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, we’re 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 haven’t said so explicitly.
Here’s another example, from page 35 of our Word Problems book:
If Kelly received 1/3 more votes than Mike in a student election
If we say that 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. (That is, Mike’s votes plus Kelly’s votes = total votes.)
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, let’s say I ask you:
The ratio 3 to 1/2 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 3: 1/2, and then they ask what that’s equal to, so I know to use an equals sign:
3: ½ = ?:?
Bingo — translated! But they told us the word ratio, so if we’ve studied ratios, then we should know how to write them. How does this get harder? Take a look at this one:
It takes 3 times as long to clean the blender as it takes to make a milkshake.
The word ratio isn’t there, but it is actually describing a ratio! For every 3 parts spent cleaning, there is 1 part of making: the ratio is 3:1 (cleaning: making). How do we know this? Well, they give us a relationship about the time it takes to perform the two activities (cleaning and making) 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. For a problem like this, the explanation would actually use 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!).
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 this statement, Hmm, it would have been useful to know the fraction of time spent cleaning the blender — 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, cleaning and making. The ratio of the time it takes to do each action is 3:1. If I make the milkshake and then clean the blender, I have 4 parts of time; 3 of them are used to clean the blender and 1 is used to make the milkshake in the first place. Therefore, it takes 3 parts out of the whole 4, or 3/4, of the total time to clean the blender and only 1/4 of the total time to make the milkshake. If the problem also provides any of the real times (such as how much time I spent to make the milkshake), then I can also figure out the total time for both activities or the time spent to clean the blender.
Okay, that’s all for now on translation. Check back in soon — I’ll have another article for you in which we tackle a full problem!
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 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.
(2) 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.
(3) If you’re looking for an 80th percentile score or higher, 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 harder questions on the GRE, the task becomes so much harder because of the way in which these problems are written, not just because of the content being tested.