The Hardest Easy Math Problem in the World

by

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:

frustratedA 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.

To read more about the original research, go to:

http://mitsloan.mit.edu/newsroom/newsbriefs-0605-frederick.php 

  1. Alabi Michael April 30, 2015 at 3:46 pm

    B=105 ,b=5

  2. Connor April 24, 2015 at 3:06 pm

    (From Below)

    Google lied.
    The answer is actually $0.05

  3. 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.

  4. vidyasagar December 20, 2014 at 5:56 am

    B=105,b=5

  5. 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

  6. 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

  7. Scouty November 17, 2014 at 12:53 am

    The bat is $1 while the ball is 10 cents.

  8. 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

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    […] Marketing Software Designed to Make Buying Easy. Duh! January 20, 2014 by Andrew Rudin Leave a comment A bat and a ball cost a dollar and ten cents in total. The bat costs a dollar more than the ball. Ho… […]

  10. 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 ;)

  11. 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!

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

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

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

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

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

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

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

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

  19. 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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  22. Huugh January 23, 2013 at 8:32 pm

    That was so easy

  23. 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

  24. 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….

    so I get the answer.

  25. Chazz October 29, 2012 at 12:04 am

    That was fucking easy, what are you talking about…

  26. xpucsc October 27, 2012 at 7:12 am

    I’m wondering why people would miss. It’s 5th grader math.
    B+b =110 and B = 100 +b. Just solve for B and b.