Translating Words into Math, Part 2
This is the second part of a two-part article on the topic of translating wordy quant problems into the actual math necessary to set up and solve the problem. Click here for the first part.
Last time, we discussed the basics as well as these two tactics:
- Translate everything and make it real
- Use a chart or table to organize info
Today, were going to dig a bit deeper into how the test writers can make translation really challenging.
Task 3: finding hidden constraints
The higher-level the problem, the more likely it will be to contain some kind of constraint that is not stated explicitly in the problem. For instance, I could tell you explicitly that x is a positive integer. Alternatively, I could tell you that x represents the number of children in a certain class. In the latter case, x is still a positive integer (at least I hope so!), even though I havent said so explicitly.
Heres another example:
If Kelly received 1/3 more votes than Mike in a student election
If M equals the number of votes case by Mike, then how would we represent the number of votes cast for Kelly?
Kelly equals Mikes votes plus another 1/3 of Mikes votes, or M + (1/3)M = (4/3)M.
If the question asks us something about the total number of votes cast for Mike and Kelly, what do we know? We can represent the total votes as M + (4/3)M = (7/3)M.
Interesting. Can you figure out anything of significance from that?
The total number of votes must be a multiple of 7. Why? Well, the number of votes must be an integer (hidden constraint!), and whatever that number is, it equals 7M/3. There isnt a 7 on the bottom of the fraction, so that 7 on top can never be cancelled out. Its always there so, whatever the total number of votes is, its a multiple of 7. (If youre not sure why, play around with some real numbers that fit this pattern. Prove it to yourself.)
What else? Turns out, M has to be a multiple of 3. Again, that total number of votes must be an integer. In order for that to be true, that denominator has to disappear somehow. It isnt going to be cancelled out by the 7, so it must get cancelled out by something in the M. That M, then, must contain a 3.
Task 4: how the test-writers disguise the topic
When we begin reading a new wordy problem, our first task is simply to identify what the problem is about in the first place. The test writers may use an identifying word thats relatively easy (e.g., they may use the word ratio), or they may test us on that topic without using the word making our job harder.
For instance, Problem Solving (PS) problem #20 in the Official Guide 12th Edition (OG12) asks:
The ratio 2 to 1/3 is equal to the ratio
Great they actually said the word ratio, so I know its a ratio problem. The word to in a ratio problem means to use that little colon symbol, so the first part says 2: 1/3, and then they ask what thats equal to, so I know to use an equals sign:
2: [pmath]1/3[/pmath]= ?:?
Bingo translated! But they told us the word ratio, so if weve studied ratios, then we know how to write them. How does this get harder? Take a look at Data Sufficiency (DS) problem #103 from OG12. Statement 2 says in part:
It takes 6 times as long to run the cartoon as it takes to rewind the film
The word ratio isnt there, but it is actually describing a ratio! For every 6 parts spent running, there is 1 part of rewinding: the ratio is 6:1 (running: rewinding). How do we know this? Well, they give us a relationship about the time it takes to perform the two activities (running and rewinding) without telling us any actual numbers about how long these activities take. Thats basically what we use a ratio to do: tell us some relationship between two quantities without giving us the actual quantities.
If you have studied ratios in that way (Whats the point of a ratio? Why do we use them?), then youll find it easier to spot the true significance of this wording. And even if you didnt, you still have a chance to learn after the fact: read the explanation. It actually uses the word ratio, even though the problem itself didnt! Ask yourself why and how you could have known that yourself before you read the explanation (because this is how youre going to know next time!).
Okay, so if we notice that the sentence is really describing a ratio, we have a chance to take the next leap: when thinking about the actual question asked (which I didnt give you), we might think about statement 2, Hmm, it would have been useful to know the fraction of time spent running the cartoon but they didnt give us a fraction. They gave a ratio. Next, we might remember that theres a relationship between ratios and fractions. A ratio is whats called a part-to-part relationship, while a fraction gives us a part-to-whole relationship.
In our example above, we have two parts, running and rewinding. The ratio of the time it takes to do each action is 6:1. If I run the film and then rewind it, I have 7 parts of time; 6 of them are used to run the film and 1 is used to rewind. Therefore, it takes 6 parts out of the whole 7, or 6/7, of the total time to run the film and 1/7 of the total time to rewind. If I keep going down this path (and given the other info in the problem), I can discover that this statement is actually sufficient to answer the question.
Okay, lets really test ourselves now. Heres a link to an article from a few months ago a super-hard GMATPrep Probability Problem. Theres just one thing the headline is misleading. The problem looks like a probability problem in fact, the question actually uses the word probability. But its really about something else the probability bit is just a disguise. See if you can figure out what the problem is really about, then check the rest of the article to see whether youre right.
Key Takeaways for Translating:
(1) Hidden constraints: Sometimes, the test writers will simply tell us a piece of information. Other times, those keys will be hidden in the details of the problem. Start looking for hidden constraints while youre studying. If you dont notice until after youre done with the problem, try it again, even if you got it right. Maybe noticing that hidden constraint at the beginning could have helped you spot a shortcut and answer the question more quickly.
(2) Topic disguises: If an explanation starts talking about a concept that wasnt mentioned by name in the question (and you didnt spot that the question was talking about this concept), go back and figure out how you could have known that at the start / how you will know that next time.
(3) If youre looking for a 650+ score, be aware that your primary task is NOT to do as many problems as possible really! Your task is to learn as much as you can from each problem you do such that you can apply this knowledge to other problems in future. Doing #1 and #2 above is time-consuming, but this is absolutely how you learn to recognize what to do, strip off disguises, avoid traps, and so on. When you start to get to the higher levels of the GMAT, the task becomes so much harder because of the way in which these problems are written, not just because of the content being tested.
* The text excerpted above from The Official Guide for GMAT Review 12th Edition is copyright GMAC (the Graduate Management Admissions Council). The short excerpts are quoted under fair-use statutes for scholarly or journalistic work; use of these excerpts does not imply endorsement of this article by GMAC.