This Fraction Problem Is Harder Than It Looks

by on December 12th, 2012

I’ve spoken with multiple students lately who received a disappointing (lower than they were expecting) score on the quant section and who all said that the quant felt relatively easy or straightforward. How is that possible?

First of all, thinking that a test like the GMAT is easy is actually a warning sign: things probably are not going very well. If the test was going very well, then you’d be seeing some seriously hard – next to impossible – problems.

Second, the test writers are phenomenal at writing questions that don’t seem all that complicated… but are in fact your worst nightmare. My worst nightmare is not an impossible question – I know I can’t do it, so I just pick and move on. My worst nightmare is a question that I think I can do, and I spend a decent chunk of time doing it, and then I get it wrong anyway – even though I’m sure I got it right!

Try this GMATPrep problem and you might see what I mean. Set your timer for 2 minutes…. and… GO!

* ” Of the 3,600 employees of Company X, 1/3 are clerical. If the clerical staff were to be reduced by 1/3, what percent of the total number of the remaining employees would then be clerical?

“(A) 25%

“(B) 22.2%

“(C) 20%

“(D) 12.5%

“(E) 11.1%”

What’s hard about this one? It looks completely straightforward!

Hmm so I have 3,600 employees and 1/3 are clerical. That’s easy: 1,200 are clerical. Then I need to take 1/3 of that, so that’s 400. They want the percent of the total, so 400/3600 = 4/36 = 1/9 = ugh. Let’s see, a little quick long division… right, 1/9 is 11.1%. Answer E. Done!

Where’s my mistake? Go find it. (Actually, find both of them.) Also note that my wrong answer was right there in the answer choices!!

Oh, I see what I did! I correctly found 1/3 of 1,200, which is 400, but that’s how many are getting laid off. The remaining number is 1,200 – 400 = 800. Argh! Okay, 800/3600 = 2/9 = 22.2%. The answer is B.

Sigh. Nope. Still wrong. Nightmare!

Okay, let’s try this again, step by painful step. Before I show the correct solution, see if you can figure out how to calculate the other two incorrect solutions. I’ll explain at the end.

“Of the 3,600 employees of Company X, 1/3 are clerical.”

There are 3,600 employees. 1/3 of the employees are clerical. (3600)(1/3) = 1200. So far so good: there are 1,200 clerical employees.

“If the clerical staff were to be reduced by 1/3”

Okay, this was one of my mistakes. The clerical staff is “reduced by” (1200)(1/3) = 400, so the remaining staff equals 800.

“what percent of the total number of the remaining employees would then be clerical?”

What is the total number of the remaining employees? Oh. The remaining employees. I started with 3,600, but I just laid off 400, so there are only 3,200 remaining. Here’s my second mistake. So the actual calculation should have been 800 / 3200 = 8 / 32 = 1/4. The correct answer (for real this time!) is 25%, or answer A.

Note to those who got this one right: you can see how easy it would be to make any of those other mistakes, right? Imagine the pressure of test day, you know this is the real thing, you’re worried about the timing… and all those things just make these little mistakes even more likely. So, just because you got it right this time doesn’t mean that you’re immune to making these kinds of mistakes in general.

One more note before we talk about the other two wrong answers. When working directly with students, I notice repeatedly that the most common time for careless errors comes towards the end of the problem when the student feels that s/he has already “cracked” it. I do the same thing: we know now that we’re going to get it right, so we want to finish it off as quickly as possible, and we sort of start to look forward to the next problem already. And then BAM! Careless error. (And, of course, the wrong answer is usually in the mix of answers, because the GMAC folks have done their homework and figured out what errors we’re most likely to make!)

Did you figure out how to get to wrong answers C (20%) or D (12.5%)? Answer D is a result of a mistake we’ve already seen: incorrectly using 400 instead of 800 for the remaining clerical employees. In this case, though, we would use the correct figure of 3,200 for the denominator: 400/3200 = 4/32 = 1/8 = 12.5%.

For C, 20%, I didn’t find a really obvious way to get to that error. In looking at the mix of answers, though, I think it’s possible that the question-writer thought, “Hmm, the correct answer is 25% and then the 3 wrong answers are all weird numbers with decimals. So I’d like to have another “normal” answer in the mix. Then, nobody will pick 25% just because it looks nice, since 20% will also be there. And maybe people will actually think that it must be one of the weird numbers, since there are three of those, so test-takers who have to guess will be less likely to guess 25%. Perfect! I’m done writing this problem.”

Key Takeaways for “This looks pretty easy!” Problems:

(1) Be alert: there’s a pretty good chance that the problem is not as easy as it looks. The GMAT test-writers have mastered the art of making a problem look easier than it is.

(2) Take the problem step-by-step. Write out ALL of your work. (You should be doing this anyway on all quant problems – it doesn’t take less time to write while you’re thinking, and you will save yourself many careless mistakes if you do actually write out your work.)

(3) Watch out for the “I’ve got this / I’m almost done!” distraction. When you think you’ve cracked the problem, then focus in even more – I’m going to get this and I’m not going to make a last-minute mistake! Don’t start thinking about the next problem, or where you are in the section, or whether you might not need to save a little bit of time by speeding up now that you know what you’re doing, or I can’t wait till this stupid test is over!! No. Finish this problem. Then move to the next one.

* GMATPrep® questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.

15 comments

  • What an awesome name for the distraction. Very easy to remember and recall. Only Stacey could write like this.

  • For me the problem is not difficult at all.

    I performed it well. The most difficult part is to understand what the problem , actually , ask you.

    translating: what % is X/100 of the remaining employees (this is difficult to figure out)  -  would then be clerical means = 8, in concrete !

    x/100 * (I m not sure about this number) = 8 -----> X= 800 / I do not know the number.

    So is not difficult (I think is an upper level question, indeed) per se, but to figure out the denominator.

    Do you have a suggestion Stacey ??'

    Thanks. I follow on a regular basis your articles because you are a strong motivation for me.

    Thanks, again.

    • We do write "what percent" as x/100, yes. I'm not sure how you calculated the number 8 though - can you explain your reasoning to me?

  • This is an ambiguous problem, and I don't think it could be asked on a real GMAT. We don't know whether "Clerical Staff" is a sub commitee or sub group or what not. They need to clarify that they were FIRED straight up, not merely relocated to another group within the company, or else they could still be part of the company.

    • This is a real problem from the real GMAT. This is from GMATPrep, which consists entirely of probelms that were once given on the real GMAT and were then retired and released as study aids.

  • Hi Stacey,

    This question sounded a little ambiguous to me because we are relying on the reasoning that "If the clerical staff were to be reduced by 1/3" means they are getting fired. I got the numerator correct but for the denominator I used 2800 (3600 - 800) since they say remaining employees (which means total minus clerics) and we know the clerics are 800 in number.
    Thanks Karthik !

    • Well, getting fired, getting laid off - the exact details don't matter, but the idea is that these people are gone - and that's what "reduced" really means. :)

      Oh, I see what you did with the denominator. And then your only clue would have been that your answer wasn't among the 5 answers, so you went back and read it again?

  • Yes :(

    Also Stacey I have a request, there is a Set problem which has gone through multiple threads but the explanations don't seem to be easily understandable. Would you be willing to explain it "your way" ?
    http://www.beatthegmat.com/least-value-venn-diagram-t46464.html
    Thanks
    Karthik

    • I just took a look at the problem. I think Stuart's answer is the best "official math" answer and I can't add to that. In terms of how I'd actually do it, though... I wouldn't do the official math. This is what I did:
      I drew a number line and labeled it 0 on one end and 100 on the other. Those are my 100 people Then I marked off from 0 to 70 and labeled it "apples" - these 70 people like apples. Now, before I do the second fruit, what's my goal?

      I want to *minimize* overlap between the groups. So for my second group, the 75 who like bananas, I start at the other end of the number line. I mark off from 100 down to 25 and label that "bananas."

      Now, how many like both? The overlap is 50 people; so far, 50 people like both apples and bananas.

      Now come the cherries. I need to mark off 80 people who like cherries, but as much as I can, I don't want to assign those 80 people to the 50 who like both apples and bananas.

      So I mark off the first 0 to 25 and label them cherries. The first 25 people like apples and cherries but NOT bananas. I now have 80 - 25 = 55 cherry people left to place. I also mark off the final 70 to 100 people - they like cherries and bananas, but NOT apples. Now, I've got 55 - 30 = 25 cherry people still to place.

      The remaining 25 cherry people MUST be placed among the middle group of people who do like apples and bananas. So 25/100 (or 25%) = the minimum who must like all 3.

  • Dear Stacey,

    You never cease to amaze me !!! You are the BEST :) and I really appreciate your taking time to give this much simpler intuitive approach to solving this problem. It is one thing to solve Qs on your own and another thing to teach others how to solve them. Thats what makes you stand out from the rest.

    Coming back to this wonderful explanation, is the overlap between apples and bananas 50 or 55 (70 - 25 in the number line) ? I know this is insignificant in this problem but I am just trying to understand everything that you have just explained.

    There is another similar question posted by Stuart in the same post which I was unable to answer.
    "One quarter of the cars on a parking lot have air conditioning and one third of the trucks on the same lot have air conditioning. If there is at least one car and at least one truck on the lot, no other vehicles on the lot, and a total of 72 vehicles on the lot, what's the maximum number of cars that could have air conditioning?
    We think: we want to maximize the number of cars with AC, so we need to minimize the number of trucks with AC.

    I'll let you try that problem on your own for practice - the answer is 15. "

    Why can't the minimum no: of trucks be 1 and the remaining 71 be cars out of which 25% are ac, ie, 17.

    Thanks a ton !
    Karthik

    • Let's see - it's entirely possible that I made a mistake while typing that out. I was doing so without the benefit of the paper on which I had originally drawn everything out. :)

      Yes, let's see - there are some typos there.
      Apples = 0 up thru 70.
      Bananas = 100 down thru 26, NOT 25. So the two are overlapping between 26 and 70 (or 45 people).
      Cherries = first I do 0 thru 25 and 71 thru 100, for a total of 55 people who overlap with either apples OR bananas, but NOT both.

      The remaining 80 - 55 = 25 cherry people have to fit in that middle swath of people who also like apples and bananas.

      Re: your second question. If you have 1 truck, then how can 1/3 of the trucks have air conditioning? :) You have to make sure the 1/3 and 1/4 things give us whole numbers of cars and trucks.

  • Dear Stacey,
    I am back with more ! :) . The source of these questions is from a forum http://gmatclub.com/forum/seventy-percent-of-the-800-students-in-school-t-are-male-at-38122.html but I think they are from Ivy gmat.

    1) Seventy percent of the 800 students in School T are male. At least ten percent of the female students in
    School T participate in a sport. Fewer than thirty percent of the male students in School T do not participate
    in a sport. What is the maximum possible number of students in School T who do not participate in a sport?
    I took around to 4 mins to get this one so wanted to know if you could suggest any shortcut/more efficient way ?

    2) For Qs which require a 2x 2 matrix to be set up I am getting the set up wrong for statements such as "25%
    of the people who do not have the job of their choice have a university diploma". In the solution this value is set up in the matrix as the intersection between No Job and Have Choice of Diploma. But I put the value under total no: of people with Choice of Diploma because I read it as 25% of No Job = no: with Choice With Diploma. I got this wrong couple of times so wanted to get your thoughts on it before I "learn" to do it this way !

    Thanks as always
    Karthik

    • Hi, Karthik

      I'm sorry, but we're not supposed to use the comments section of articles as though these were the forums - there's already a place to discuss other questions like these.

      I know you're hoping to get my input here because I don't participate on the BTG forums, but there's a reason why I don't - I have way too many other responsibilities and not enough time in the day. :)

      Also, I don't typically respond to (or study from) questions whose source I don't know, and I also don't discuss (or study from) questions that are not from a highly reputable source. You don't have to hold yourself to the same standards... just FYI.

      For your 2nd more general question:
      "25% of the people who do not have the job of their choice have a university diploma".

      The "who" part is a modifier and it tells you which group is the main group on which the 25% figure is taken. The start parting with "have" (the verb) is telling you something about the smaller 25% group.

      Let's say there are 60 people who do not have a job of their choice (whether or not they have a diploma).
      25% OF this group...
      25% of 60 = 15
      15 people have a university diploma AND a job of their choice.

  • Hi Stacey,

    I am sorry about that. I am aware that you do not respond to entirely different Qs on the comments and that you dont participate on the forum discussions ! :) And like you rightly said my intention was to get more insight into the topic than get an answer to the Q which I would have got from the forum. But thanks for responding, as always ! You are the best.

    Thanks
    Karthik

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