{"id":10292,"date":"2017-05-03T16:54:08","date_gmt":"2017-05-03T16:54:08","guid":{"rendered":"http:\/\/www.manhattanprep.com\/gre\/?p=10292"},"modified":"2019-08-30T16:38:49","modified_gmt":"2019-08-30T16:38:49","slug":"gre-math-for-people-who-hate-math-what-is-a-variable-really","status":"publish","type":"post","link":"https:\/\/www.manhattanprep.com\/gre\/blog\/gre-math-for-people-who-hate-math-what-is-a-variable-really\/","title":{"rendered":"GRE Math for People Who Hate Math: What Is a Variable, Really?"},"content":{"rendered":"

\"Manhattan<\/p>\n

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<\/i><\/b>Imagine a world where every conversation went like this:<\/span><\/p>\n

Student<\/b>: When is our final project due? <\/span><\/p>\n

Professor<\/b>: Three days after the first Wednesday after your rough draft is due. <\/span><\/p>\n

Student<\/b>: What?<\/span><\/p>\n

Professor<\/b>: The rough draft is due 15 days after the date 6 days before May 14. <\/span><\/p>\n

Solving a GRE math word problem is a little bit like having this kind of conversation. That\u2019s why word problems can be so infuriating. The problem isn\u2019t <\/span>lying <\/span><\/i>to you. It\u2019s just telling you the truth in a really annoying, backwards way. (Reading Comprehension problems do that too\u2014it\u2019s not just a Quant thing.)<\/span><\/p>\n

In the conversation above, how would you work out the due date of the final project? Personally, I\u2019d start by getting out my calendar. I\u2019d start at May 14, then count 6 days backwards. Then, I\u2019d count 15 days forwards, put a star on the calendar, and mark it \u2018rough draft.\u2019 Then I\u2019d find the first Wednesday after that date, and finally, I\u2019d count three days forward from there. That would give me my answer. <\/span><\/p>\n

I <\/span>didn\u2019t<\/span><\/i> 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\u2019s an example that works that way: <\/span><\/p>\n

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?<\/span><\/i><\/p>\n

(A) 33%
\n<\/span>(B) 44%
\n<\/span>(C) 50%
\n<\/span>(D) 56%
\n<\/span>(E) 67%<\/span><\/p>\n

When you first read it, you don\u2019t know whether you\u2019ll need variables and equations. Start with a useful-looking fact, and see what you can get out of it. \u201cJane has a 40-ounce mixture of apple juice and seltzer that is 30% apple juice.\u201d Okay, that\u2019s 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 \u2013 12 = 28 ounces. <\/span><\/p>\n

Can you figure anything else out? Sure. Jane pours 10 more ounces of apple juice into the mixture. They <\/span>should\u2019ve<\/span><\/i> just told you how much apple juice she ended up with, but this\u2019ll 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%.) <\/span><\/p>\n

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\u2019m telling you that variables aren\u2019t always necessary! Sometimes, numbers are enough. Here\u2019s one more where you don\u2019t need any variables at all (the answer is at the end of the article*): <\/span><\/p>\n

The number that is 50% greater than 60 is what percent less than the number that is 20% less than 150? <\/span><\/i><\/p>\n

(A) 5%
\n<\/span>(B) 10%
\n<\/span>(C) 15%
\n<\/span>(D) 20%
\n<\/span>(E) 25%<\/span><\/p>\n

However, variables sometimes <\/span>are<\/span><\/i> necessary. Here\u2019s a slightly different conversation:<\/span><\/p>\n

Alex:<\/b> How old is your daughter?<\/span><\/p>\n

Beryl:<\/b> She\u2019s six years younger than twice my son\u2019s age.<\/span><\/p>\n

Alex:<\/b> What? How old is your son?<\/span><\/p>\n

Beryl:<\/b> He\u2019s four years younger than my daughter. <\/span><\/p>\n

Unfortunately, none of Beryl\u2019s statements is very useful. If she had said <\/span>Three years ago, my son was seven years old<\/span><\/i>, that might have helped. But here, she\u2019s defined her daughter\u2019s age based on her son\u2019s age, and then turned around and defined her son\u2019s age based on her daughter\u2019s age! <\/span>GRE math does this frequently<\/span><\/i>. You\u2019ll know it\u2019s happening if you can\u2019t seem to calculate anything useful based on the facts you\u2019re given. This is one scenario in which you\u2019ll need variables. (Another scenario will be discussed in a followup post in a few weeks.) <\/span><\/p>\n

Think of a variable as <\/span>something you wish you knew<\/span><\/i>. What does Alex wish he knew? Well, he asks Beryl right away what her son\u2019s age is. That\u2019s a sign that the son\u2019s age would make a good variable. Same with the daughter\u2019s age. You\u2019d 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: <\/span><\/p>\n

daughter = 6 yrs younger than twice son<\/span><\/p>\n

d = (twice son) \u2013 6<\/span><\/p>\n

d = 2s \u2013 6<\/span><\/p>\n

son = 4 yrs younger than daughter<\/span><\/p>\n

s = d \u2013 4<\/span><\/p>\n

Solve those two equations to find that Beryl\u2019s children are 10 and 14 years old. <\/span><\/p>\n

Here\u2019s a more GRE-like problem that works in the same way. <\/span><\/p>\n

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?<\/span><\/i><\/p>\n

(A) 5
\n<\/span>(B) 10
\n<\/span>(C) 15
\n<\/span>(D) 20
\n<\/span>(E) 25<\/span><\/p>\n

The first fact relates Anke\u2019s candy to Conrad\u2019s 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\u2019s your clue to start constructing equations. <\/span><\/p>\n

a = c \u2013 5<\/span><\/p>\n

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 \u2013 5, and Conrad will have c + 5. Conrad will <\/span>then<\/span><\/i> have 4 times as many as Anke. That translates to this:<\/span><\/p>\n

c + 5 = 4(a \u2013 5)<\/span><\/p>\n

You can combine the two equations and solve to learn that Anke has 10 candies. (Conrad has 15). <\/span><\/p>\n

Here are the major points:<\/span><\/p>\n