than solve a problem or enlighten its audience.

Let me put the same question in a different way by injecting a third item(glove) in place of the second sentence and changing the word “Together” to Equivalent(the same), for you to see if it make sense:

A ball, a glove and a bat cost 110 cents. The “Glove” cost 100 cents more than the bat and the ball which values are equivalent or the same. How much does the ball cost?

lalithashreeMarch 22, 2018With reference to the bat and ball problem: ‘1 more than…’ is different from ‘more than 1’.

RIlApril 17, 2018Looking at the question over and over again I have realized that the question is wrong. The question needs to mention that the ball is exactly $1.00 over the ball. The one word “exactly” is missing in the question and that confuses all of us.

John GoeckermannMay 14, 2018No, it is not confusing if you pay attention… it states quite clearly bat costs ONE DOLLAR MORE than the ball. And it states the TOTAL must come to $1.10. Nowhere does it say the COST of either. There is no need for the word “exact”!! It is a number…. it can only mean ONE THING. TOTAL = $1.10. Difference between ball price and bat price must be ONE DOLLAR MORE for the bat. And the combined prices must be total $1.10. Those are both exact numbers, without telling you what the prices of either bat nor ball have to be. Quit trying to make it the fault of the question: it is your insistence on misinterpreting the data that is at fault.

jacque zidaneMay 28, 2018In response to Umar Khaled’s and Dan Beigel’s question(s): “please solve the Rolls-Royce & Ferrari problem…”?

Let’s adapt John Wright’s following formulaic approach to resolve the Rolls-Royce & Ferrari problem…

John Wright’s Formula…

(Bat + 1) + ball = 1.10

-1.00. = – 1.00

= Bat + Ball = 0.10

.05 + .05 = 0.10 so ball =.05

= (.05 + 1) + .05 = 1.10

-.05 = – .05

= (.05 + 1) = 1.05

= Bat = 1.05

Bat + Ball = 1.10

1.05 + .05 = 1.10

adapted formula to Rolls-Royce & Ferrari

“A Rolls-Royce and a Ferrari together cost $190,000. The Rolls-Royce costs $100,000 more than the Ferrari. How much does the Ferrari cost?”

(RR + 100) + Ferr = 190

-100 = – 100

= RR + Ferr = 90

45 + 45 = 90 so Ferr = 45

= (45 + 100) + 45 = 190

-45 = – 45

= (45 + 100) = 145

= RR = 145

RR + FERR = 190

145 + 45 = 190

Bill ReynoldsMay 29, 2018I found it easier to put together a spreadsheet to visualize the differences between the prices for the items in this article.

https://docs.google.com/spreadsheets/d/18wWax1UaVHnVAAXQ4yWUmn1qAlqkyg8hRwS3vNrazDU/edit?usp=sharing

So, if the ball cost 10 cents, and the bat cost $1, then clearly they both total up to $1.10, but the difference between the two items is 90 cents. That is why the ball must cost 5 cents, and the bat must cost $1.05.

Using the same logic, the Ferrari must cost $45,000, and the Rolls Royce must cost $145,000 – the difference between the two values is $100,000, and the price of both cars totals up to $190,000.

EhJune 28, 2018It’s more that “more than” is not usually used in this way in most word problems, I think. It’s usually used as just a phrase to indicate that you should use simple addition in a lot of math books, if you recall your grade school grades. So, the problem actually lies in the way the puzzle is worded. It’s meant to beguile and confuse via poor word choice, and thus a poor measure of intelligence. There are honestly far better worded math puzzles out there than this one…and many are more clever, too.

Keith MJuly 11, 2018To the people who say that question should include the word “exactly,” I say hooey (or “BS”).

$1.00 is an exact number. If the ball cost $0.04 and the bat cost $1.06, then the bat is $1.02 more than the ball. $1.02 is not $1.00, so the answer is wrong, even without the word “exactly.”

apJuly 19, 2018It seems to me that the incorrect answers such as the .10c and 90,000 ones stem from attempts to answer a different type of question wherein the total has to be split between two costs totaling the given amount (and thereby addressing the question) instead of producing the answer by way of realizing the difference in cost between two things . So in a way of seeing it there’s an ambiguity in the question. The hinted/given difference (a dollar more or a hundred thousand more) doesn’t seem to indicate/register that it constitutes an inclusion to a figure at the least both things/costs share. So in their heads (ones who incorrectly answer) 90,000 isn’t included with another amount (the 100,000) whereby it turns into 190,000 + 90,000 bringing the total over the given one of 190,000. All in all, and I say this knowing it’s merely a riddle, the foundation isn’t clarified to an answerer. Maybe a use of number line (as a form of diagram) to illustrate the answer to this sort of question could help. This of course could help to know why those incorrect answers are incorrect, but not yet offer a method to realizing why a correct answer is a correct one. One baby step at a time is helpful.

ARTHUR SPEARSAugust 13, 2018This isn’t a psychology problem. It’s an algebra problem. If one takes the time to write out the problem, then there’s only ever the correct answer to give. The issue is the “fast math” of people bad at algebra.

X + Y = 190000

X = Y + 100000

Y + 100000 + Y = 190000

2Y = 90000

Y = 45000

X = 145000

People don’t generally do multivariable analysis when asked about the cost of bats and balls and cars. The phrasing of the question lends itself to be unwittingly simple when it is not. Simply knowing this trick ahead of time would eliminate most wrong answers, imo.

DEs KennyNovember 18, 2018If you haid paid attention in high school Algebra you would instantly recongognise this is a problem in Simultaneous Equations. We have two Variables B (Bat) and b (ball) and two equations:

1. b + B = 1.10

2. B -b = 1.00

The principle of simultaneous equations is; If the number of variables is equal to the number of equations then there is a solution to find the value of all the variables.

So I will leave you to do the Algebra, It is quite simple.

Des kennyNovember 18, 2018This is high school algebra. There are two variables B and b. Write two simultaneous equations involving the two variables and then do the Algebra. If the number of variables equals the number of equations there is always a solution to a problem of simultaneous equations. Remember high school algebra? Yes, so it is quite easy then 🙂

APDecember 7, 2018Now, here, I think it’s important to hit at what makes “more than” fail to be interpreted correctly upon first or many encounters: ‘more’ contains the concept of ‘largeness’ implicitly as it does of ‘comparison’, and there seems to me to be a shift that occurs where the stress may be less or none on the latter than on the former. Because we are immersed in a non-logical use of language, the primary sense of ‘comparison’ in ‘more’ likely doesn’t occur. However, this is unlikely saying enough, as the process requires that we take into account the ‘other’ object of comparison. Even if we grasp the logical form of ‘more’, we may not realize another contained concept – ie ‘likeness’ – two objects of comparison must share for the difference of one against the other to be understood. Even this may not readily be useful for detecting the difference unless we recognize which likeness is to be accepted – here, it must be in figure/amount/number, not likeness or lack of likeness between two objects (cup and water, bat and ball, ferari and rolls-royce), that is, in characteristics or nature. All this considered, it is curious why a certain sense of ‘how much more of’ or ‘how much less of’ fails to occur in the mind of the interpreter. The bat and ball should share five cents in common. The ferari and ford should share the figure of 45,000 in common. I suppose this is because the ‘than’ after ‘more’ should not elicit the object (ball or ford) in mind, but the not yet known figure/number that represents the object in cost.

APDecember 7, 2018Let’s see what happens when we modify the question: “A bat and a ball cost $1.10 in total. The bat costs $1.00 more than (the cost of) the ball. How much does the ball cost?”

On tests like GMAT, one would be punished for not understanding the question. However, it wouldn’t be the test takers’ fault for they did not get to phrase the questions.

I wish to add that I believe there are words as units of language and there are unsaid and unsayable things behind the words in the exact moment of conveyance. Among those unsaid things are concepts as much as assumptions, which we take for granted in a given moment. In a given moment it isn’t easy to checklist all the various ways a word may be ambiguated in order to fit one variation on standard logical grounds with a given word. Even if this were possible, the checklisting process would comprise singular moments, too.