Translating Words into Math
I’ve spoken with several students recently who are struggling with translating wordy quant problems into the actual math necessary to set up and solve the problem. Some people make too many mistakes when doing this, and others find that, though generally accurate, they take more time than they can afford. In the next two articles (this is part 1!), we’re going to talk about how to translate efficiently and effectively.
We’re going to do this by example: I’ll provide short excerpts from OG or GMATPrep problems, and then we’ll discuss how to know what to do, how to do the actual translation, and how to do so efficiently. Note that I’m not going to provide the full text of problems “ and, therefore, we’re not going to solve fully. That’s not our goal today.
The Basics
Before we dive into more advanced issues, there are some basics we all need to know. We’re not going to spend a lot of time on the basics because all GMAT books out there already explain this; I’ll give a quick introduction and, if you need more, seek out one of the standard books on this topic (in ManhattanGMAT’s books, you’ll find this info in the Algebraic Translations chapter of the Word Translations Strategy Guide).
First, when the problem introduces certain people, objects or other things, we will likely need to assign variables. Cindy can become C and Bob can become B. Next, the words will give us some kind of relationship between variables.
For instance, a sentence might tell us that Cindy is five years older than Bob. We’ve already decided to use C for Cindy and B for B. Next, the is represents an equals sign. Five, of course, represents the number 5. Finally older than indicates addition; we need a plus sign. Our translated equation becomes C = 5+B. (Another very common word is of, which typically means to multiply. For example, ½ of 6 would be written: ½ × 6.)
Notice a couple of things about this equation. We have two unknowns in the sentence, so we should expect to have two variables in the equation. Also, how can we quickly check the equation to see that it makes sense? There are two common ways. We can plug in some simple numbers to test the equation “ this might take a little bit longer, but it’s the more certain method. Or we can think about the concepts that have been presented. Who’s older and who’s younger? To which person do we need to add years in order to make their ages equal? We want to add to the younger in order to equal the older. Bob’s the younger one, so we want to add to his age. Does the equation do that?
Here’s an excerpt from an official question, Problem Solving (PS) problem #120 in the Official Guide 12th Edition (OG12):
David has d books, which is 3 times as many as Jeff and ½ as many as Paula.
They’ve already defined one variable for us: d for the number of books David has. Let’s use j for Jeff’s books and p for Paula’s books.
Next, take each piece of info separately:
David has d books, and d is (=) 3 times as many as (multiply) j, or d = 3j.
David has d books, and d is (=) ½ as many as (multiply) p, or d = ½p.
Task 1: translate everything and make it real
In OG12, PS problem #91 first tells us that a store sells all of its maps at one specific price and all of its books at another specific price. It also tells us:
On Monday, the store sold 12 maps and 10 books for a total of $38.00, and on Tuesday the store sold 20 maps and 15 books for a total of $60.00.
What should we do? First, set variables. Let m = the price for one map and let b = the price for one book. Then, pretend you own the store and a customer walks up with 12 maps and 10 books. What do you do? Make it real “ actually visualize (or draw out) what needs to happen.
First, I’d figure out how much I need to charge for the maps: $m each for 12 = 12m. Similarly, the books would cost 10b. You want to buy all of them? Excellent! You owe me 12m + 10b = 38. If we do the same thing with the second half of the sentence quoted above, we get 20m + 15b = 60.
So, we’re done with that “ now, we need to solve for m and b, right? Not so fast! Read the actual question first:
How much less does a map sell for than a book?
Hmm. They’re not just asking for the price of a map or the price of a book. They’re asking for the difference (less than) between the two. Which one costs more and which one costs less?
The sentence is telling us that the map is cheaper. Okay, so if I want the difference in cost between a book and a map, and the map is the cheaper item, how do I do that subtraction? Right, b “ m. I actually want to solve for that overall combination (b “ m); if I can find a way to do that without solving for b and m individually first, I can save time! (That topic, however, we’ll save for another time.)
Task 2: Where appropriate, use a chart or table to organize
Let’s try another; this is excerpted from PS #154 from OG12:
An empty pool being filled with water at a constant rate takes 8 hours to fill to 3/5 of its capacity.
Again, visualize “ you’re standing there (for 8 hours!) with the hose, watching the pool fill. How does it work? RTW: Rate × Time = Work. Make a chart:
|
Rate |
Time |
Work |
|
| First sentence: |
R |
8 |
3/5 |
Okay, so we have one formula: R × 8 = 3/5. The next sentence says:
How much more time will it take to finish filling the pool?
To finish filling hmm, how much more do we have to fill? An entire job = 1. We’ve filled 3/5, so we have 2/5 to go, right? Add another row to your chart:
|
Rate |
Time |
Work |
|
| First sentence: |
R |
8 |
3/5 |
| Question |
R |
T |
2/5 |
Hey, we’ve got another formula: RT = 2/5. We can use the first one to solve for R, and, since the rate stays the same, we can then plug into the second to solve for T.
One more! Let’s try this excerpt from PS #153 from OG12:
Jack is now 14 years older than Bill. If in 10 years Jack will be twice as old as Bill, how old will Jack be in 5 years?
First, set a chart up. We need a row for each person in the problem, and we also need to represent all of the timeframes that are discussed. Careful “ there are three timeframes, not two!
|
Now |
+5 y |
+10 y |
|
| Jack |
J |
||
| Bill |
B |
Assign variables “ decide whether to use one variable or two and decide when to set each base variable (most of the time, we’ll set the base variable to the Now timeframe). In the above chart, I’ve set two variables in the Now timeframe.
Next, if you want to use one variable, try to use the simplest piece of information given in the problem to simplify to one variable. In this case, the first sentence is the simplest info because it is set in the Now timeframe for both Jack and Bill.
Jack is now 14 years older than Bill.
J = 14 + B
Remember, is means equals and older than means add. Do you remember how to check your equation quickly to make sure it makes sense?
Who’s older, Jack or Bill? According to the sentence, Jack. The equation adds the 14 to the younger person, Bill. That makes sense.
Okay, so we can either remove the J from our table and insert 14 + B instead, or we can flip the equation around (to J “ 14 = B), then remove B from the table and insert J “ 14 instead. Does it matter? Mathematically, no, but practically speaking, yes “ make your life easy by keeping the variable for which you want to solve! We want to solve for Jack, so our new table looks like this:
|
Now |
+5 y |
+10 y |
|
| Jack |
J |
||
| Bill |
J – 14 |
Now fill in the remaining timeframes (you have the info to do this already “ just add 5 for the middle column and 10 for the final column!):
|
Now |
+5 y |
+10 y |
|
| Jack |
J |
J + 5 |
J + 10 |
| Bill |
J – 14 |
J “ 9 |
J – 4 |
What now? Oh, right “ now we have that harder second statement to translate:
If in 10 years Jack will be twice as old as Bill
Okay, what timeframe do we need to use? in 10 years “ okay, go to that column. In 10 years, Jack is J + 10 and Bill is J “ 4. Make sure to use these as you translate.
Next, will be is a variation of is and means equals. Twice means 2, and as old as means multiply. Here’s the translated equation:
J + 10 = 2(J “ 4)
Hey, we have an equation with one variable! Now we can solve.
That’s all for today; make sure to check back in next week for more, including how the test-writers will disguise the topic area being tested (and how we can recognize what to do anyway!).
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 haven’t said so explicitly.
Here’s 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 Mike’s votes plus another 1/3 of Mike’s 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 isn’t a 7 on the bottom of the fraction, so that 7 on top can never be cancelled out. It’s always there so, whatever the total number of votes is, it’s a multiple of 7. (If you’re 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 isn’t 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 that’s 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 it’s 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 that’s equal to, so I know to use an equals sign:
2: = ?:?
Bingo “ translated! But they told us the word ratio, so if we’ve 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 isn’t 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. That’s 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 (What’s the point of a ratio? Why do we use them?), then you’ll find it easier to spot the true significance of this wording. And even if you didn’t, you still have a chance to learn after the fact: read the explanation. It actually uses the word ratio, even though the problem itself didn’t! Ask yourself why and how you could have known that yourself before you read the explanation (because this is how you’re 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 didn’t 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 didn’t give us a fraction. They gave a ratio. Next, we might remember that there’s a relationship between ratios and fractions. A ratio is what’s 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, let’s really test ourselves now. Here’s a link to an article from a few months ago “ a super-hard GMATPrep Probability Problem. There’s just one thing the headline is misleading. The problem looks like a probability problem “ in fact, the question actually uses the word probability. But it’s 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 you’re right.
Key Takeaways for Translating:
(1) Know the basics. Certain words consistently mean the same thing (for example, forms of the verb to be generally mean equals). There are lots of great resources out there already that will give you the basics.
(2) Those annoying wordy problems have a lot going on. Make sure you are translating every last thing, and also try to make it real! Insert yourself into the situation; imagine that you are the one doing whatever’s happening and ask yourself what you’d have to do at each step along the way.
(3) When there are multiple variables, multiple timeframes, or other kinds of moving parts, use a chart or table to organize your info. Label everything clearly and only then start filling in.
(4) 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 you’re studying. If you don’t notice until after you’re 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.
(5) Topic disguises: If an explanation starts talking about a concept that wasn’t mentioned by name in the question (and you didn’t 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.
(6) If you’re 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.
* GMATPrep text courtesy of the Graduate Management Admissions Council. Usage of this text does not imply endorsement by GMAC.
* 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.