How Do I Translate This GMATPrep problem?
Ive been speaking with a lot of students recently who are really struggling with translation problems even when they can figure out how to translate, they end up taking way too much time on the problem.
So lets try this GMATPrep problem; set your timer for 2 minutes and GO!
If Bob produces 36 or fewer items in a week, he is paid x dollars per item. If Bob produces more than 36 items in a week, he is paid x dollars per item for the first 36 items and 1[pmath]1/2x[/pmath] times that amount for each additional item. How many items did Bob produce last week?(1) Last week Bob was paid a total of $480 for the items that he produced that week.
(2) This week Bob produced 2 more items than last week and was paid a total of $510 for the items that he produced this week.
Ugh. Okay, obviously were going to have to translate, because weve got a story going on here. It also looks like theres going to be some algebra involved. Lets dig in.
Im now Bob. (Put yourself in the story; thatll make things a little bit easier.) I can make either 36 or fewer items in one week or more than 36 items. How am I going to get paid? For the first scenario, I can figure out my pay by multiplying the number of items by x. If I make exactly 36 items, Ill get paid 36x. If I make 33 items, I'll get paid 33x. Hmm.I guess I should assign a variable for the number of items I make; lets call thatN.
For less than or equal to 36 items, Ill make Nx dollars. I'd write this down:
When N < 36, pay = Nx
What about if I make more than 36 items? Hmm. For the first 36, itll just be 36x. If I make 37 items, though, then Ill get paid 1.5x for the 37thitem only. How am I going to write that out?
WhenN> 36, pay = 36x + (something)
I still have to figure out that "something." Lets say I made 40 items. Then Id get paid 36x for the first 36, and (4)(1.5x) for the rest. Where did that number 4 come from? Right, its 40 36; the calculation is really 36x + (40-36)(1.5x). 36 is a fixed number in this calculation, but I dont know how many more I will make it could be 40 or 37 or 52. I can re-use my N variable here.
for >36 items, pay = 36x + (N- 36)(1.5x)
My key takeaway here? I had to introduce a second variable; thats probably going to be important.
Okay, finally, they want to know how many items (thats my N) Bob produced last week. Uh oh yep, a quick glance at the statements confirms what I feared: weve also got multiple timeframes. One of the statements is about last week and one is about this week. So there are two different potential N values: the one for last week and the one for this week. The x value stays the same from week to week, so we have three different variables here: x (the $ amount paid), [pmath]N_1[/pmath](the # of items made last week), and [pmath]N_2[/pmath](the # made this week). I want to solve for [pmath]N_1[/pmath]. Sigh.
[Note: at this point, Im thinking to myself: are you serious? Theres got to be an easier way. Im going to plow forward first with the actual algebra, which most people would try to do at this point. But there is an easier way; well discuss it down below.]
Am I ready for the statements finally?
(1) Last week Bob was paid a total of $480 for the items that he produced that week.
Did I make more than 36? I dont know. So I dont know which formula to use. Its either:
[pmath]N_1[/pmath]x = 480, or
36x + ([pmath]N_1[/pmath] - 36)(1.5x) = 480
I dont know which formula to use, so I cant solve. Even if I did, Ive got two variables and just one formula, so I still cant solve. This statement is insufficient; eliminate answers A and D.
Moving on to statement 2:
(2) This week Bob produced 2 more items than last week and was paid a total of $510 for the items that he produced this week.
Lets see, now were dealing with both this week and last week so Im going to have to be extra-careful to keep everything straight.
Last week:[pmath]N_1[/pmath]
This week: [pmath]N_2[/pmath]= [pmath]N_1[/pmath]+2
If N < 36, then the formula is: [pmath]N_2[/pmath]x = 510
If N > 36, then the formula is: 36x + ([pmath]N_2[/pmath] - 36)(1.5x) = 510 (note: I substituted in for [pmath]N_2[/pmath]= [pmath]N_1[/pmath]+ 2)
I have two different possible equations again; I dont know which to use and, even if I did, I'd still have one equation with two variables. Eliminate answer B.
Next I can try the two statements together. Its probably going to work, right? Each time, I had one equation with two variables. Now I should have two equations with two variables! Lets see.
From statement 1: 36x + ([pmath]N_1[/pmath] - 36)(1.5x) = 480
From statement 2: 36x + ([pmath]N_1[/pmath]+ 2 - 36)(1.5x) = 510
Wait. Stop. I didnt just have two equations with two variables for the individual statements. I had two different possible equations each time and didnt know which one to use! The two equations I listed above are for the case where N > 36. But I dont that thats the case! I could also need to use the other equations:
From statement 1: [pmath]N_1[/pmath]x = 480
From statement 2: ([pmath]N_1[/pmath]+ 2)x = 510
If I knew that it was one or the other (< 36 or > 36), then I would know which set of equations to use. But I dont even putting the two statements together is NOT sufficient to answer the question.
The correct answer is E.
That was pretty painful, wasnt it? Isnt there an easier way? Possibly, yes. Were going to start thinking more logically and were also going to start thinking more real world.
Right from the beginning, I can figure out this general fact without actually writing down the formulas: there are two possible formulas one where N is less than or equal to 36 and another where N is greater than 36. So, unless Im told which formula to use, there are two possible solution paths from the start which isnt good enough on data sufficiency.
Time to put myself back in the problem again. The first statement gives me the total pay for last week, but I have any number of ways I can get there. Try a couple. Maybe I get paid $480 per item and I made just one item. Alternatively, maybe I get paid $240 per item and I made two items. Both are possible, so statement 1 isnt sufficient and I can eliminate answers A and D.
Statement 2 is more complicated, but it does tell me I was paid $510 this week and also that I must have made at least two items this week (because I made two more than last week). So, great, lets say that I did make 2 items this week and I was paid $255 per item (note: I dont actually have to figure out the $255 number). Last week, then, Id have made 0 items. Alternatively, lets say that I made 3 items this week at a rate of $510/3 per item, so then Id have made 1 item the week before. Both are possible, so I can eliminate answer B.
What about the two statements together? Last week, I was paid $480; this week, I was paid $510 and made 2 more items. Those extra two items, then, earned me an extra $30 total, or $15 per item. Hey, isnt that enough to give mex? Oh, wait once again, I have to remember that there are two different solution paths. If I made 36 or fewer items, then x = 15, and I could solve to find the number of items. If, however, I made more than 36 items, then 1.5x = 15. I could also use that to solve, but I would get a different number of items. Heres where logic comes roaring back in: these two scenarios MUST result in a different number of items, because the first one is valid only when N < 36, and the second one is valid only when N > 36. By definition, N must be a different number in each scenario!
In other words, I dont have one definitive value for an answer, so statements 1 and 2 together are not sufficient; eliminate C and choose E.
That last bit is a pretty sophisticated piece of reasoning. If you cant get that far or if you understood it when you read it, but think youd never think of that yourself during the real test thats okay. Youve narrowed it to C and E, a 50/50 shot. Guess and move on. (Also, at this point, youll have realized youre dealing with a pretty complicated problem. Guess the one that seems least likely to be right. On this one, Id guess E because my instincts would be telling me sure, that looks like it could be done! But I also know that I can't really see how it would be done... so I'm probably wrong!)
Key Takeaways:
(1) Know how to translate from words to math and follow that path at first, but keep two things in mind: first, insert yourself into the problem to make it more real world (so you can use normal logic and reasoning to help set things up), and second, if the math starts getting way too complicated, consider using logic and / or testing numbers instead.
(2) On really hard problems, you might not be able to get all the way to the end using either translation or logic. Thats okay get as far as you can and know how to make a guess when needed. Dont, though, spend 3 or 4 minutes trying to get a problem like this right. If you need that long, chances are youll make a mistake anyway and now youll be behind on time as well.
* All quotes copyright and courtesy of the Graduate Management Admissions Council. Usage of this material does not imply endorsement by GMAC.