A picture of a baseball and bat.

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

The 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.

I 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?!

This 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.

No, 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.

As 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

No, 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.
🙂

because, the bat and the ball has to cost $1.10

The 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.

Because 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.

Except 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.

I simply don’t understand why life has to be so complicated.
This goes to confirm why I would rather stick to linguistics!

Linguistics 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

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?

The 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.)

The 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.)

If 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?

Bascally 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.

So 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?

“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.

A 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.

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