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Last week, we took a look at how to analyze a Critical Reasoning question. This week, we’re going to do the same with a Problem Solving question. The GMATPrep® problem we’re using this week is one that we’ve already discussed how to solve in a previous article; click here to read that article and try the problem first.

Here’s the problem again; if you didn’t read the first article and try the problem already, then try this problem now:

Three boxes of supplies have an average (arithmetic mean) weight of 7 kilograms and a median weight of 9 kilograms. What is the maximum possible weight, in kilograms, of the lightest box?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

After trying the problem, checking the answer, and reading and understanding the solution (read the original article, linked above), I try to answer these questions:

1. Did I know WHAT they were trying to test?

Was I able to CATEGORIZE this question by topic and subtopic? By process / technique? If I had to look something up in my books, would I know exactly where to go?

• The question is a Problem Solving question from the Statistics chapter of my Word Translations book. It’s specifically testing the concept of average (arithmetic mean) and median, plus it’s a max / min problem. There are real numbers, so I won’t be using lots of variables or picking my own numbers here – I will have to somehow set up and solve using the numbers they gave me.

Did I COMPREHEND the symbols, text, questions, statements, and answer choices? Can I comprehend it all now, when I have lots of time to think about it? What do I need to do to make sure that I do comprehend everything here? How am I going to remember whatever I’ve just learned for future?

• Average / mean is characterized by the formula A = S/n, where A is the average of the set, S is the sum of the items in the set, and n is the number of items in the set. The median is the middle number in a set with an odd number of terms (which is the case in this problem: there are 3 boxes). (How would I have handled it if there had been an even number of terms? If I don’t remember, I go check my book, even though I don’t need to know for this specific problem.)

Did I understand the actual CONTENT (facts, knowledge) being tested?

• See last answer. Also, they want me to find the maximum (highest) possible weight of one particular box. The problem uses kilograms throughout, so I don’t need to worry about keeping track of units or converting anything (but I still needed to check!).

2. How well did I HANDLE what they were trying to test?

Did I choose the best APPROACH? Or is there a better way to do the problem? (There’s almost always a better way!) What is that better way? How am I going to remember this better approach the next time I see a similar problem?

• (See the original article, linked at the top, for a detailed discussion of the best approach. Here, I’ll pretend that I didn’t use the best approach.) Max / min questions require me to figure out what I have to maximize and what I have to minimize. I should have been thinking about that from the moment I saw the word “maximum” in the problem, but I forgot about that and just tested various numbers for the heaviest box instead, so I took longer than I should have to solve. Next time I see a max / min problem, I’m going to write down the relevant word (max or min) and put a circle around it immediately.

Did I have the SKILLS to follow through? Or did I fall short on anything?

• I did get it right eventually, but I took too long, because I didn’t think the problem through from the point of view of what needed to be minimized in order to maximize the lightest box. I need to do a few max / min problems to drill the process so that I can do this problem as efficiently as possible.

Did I make any careless mistakes? If so, WHY did I make each mistake? What habits could I make or break to minimize the chances of repeating that careless mistake in future?

• I didn’t make any mistakes that caused me to get this one wrong, but I can guess what kinds of mistakes people might make. People might not notice that the information about the median tells us the exact weight of the middle box, so that figure cannot change. The only two that we don’t know for sure are the lightest and the heaviest. Alternatively, even if I had used the proper max / min approach, I might have made the mistake of thinking that the heaviest box had to be 10 instead of 9 (that is, I would have made the mistake of thinking the “heaviest box” had to be heavier than the middle one, when they could actually have the same weight). That would have led me to pick wrong answer B.

Am I comfortable with OTHER STRATEGIES that would have worked, at least partially? How should I have made an educated guess?

• I tried testing random numbers for the weight of the heaviest box. If I’m going to pick numbers anyway, I should let the problem guide me. The five answers are all small integers; I could have just tried those to see which one worked. When plugging the answers back into the problem, I want to start with either B or D (it doesn’t matter which; pick whichever one looks like it will be the easiest to calculate). Let’s try D first. The lightest box is 4 kg and the middle box is still 9 kg; together, they add up to 13. The total weight for all three boxes is 21, so the heaviest box would be 21 – 13 = 8. Is that okay? No – that makes the “heaviest” box lighter than the middle box. So 4 kg is too heavy for the lightest box; cross off choices D and E (because E is even heavier than D!). Now, try B.

• If the lightest box is 2 kg, and the middle box is 9 kg, then the two boxes add up to 11. The total weight for all three boxes is 21, so that would mean the heaviest box would be 21 – 11 = 10 kg. Is that possible? Sure, because 10 is greater than or equal to 9. Could I make the lightest box even heavier, though? Possibly; I should try answer C. If the lightest box is 3 kg, that will make the heaviest 9 kg. Is that possible? Yep! And that’s the largest remaining answer choice, so it’s the right answer.

Do I understand every TRAP & TRICK that the writer built into the question, including wrong answers?

• Someone might pick E because the question asks for the maximum, and E represents the largest of the five answer choices; likewise, someone might mis-read the question and pick A because the question asks us to calculate the weight of the lightest box and A represents the smallest of the answer choices. I already figured out the trap for B earlier: thinking that the largest box has to be heavier than the middle box. I would guess B is the biggest trap – the most common mistake.

3. How well did I or could I RECOGNIZE what was going on?

Did I make a CONNECTION to previous experience? If so, what problem(s) did this remind me of and what, precisely, was similar? Or did I have to do it all from scratch? If so, see the next bullet.

Can I make any CONNECTIONS now, while I’m analyzing the problem? What have I done in the past that is similar to this one? How are they similar? How could that recognition have helped me to do this problem more efficiently or effectively? (This may involve looking up some past problem and making comparisons between the two!)

• Yes. I recognized that this was a mean + median question and I knew how to deal with both of those concepts. I could have done better with my handling of max / min though; I did know how to deal with max / min problems already, but I glossed over the word “maximum” when I first read this problem.

HOW will I recognize similar problems in the future? What can I do now to maximize the chances that I will remember and be able to use lessons learned from this problem the next time I see a new problem that tests something similar?

• I need to do everything I already described in my notes above. I’m also going to re-do this problem – actually make myself write out the best way to do it, alternate ways to do it, and so on, so that I really remember the lessons. Then, because my big problem on this one was with the max / min stuff, I’m going to find another max / min problem and practice: (1) immediately writing max or min down and circling it, (2) thinking through the problem using the max / min process from the start, and (3) if appropriate, try to use the answer choices to solve. (And I’m going to do the full analysis on any new problems I do, too.) I’ll also find some other problems of different types in which the answer choices are small easy numbers and practice plugging in the answers on those as well.

And that’s it! Note that, of course, the details above are specific to each individual person – such a write-up would be different for every single one of you, depending upon your particular strengths, weaknesses, and mistakes. Hopefully, though, this gives you a better idea of the way to analyze a problem. This framework also gives you a valuable way to discuss problems with fellow online students or in study groups – this is the kind of discussion that really helps to maximize scores.

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