Story Problems: Make Them Real - Part 1
Ive been on a story problem kick lately. People have a love / hate relationship with these. On the one hand, its a story! It should be easier than pure math! We should be able to figure it out!
On the other hand, we have to figure out what theyre talking about, and then we have to translate the words into math, and then we have to come up with an approach. Thats where story problems start to go off the rails.
You know what I mean, right? Those ones where you think itll be fine, and then youre about 2 minutes in and you realize that everything youve written down so far doesnt make sense, but youre sure that you can set it up, so you try again, and you get an answer but its not in the answer choices, and now youre at 3.5 minutes or so argh!
So lets talk about how to make story problems REAL. Theyre no longer going to be abstract math problems. Youre riding Train X as it approaches Train Y. Youre the store manager figuring out how many hours to give Sue so that shell still make the same amount of money now that her hourly wage has gone up.
Try this GMATPrep problem:
* Six machines, each working at the same constant rate, together can complete a certain job in 12 days. How many additional machines, each working at the same constant rate, will be needed to complete the job in 8 days?(A) 2
(B) 3
(C) 4
(D) 6
(E) 8
Yuck. A work problem.
Except heres the cool thing. The vast majority of rate and work problems have awesome shortcuts. This is so true that, nowadays, if I look at a rate or work problem and the only solution idea I have is that old, annoying RTD (or RTW) chart Im probably going to skip the problem entirely. Its not worth my time or mental energy.
This problem is no exceptionin fact, this one is an amazing example of a complicated problem with a 20-second solution. Seriously20 seconds!
You own a factory now (lucky you!). Your factory has 6 machines in it. At the beginning of the first day, you turn on all 6 machines and they start pumping out their widgets. After 12 continuous days of this, the machines have produced all of the widgets you need, so you turn them off again.
Lets say that, on day 1, you turned them all on, but then you turned them off at the end of that day. What proportion of the job did your machines finish that day? They did 1/12 of the job.
Now, heres a key turning point. Most people will then try to figure out how much work one machine does on one day. (Many people will even make the mistake of thinking that one machine does 1/12 of the job in one day.) But dont go in that direction in the first place! If you were really the factory owner, you wouldnt start writing equations at this point. Youd figure out what you need by testing some scenarios.
Six machines do the entire job in 12 days. I want to do the job in 8 days instead. First, how much of the job can my existing six machines do in 8 days? If those six machines worked just for 8 days, they would do 8/12 = 2/3 of the job. So I need to buy enough extra machines to do another 1/3 of the job during that 8 days.
Wait a second! If 6 machines can do 2/3 of the job in 8 days then 3 machines can do half of that, or 1/3 of the job, in 8 days. Thats it! Im done! Thats the 20-second solution. And its a solution I would never have found without simply thinking about how Id figure this out in the real world, not on a standardized test.
The answer is (B): 3 machines.
Lets try another GMATPrep problem.
* A used-car dealer sold one car at a profit of 25 percent of the dealers purchase price for that car and sold another car at a loss of 20 percent of the dealers purchase price for that car. If the dealer sold each car for $20,000, what was the dealers total profit or loss, in dollars, for the two transactions combined?(A) $1,000 profit
(B) $2,000 profit
(C) $1,000 loss
(D) $2,000 loss
(E) $3,334 loss
Okay, you know the drill: youre the used-car dealer! One of your employees just came to you and told you that she sold a car for $20,000. Youre excited because you know that represents a 25% profit on what you paid for that particular car.
How much did you pay for it in the first place? Darn, you lost the record. If this were really happening, would you whip out a piece of paper and start writing equations? Of course not! Youd just try some numbers till you zeroed in on the answer. Try it yourself before you read the next paragraph.
Lets see was it $15,000? No, that would be a $5,000 profit, which is 33.3%. Is it a larger or smaller number? You want the profit to go down (it was only 25%) so you need the cost to be higher. How about $16,000? Lets see, then the profit would be $4,000 and, bingo, $4k is indeed 25% of $16k! The first car gave you a $4,000 profit.
Okay, then another employee just told you that she finally got rid of that car thats been sitting on the lot for ages. She was only able to get $20,000, though, even though you paid more for it.
This time, the car represents a 20% loss. How much did you pay for it? Try some numbers till you figure it out.
Lets see. $22,000 would be a $2,000 loss but $2k is nowhere near 20% of $22k. What about $25,000? Thatd be a $5,000 loss. And, yes, $5k is 20% of $25k! Alright, so the second car cost you $25,000 and gave you a $5,000 loss.
Hmm. Not such a great day so far, huh? The first car was a $4,000 gain but the second was a $5,000 loss. Thats a net loss of $1,000.
The answer is (C): $1,000 loss.
The beauty of this "make it real" method is two-fold. First, when you can do the problem, it's a whole lot easier to do it "logically" rather than in the "textbook math" fashion. Second, it'll be much more obvious when you can't do the problem, so it will be easier to let go and move to the next problem.
Okay, got all that? Try a harder problem.
Key Takeaways for Story Problems
(1) Whenever you read a story, ask yourself what you would do if you had to figure this out in the real world. 99% of the time, youd never think of writing equations. Instead, youd logic it out. Where will the train be after 1 hour? 2 hours? If you increase Sues wage by 10%, what will happen? 20%?
(2) This will feel slow and funny at first, because youre not used to treating standardized tests this way. Also, you may need to develop something that math teachers call numbers sense or math sense. Math sense is the ability to do the kind of back-of-the-envelope thinking that was demonstrated in this article. In a lovely bit of symmetry, this skill will be really useful in b-school and in your business career.
(3) Practice this same kind of thinking in the real world, with real scenarios or made-up ones. Whats the easiest way to approximate that 18% tip that you want to leave? You and a friend are on different trains heading straight towards each other (on different tracks!). You want to wave to your friend when the two trains pass each other. When is that going to happen? (This last one is basically the same scenario depicted in OG Quant Supplement #119!)
* GMATPrep questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.