Story Problems: Make Them Real
I’ve been on a story problem kick lately. People have a love / hate relationship with these. On the one hand, it’s 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 they’re talking about, and then we have to translate the words into math, and then we have to come up with an approach. That’s where story problems start to go off the rails.
You know what I mean, right? Those ones where you think it’ll be fine, and then you’re a minute or two in and you realize that everything you’ve written down so far doesn’t make sense, but you’re sure that you can set it up, so you try again, and you get an answer but it’s not in the answer choices, and now you’ve crossed the 3-minute mark…argh!
So let’s talk about how to make story problems REAL. They’re no longer going to be abstract math problems. You’re riding Train X as it approaches Train Y. You’re the store manager figuring out how many hours to give Sue so that she’ll still make the same amount of money now that her hourly wage has gone up.
Note: I’ve used GMAT problems in this article because the makers of the GRE don’t allow us to re-publish their problems. I’d rather work from actual problems written by standardized test-writers, just to show you how well this technique does work on real problems. I’ve chosen two problems that could just as easily be seen on the GRE.
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… here’s 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… I’m probably going to skip the problem entirely. It’s not worth my time or mental energy.
This problem is no exception—in fact, this one is an amazing example of a complicated problem with a 20-second solution. Seriously—20 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.
Let’s 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, here’s a key turning point. Most people will then try to figure out how much work one machine does on one day. (Many people will make the mistake of thinking that one machine does 1/12 of the job in one day.) But don’t go in that direction in the first place! If you were really the factory owner, you wouldn’t start writing equations at this point. You’d 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. That’s it! I’m done! That’s the 20-second solution. And it’s a solution I would never have found without simply thinking about how I’d figure this out in the real world, not on a standardized test.
The answer is (B): 3 machines.
Let’s try another GMATPrep problem.
* ” A used-car dealer sold one car at a profit of 25 percent of the dealer’s purchase price for that car and sold another car at a loss of 20 percent of the dealer’s purchase price for that car. If the dealer sold each car for $20,000, what was the dealer’s 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: you’re the used-car dealer! One of your employees just came to you and told you that she sold a car for $20,000. You’re 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! You’d just try some numbers till you zeroed in on the answer. Try it yourself before you read the next paragraph.
Let’s 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? Let’s 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 that’s 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.
Let’s see. $22,000 would be a $2,000 loss… but $2k is nowhere near 20% of $22k. What about $25,000? That’d 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. That’s 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.
*The textbook math solution for the second problem isn’t that bad, as long as you feel comfortable with that kind of math. I will tell you this, though: I know exactly how to do the textbook method but I would still use the “make it real” method because the numbers are easy. $20,000 is a nice round number and the answer choices are all “easy” numbers as well. So why risk making a mistake with the textbook math?
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, you’d never think of writing equations. Instead, you’d “logic” it out. Where will the train be after 1 hour? 2 hours? If you increase Sue’s wage by 10%, what will happen? 20%?
(2) This will feel slow and funny at first, because you’re 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 graduate school and in your career!
(3) Practice this same kind of thinking in the real world, with real scenarios or made-up ones. What’s 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?
* GMATPrep® questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.