### The Hardest Easy Math Problem in the World

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I was hanging out with a friend of mine the other day. She is a graduate student, and she asked me a question that she had come across during her research:
A bat and a ball cost a dollar and ten cents in total. The bat costs a dollar more than the ball. How much does the ball cost?

If you get it wrong, by the way, you’re in good company; this question was asked of a random sample of Princeton undergrads and University of Michigan undergrads, and roughly half of each group got the question wrong. I was totally fascinated by this result, because it really opened my eyes to how the GMAT can trick people so consistently with seemingly simple questions. The Quant section isn’t hard because the math is confusing; it’s hard because the language is confusing. The corollary here is that you can’t study for GMAT Quant just by studying formulas; you really need to study language as well.

So, let’s study the language in the above problem. To do so, it’s important to realize how humans process information; we try to organize information as we take it in. Our brains are pretty amazing machines this way, but sometimes they get us into trouble. For instance, we hear “A bat and a ball cost a dollar and ten cents in total” and we (correctly) think, “This question is probably about the individual cost of either the bat or the ball.” We’ve created a knowledge gap that we want desperately to fill. Then we hear “The bat costs a dollar,” and we automatically translate that to the ball must therefore cost the remaining ten cents, because the first sentence refers to a dollar and ten cents — two separate things. We also probably know now that the question will ask about the price of the ball. In fact, that’s really the only information we care about at this point since we know everything else. And, in the end, the question asks exactly this, so our brains are satisfied.

Do you see the tricks yet?

The modifier “more than the ball” critically alters the meaning of  “the bat costs a dollar…” It turns an absolute into a comparison. But because it comes mid-sentence, our brain registers it as unimportant, especially since we feel at this point that we have all the information we need to solve the problem. Furthermore, because the phrase “a dollar and ten cents” sounds like two separate things, we assign the dollar to one object and the ten cents to another object. If instead of $1.10, the question had said$3, it would be very easy to solve.

Ever since my friend told me this question, I’ve found several GMAT problems where the trap answer hinges on skipping a modifier, two sentences that look similar but are slightly different, or some other type of misreading. Make sure that when you read through a GMAT problem, you try and register all the information available before you start combining the various pieces to solve the problem!

By the way, the correct answer is 5 cents.

Don’t forget that you can attend the first session of any of our online or in-person GMAT courses absolutely free! Seriously. Check out our upcoming courses here.

Ryan Jacobs is a Manhattan Prep instructor based in San Francisco, California. Ryan holds an MBA from the University of California, San Diego. He has a perfect SAT score and a 760 GMAT. He has worked as a professional musician and as an administrator at Stanford University. During his MBA program, he completed a capstone project exploring alternative pricing models for online music lessons. Although he barely knows how to ice skate, Ryan is also a long-time San Jose Sharks fan. He’ll give you extra credit if you wear your gear to class. Check out Ryan’s upcoming GMAT courses here.

1. clinton January 8, 2016 at 9:34 pm

that was some maths, eh

2. Natashialou November 19, 2015 at 4:43 pm

I don’t understand why, but I got it correct try simply by seeing that a dollar more means exactly that, 1.00 more, so because the ball has a cost then it has to be .05 because only 1.05 and .05 would be only solution to total cost of 1.10. I don’t know if it’s due to being possibly simple minded outside the box, but as an adult now I can understand things that in HS I had no and wanted no understanding of. Now my 12 year old took a look as well at just the question, and before reading whole article, he switched from xbox to tv, and laughed while saying ” yup, your old mom, totally a trick question, it’s 1.05″. Well there goes my feeling of superiority and age defying beauty serum.

3. Mark November 10, 2015 at 9:55 pm

jaynebaby got it through my left brain lets see if I can help anyone else who is still stuck. To those writing formulas you don’t understand the language problem. Your formulas aren’t helping the mathematically challenged here. Language and math are both logical structures but language is sloppy and you have to take that into account. Sometimes the error has to be understood and explained. The problem says, “The bat costs a dollar more than the ball.” Those of us who are getting this wrong end up interpreting this statement as “The total costs a dollar more than the ball” and that’s NOT what it says.

So if the ball is ten cents and the bat costs a Dollar more than the ball that would be $1.10 for the bat. Add bat and ball and you get$1.20 (BZZZT that’s wrong).

The ball costs five cents, the bat costs a Dollar more, $1.05 add those two up$.05 + $1.05 =$1.10

So any of my fellow left brainers still stuck at a Dime for the ball?

4. youstupid September 30, 2015 at 10:37 pm

Yall are all stupid if you read it then you automatically know they are $1.10 together then you know just divide the .10 by 2 and you get .05. no formulas or big thinking. This is third grade math. 5. allison May 19, 2015 at 4:49 am You don’t even need to set up any equations. I did it in my head within the first 3 seconds of reading the question. (I didn’t read anything else on the page.) 6. Alabi Michael April 30, 2015 at 3:46 pm B=105 ,b=5 7. Connor April 24, 2015 at 3:06 pm (From Below) Google lied. The answer is actually$0.05

8. Connor April 24, 2015 at 2:57 pm

A bat and a ball cost a dollar and ten cents in total:
$1.10 The bat costs a dollar more than the ball: Bat must be$1.00 more than what the ball costs within $1.10 So: How much does the ball cost?: BAT : BALL$1.00 = $0.10$1.01 = $0.09$1.02 = $0.08$1.03 = $0.07$1.04 = $0.06$1.05 = $0.05$1.06 = $0.04$1.07 = $0.03$1.08 = $0.02$1.09 = $0.01 All = A dollar more than the ball. ($1.10 in total)

Why specifically $0.05? The answer is not hard, the question is just incorrect. 9. vidyasagar December 20, 2014 at 5:56 am B=105,b=5 10. JJ November 22, 2014 at 2:51 am Lol, It was simple xD just do (1.00+x)+x=1.10; simplify that to 1.00+2x=1.10, then to solve subtract 1 fro both sides to get 2x=0.10, then isolate by doing th inverse of 2x, and you’ll get 0.05, hence 1.00 + 0. 05 is 1.05 and the all 0.05 11. Kibet November 17, 2014 at 3:16 pm That problem should take less than 10 seconds to solve. Just form a system of two equations: x+y=1.10 y+1=x y+1+y=1.10 y=.05 12. Scouty November 17, 2014 at 12:53 am The bat is$1 while the ball is 10 cents.

13. twincamturbo April 10, 2014 at 9:10 pm

It’s an age old problem relating to extracting the correct information from the paragraph (comprehension) and expressing it in Mathematical terms. Once that has been done it’s relatively straight forward, however I can fully understand peoples errors trying to solve this from a verbal reasoning side (You actually will struggle as it’s not too clear, rather ambiguous in fact). This is where Maths comes into play. But there is only one solution and they have been post above already.
Twincamturbo UK

14. julesdb January 2, 2014 at 7:54 am

you took the long way
you can just do:
BAT= x+1,00
BALL= x
x+(x+1,00)=1,10
2x+1,00=1,10
2x=0,10
x=0,05
So the ball costs 5 cents and the bat 1,05 cents
Not bad for a 16 year old boy, eh 😉

15. jaynebaby June 25, 2013 at 10:14 am

If together 1 bat and 1 ball cost one dollar and ten cents. And in this the bat cost 1 dollar more than 1 ball..then that means the ball must cost 5 cents as then the bat would cost 1 dollar 5 cents which works out exactly one dollar more than the ball.
BOOM!

16. Gourav jeet nair April 25, 2013 at 11:24 pm

Solve it simonteniously ……… B=105 ,,, b=5

17. thornton April 4, 2013 at 10:21 am

it is alot simpler than your working makes it out to be…

18. thornton April 4, 2013 at 10:20 am

yeah, that was probably the easiest simultaneous equation I have ever done

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21. ibrar karim March 14, 2013 at 11:10 am

is this possible
35*35=01122570
is possible why or not why

22. A concerned englsh teacher March 7, 2013 at 10:54 pm

For the second part of the “equation” it clearly uses “the” which would mean that it is talking about the same ball and bat stated previously. That would mean that according to your logic, there would be two bats and two balls, but that is irrelevant. The ball would actually cost 10 cent because using “the” instead of another “a” would not add another ball.

P.S. Please revise before you try to trick people and be clever!

Amen

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25. Huugh January 23, 2013 at 8:32 pm

That was so easy

26. mrinmoy mondal January 20, 2013 at 9:45 am

B+b=110
B=100+b

=> B=100+b
=>B+b=100+b+b
=>B+b=100+2b
Then,
=>100+2b=110 [B+b=110]
=>2b=110-100
=>2b=10
=>b=5
B=100+b
=>B=100+5
=>B=105
Ans:B=105,b=5

27. Chen January 12, 2013 at 4:20 pm

I was wanted to give the wrong answer initially, then I realized the title does implies it’s not that easy….