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 actual problems, and then we’ll discuss how to know what to do, how to do the actual translation, and how to do everything efficiently. Note that I’m not necessarily 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 GRE 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 Manhattan Prep’s books, you’ll find this info in the Algebraic Translations chapter of the Word Problems 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 another one; try to translate it yourself before you keep reading.
Sarah has 4 times as many books as does Beth and ½ as many as Adam.
Each person’s name starts with a different letter; they’ll usually do this to make it easier for us to assign variables. We want to use variables that will remind us of the original person, or item, or whatever it might be, so that we’re less likely to make mistakes at the end of the problem (by, for example, solving for the wrong person). Let’s use s for Sarah’s books, b for Beth’s books and a for Adam’s books.
Next, take each piece of info separately:
Sarah (s) has (=) 4 times as many as (multiply) b, or s = 4b.
Sarah (s) has (=) [latex]frac{1}{2}[/latex] as many as (multiply) a, or s = [latex]frac{1}{2}[/latex]a.
Task 1: translate everything and make it real
In our Word Problems book, page 39, we have this problem:
Caleb spends $72.50 on 50 hamburgers for the marching band. If single burgers cost $1.00 each and double burgers cost $1.50 each, how many double burgers did he buy?
What should we do? First, set variables. Let s = the number of single burgers and let d = the number of double burgers. Then, pretend you’re Caleb and you’re buying the burgers. What do you do? Make it real ” actually visualize (or draw out) what needs to happen.
First, I need to order 50 burgers hmm, I can write an equation with that. The number of single burgers plus the number of double burgers needs to add up to 50: s + d = 50.
Next, I need to pay for my 50 burgers. How much do I owe? I pay a dollar for each single burger and $1.50 for each double burger I can write that as 1s + 1.5d. And the problem told me that I spent $72.50, so I can finish off that equation: 1s + 1.5d = 72.5.
So, we’re done with that ” now, we need to solve for s and d, right? Not so fast! Read the actual question first:
How many double burgers did he buy?
Hmm. They’re only asking for the value of d, not s. I don’t necessarily have to solve for s then. If I can find a way to solve directly for d, I’ll save time. (Note: this is a very common point where people lose time. From school tests, we’re used to having to solve for all of the given variables. We rarely have to do that on a standardized test like the GRE. So learn to break the habit of solving for everything ” you’re just wasting precious time!)
Task 2: Where appropriate, use a chart or table to organize
Let’s try another:
If it takes Anne 5 hours to paint 1.5 houses, and she has been working for 7 hours, how many houses has she painted?
Again, visualize ” you’re standing there (for 7 hours!) painting the house. How does it work? RTW: Rate × Time = Work. Make a chart:
| Rate | Time | Work | |
|---|---|---|---|
| Given info: | R | 5 | 1.5 |
Okay, so we have one formula: R × 5 = 1.5. The rest of the sentence says:
and she has been working for 7 hours, how many houses has she painted?
Hmm, this is a different job from the first part. I can start a new row on the chart. I’m going to keep painting at the same rate, so I can use the same variable R there.
| Rate | Time | Work | |
|---|---|---|---|
| Given info: | R | 5 | 1.5 |
| Question: | R | 7 | W |
Hey, we’ve got another formula: R × 7 = W. We can use the first equation, above, to solve for R, and, since the rate stays the same, we can then plug R into the second equation to solve for W.
One more! Let’s try this excerpt from our Word Problems book, page 30:
Jack is 13 years older than Ben. In 7 years, he will be twice as old as Ben. How old is Jack now?
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.
| Now | +7 y | |
|---|---|---|
| Jack | J | |
| Ben | 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 Ben.
Jack is 13 years older than Ben.
J = 13 + 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 Ben? According to the sentence, Jack. The equation adds the 13 to the younger person, Ben. That makes sense.
Okay, so we can either remove the J from our table and insert 13 + B instead, or we can flip the equation around (to J “ 13 = B), then remove B from the table and insert J “ 13 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 | +7 y | |
|---|---|---|
| Jack | J | |
| Ben | J – 13 |
Now fill in the remaining timeframe (you have the info to do this already “ just add 7!):
| Now | +7 y | |
|---|---|---|
| Jack | J | J + 7 |
| Ben | J – 13 | J – 6 |
What now? Oh, right ” now we have the rest of that second statement to translate:
In 7 years he will be twice as old as Ben
Okay, what timeframe do we need to use? in 7 years ” okay, go to that column. In 7 years, Jack is J + 7 and Ben is J “ 6. 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 + 7 = 2(J “ 6)
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 still recognize what to do!).
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.