GRE Math for People Who Hate Math: What Is a Variable, Really?
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Imagine a world where every conversation went like this:
Student: When is our final project due?
Professor: Three days after the first Wednesday after your rough draft is due.
Student: What?
Professor: The rough draft is due 15 days after the date 6 days before May 14.
Solving a GRE math word problem is a little bit like having this kind of conversation. That’s why word problems can be so infuriating. The problem isn’t lying to you. It’s just telling you the truth in a really annoying, backwards way. (Reading Comprehension problems do that too—it’s not just a Quant thing.)
In the conversation above, how would you work out the due date of the final project? Personally, I’d start by getting out my calendar. I’d start at May 14, then count 6 days backwards. Then, I’d count 15 days forwards, put a star on the calendar, and mark it ‘rough draft.’ Then I’d find the first Wednesday after that date, and finally, I’d count three days forward from there. That would give me my answer.
I didn’t start by writing equations. Instead, I started by gleaning as much information as I possibly could from what the professor told me. In fact, many GRE math word problems work this way. If you start by squeezing as much as you can out of the given information, you may never have to write an equation at all. Here’s an example that works that way:
Jane has a 40-ounce mixture of apple juice and seltzer that is 30% apple juice. If she pours 10 more ounces of apple juice into the mixture, what percent of the mixture will be seltzer?
(A) 33%
(B) 44%
(C) 50%
(D) 56%
(E) 67%
When you first read it, you don’t know whether you’ll need variables and equations. Start with a useful-looking fact, and see what you can get out of it. “Jane has a 40-ounce mixture of apple juice and seltzer that is 30% apple juice.” Okay, that’s just an obnoxious way of telling you how much apple juice Jane has. Plug in (0.3)(40) to your calculator, to find that Jane has 12 ounces of apple juice. That also tells you how much seltzer she has: 40 – 12 = 28 ounces.
Can you figure anything else out? Sure. Jane pours 10 more ounces of apple juice into the mixture. They should’ve just told you how much apple juice she ended up with, but this’ll do just as well. When you add 10 ounces to her current 12 ounces, she ends up with 22 ounces in total. 22 ounces of apple juice and 28 ounces of seltzer. Is that enough to answer the question? It sure is. (The answer is 28/(22+28) = 56%.)
The solution above might seem obvious. But this way of approaching problems is different from what a lot of people do instinctively. You may have learned in school that word problems are all about variables and equations. But I’m telling you that variables aren’t always necessary! Sometimes, numbers are enough. Here’s one more where you don’t need any variables at all (the answer is at the end of the article*):
The number that is 50% greater than 60 is what percent less than the number that is 20% less than 150?
(A) 5%
(B) 10%
(C) 15%
(D) 20%
(E) 25%
However, variables sometimes are necessary. Here’s a slightly different conversation:
Alex: How old is your daughter?
Beryl: She’s six years younger than twice my son’s age.
Alex: What? How old is your son?
Beryl: He’s four years younger than my daughter.
Unfortunately, none of Beryl’s statements is very useful. If she had said Three years ago, my son was seven years old, that might have helped. But here, she’s defined her daughter’s age based on her son’s age, and then turned around and defined her son’s age based on her daughter’s age! GRE math does this frequently. You’ll know it’s happening if you can’t seem to calculate anything useful based on the facts you’re given. This is one scenario in which you’ll need variables. (Another scenario will be discussed in a followup post in a few weeks.)
Think of a variable as something you wish you knew. What does Alex wish he knew? Well, he asks Beryl right away what her son’s age is. That’s a sign that the son’s age would make a good variable. Same with the daughter’s age. You’d then write out two equations. I like building my equations piece by piece, instead of trying to construct them all at once. My mental process would look something like this:
daughter = 6 yrs younger than twice son
d = (twice son) – 6
d = 2s – 6
son = 4 yrs younger than daughter
s = d – 4
Solve those two equations to find that Beryl’s children are 10 and 14 years old.
Here’s a more GRE-like problem that works in the same way.
Anke has 5 fewer candies than Conrad. If Anke gives Conrad 5 candies, Conrad will then have 4 times as many candies as Anke. How many candies does Anke have?
(A) 5
(B) 10
(C) 15
(D) 20
(E) 25
The first fact relates Anke’s candy to Conrad’s candy. So does the second fact. That means that neither of them is very useful on its own: you have to know one of the two pieces of information in order to calculate the other. That’s your clue to start constructing equations.
a = c – 5
The second equation is a little more complex. Build it up piece by piece again. If Anke gives Conrad 5 candies, what will happen? Anke will have a – 5, and Conrad will have c + 5. Conrad will then have 4 times as many as Anke. That translates to this:
c + 5 = 4(a – 5)
You can combine the two equations and solve to learn that Anke has 10 candies. (Conrad has 15).
Here are the major points:
- Word problems consist of facts and numbers. The facts are always true, but the problem will sometimes present them in an intentionally confusing way.
- When you’re working out a word problem, think about having a conversation with a truthful but uncooperative person. They’re giving you all of the information, but it’s your job to figure out what they mean by it.
- A good first step is to calculate as many numbers as you can.
- However, if none of the facts seems useful, start asking yourself which numbers you wish you knew. These are your variables—use them!
In the next article in this series, we’ll go over one more reason to work with variables while solving a word problem. Until then, why not check out the Word Problems chapter in the 5lb. Book of GRE Practice Problems? ?
*The answer to the problem is (E)!
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Chelsey Cooley
is a Manhattan Prep instructor based in Seattle, Washington. Chelsey always followed her heart when it came to her education. Luckily, her heart led her straight to the perfect background for GMAT and GRE teaching: she has undergraduate degrees in mathematics and history, a master’s degree in linguistics, a 790 on the GMAT, and a perfect 170Q/170V on the GRE. Check out Chelsey’s upcoming GRE prep offerings here.
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