Reorient your View on Math Problems, Part 1
The Quant section of the GMAT is not a math test. Really, it isn’t! It just looks like one on the surface. In reality, they’re testing us on how we think.
As such, they write many math problems in a way that hides what’s really going on or even implies a solution method that is not the best solution method. Assume nothing and do not accept that what they give you is your best starting point!
In short, learn to reorient your view on math problems. When I look at a new problem, one of my first thoughts is, “What did they give me and how could it be made easier?” In particular, I look for things that I find annoying, as in, “Ugh, why did they give it to me in that form?” or “Ugh, I really don’t want to do that calculation.” My next question is how I can get rid of or get around that annoying part.
What do I mean? Here’s an example from the free set of questions that comes with the GMATPrep software. Try it!
* ” If ½ of the money in a certain trust fund was invested in stocks, ¼ in bonds, 1/5 in a mutual fund, and the remaining $10,000 in a government certificate, what was the total amount of the trust fund?
“(A) $100,000
“(B) $150,000
“(C) $200,000
“(D) $500,000
“(E) $2,000,000”
What did you get?
Here’s my thought process:
(1) Glance (before I start reading). It’s a PS word problem. The answers are round / whole numbers, and they’re mostly spread pretty far apart. I might be able to estimate to get the answer and I should at least be able to tell whether it’s closer to (A) or (E).
(2) Read and Jot. As I read, I jot down numbers (and label them!):
S = 1/2
B = 1/4
F = 1/5
C = 10,000
(3) Reflect and Organize. Let’s see. The four things should add up to the total amount. Three of those are fractions. Oh, I see—if I had four fractions, they should all add up to 1. So if I take those three and add them, and then subtract that from 1, that’ll give me the fractional amount for the C. Since I know the real value for C, I can then figure out the total.
But, ugh, adding fractions is annoying! You need common denominators. I’m capable of doing this, of course, but I really don’t want to! Isn’t there an easier way?
In this case, yes! Adding decimals or percents is really easy. Adding fractions is annoying. Plus, check it out, the fractions given are all common ones that we (should) have memorized. So change those fractions to percents (or decimals)!
(4) Work. Let’s do it!
S = 1/2 = 50%
B = 1/4 = 25%
F = 1/5 = 20%
C = 10,000
Wow, this is a lot easier. I know that 50 + 25 + 25 would equal 100, but I’ve only got 50 + 25 + 20, so the total is 5 short of 100. The final value, C, then must be 5% of the total.
So let’s see… if C = 10,000 = 5%, then 10% would be twice as much, or 20,000. And I just need to add a zero to get to 100%, or 200,000. Done! Read more
GMAT Challenge Problem Showdown: October 21, 2013
We invite you to test your GMAT knowledge for a chance to win! Each week, we will post a new Challenge Problem for you to attempt. If you submit the correct answer, you will be entered into that week’s drawing for a free Manhattan GMAT Prep item. Tell your friends to get out their scrap paper and start solving!
Here is this week’s problem:
A sheet of paper ABDE is a 12-by-18-inch rectangle, as shown in Figure 1. The sheet is then folded along the segment CF so that points A and D coincide after the paper is folded, as shown in Figure 2 (The shaded area represents a portion of the back side of the paper, not visible in Figure 1). What is the area, in square inches, of the shaded triangle shown?
GMAT Challenge Problem Showdown: October 14, 2013
We invite you to test your GMAT knowledge for a chance to win! Each week, we will post a new Challenge Problem for you to attempt. If you submit the correct answer, you will be entered into that week’s drawing for a free Manhattan GMAT Prep item. Tell your friends to get out their scrap paper and start solving!
Here is this week’s problem:
If a, c, d, x, and y are positive integers such that ay < x and
is the lowest-terms representation of the fraction
, then c is how much greater than d? (If
is an integer, let d = 1.)
(1)
is an odd integer.
(2) a = 4
GMAT Challenge Problem Showdown: October 7, 2013
We invite you to test your GMAT knowledge for a chance to win! Each week, we will post a new Challenge Problem for you to attempt. If you submit the correct answer, you will be entered into that week’s drawing for a free Manhattan GMAT Prep item. Tell your friends to get out their scrap paper and start solving!
Here is this week’s problem:
The positive number a is q percent greater than the positive number b, which is p percent less than a itself. If a is increased by p percent, and the result is then decreased by q percent to produce a positive number c, which of the following could be true?
I. c > a
II. c = a
III. c < a
GMAT Challenge Problem Showdown: September 30, 2013
We invite you to test your GMAT knowledge for a chance to win! Each week, we will post a new Challenge Problem for you to attempt. If you submit the correct answer, you will be entered into that week’s drawing for a free Manhattan GMAT Prep item. Tell your friends to get out their scrap paper and start solving!
Here is this week’s problem:
For how many different pairs of positive integers (a, b) can the fraction
be written as the sum
?
Story Problems: Make Them Real (Part 2)
Last time, we talked about how to make story problems real; if you haven’t read that article yet, go take a look before you continue with this one.
I’ve got another one for you that’s in that same vein: the math topic is different, but the “story” idea still hold in general. This one has something extra though: you need to know how a certain math topic (standard deviation) works in general. Otherwise, you won’t be able to think your way through the problem.
Try this GMATPrep® problem:
* ” During an experiment, some water was removed from each of 6 water tanks. If the standard deviation of the volumes of water in the tanks at the beginning of the experiment was 10 gallons, what was the standard deviation of the volumes of water in the tanks at the end of the experiment?
“(1) For each tank, 30 percent of the volume of water that was in the tank at the beginning of the experiment was removed during the experiment.
“(2) The average (arithmetic mean) volume of water in the tanks at the end of the experiment was 63 gallons.”
Standard deviation! Ugh. : )
Okay, it’s no accident that they’re using a DS-format problem for this one. It’s not possible to calculate a standard deviation in 2 minutes without a calculator (unless, perhaps, that standard deviation is zero!). They never expect us to calculate standard deviation on this test, but they do want to know whether we understand the concept in general.
So what is standard deviation? Try to answer—aloud—in your own words before you continue reading.
(Why did I say “aloud”? Often, we tell ourselves that we can explain something, but not until we actually try do we realize that we need a refresher on the concept. Giving an explanation aloud forces you to prove that you really do know how to explain the concept. If you don’t, you’ll hear your uncertainty in your own explanation.)
Standard deviation is the measure of how spread apart a set of data points is. For example, let’s say you have the following 5 numbers in a set: {3, 3, 3, 3, 3}. The standard deviation is zero because the numbers are all exactly the same—there is no “spread” at all in the set.
Which of the following two sets has a larger standard deviation?
{1, 2, 3, 4, 5}
{1, 10, 20, 80, 2,000}
The second one! The numbers are much more spread apart than in the first set.
Right now, some of you are wondering: okay, but what’s the actual standard deviation of those two sets?
I don’t know. I could calculate it—I’m sure there are many online “standard deviation” calculators I could use. But I don’t care. The real test is never going to make me calculate this! (And that’s why I haven’t gotten into the actual calculation method here… nor will I.)
There are a few concepts that we should know, though, in terms of how changes to sets can affect the standard deviation. Read more
Story Problems: Make Them Real
I’ve been on a story problem kick lately. People have a love / hate relationship with these. On the one hand, it’s a story! It should be easier than “pure” math! We should be able to figure it out!
On the other hand, we have to figure out what they’re talking about, and then we have to translate the words into math, and then we have to come up with an approach. That’s where story problems start to go off the rails.
You know what I mean, right? Those ones where you think it’ll be fine, and then you’re about 2 minutes in and you realize that everything you’ve written down so far doesn’t make sense, but you’re sure that you can set it up, so you try again, and you get an answer but it’s not in the answer choices, and now you’re at 3.5 minutes or so… argh!
So let’s talk about how to make story problems REAL. They’re no longer going to be abstract math problems. You’re riding Train X as it approaches Train Y. You’re the store manager figuring out how many hours to give Sue so that she’ll still make the same amount of money now that her hourly wage has gone up.
Try this GMATPrep® problem:
* ” Six machines, each working at the same constant rate, together can complete a certain job in 12 days. How many additional machines, each working at the same constant rate, will be needed to complete the job in 8 days?
“(A) 2
“(B) 3
“(C) 4
“(D) 6
“(E) 8”
Yuck. A work problem.
Except… here’s the cool thing. The vast majority of rate and work problems have awesome shortcuts. This is so true that, nowadays, if I look at a rate or work problem and the only solution idea I have is that old, annoying RTD (or RTW) chart… I’m probably going to skip the problem entirely. It’s not worth my time or mental energy.
This problem is no exception—in fact, this one is an amazing example of a complicated problem with a 20-second solution. Seriously—20 seconds!
You own a factory now (lucky you!). Your factory has 6 machines in it. At the beginning of the first day, you turn on all 6 machines and they start pumping out their widgets. After 12 continuous days of this, the machines have produced all of the widgets you need, so you turn them off again.
Let’s say that, on day 1, you turned them all on, but then you turned them off at the end of that day. What proportion of the job did your machines finish that day? They did 1/12 of the job.
Now, here’s a key turning point. Most people will then try to figure out how much work one machine does on one day. (Many people will even make the mistake of thinking that one machine does 1/12 of the job in one day.) But don’t go in that direction in the first place! If you were really the factory owner, you wouldn’t start writing equations at this point. You’d figure out what you need by testing some scenarios. Read more
GMAT Challenge Problem Showdown: September 16, 2013
We invite you to test your GMAT knowledge for a chance to win! Each week, we will post a new Challenge Problem for you to attempt. If you submit the correct answer, you will be entered into that week’s drawing for a free Manhattan GMAT Prep item. Tell your friends to get out their scrap paper and start solving!
Here is this week’s problem:
If a and b are different nonzero integers, what is the value of b ?
(1) ab = ab
(2) ab – ab – 1 = 2
GMAT Challenge Problem Showdown: September 9, 2013
We invite you to test your GMAT knowledge for a chance to win! Each week, we will post a new Challenge Problem for you to attempt. If you submit the correct answer, you will be entered into that week’s drawing for a free Manhattan GMAT Prep item. Tell your friends to get out their scrap paper and start solving!
Here is this week’s problem:
A semicircular piece of paper has center O, as shown above. Its diameter A’A is coated with adhesive. If the adhesive is used to fuse radii OA’ and OA along their entire lengths (so that points A and A’ coincide, points P and P’ coincide, and so on), a cone is formed as shown above. If point B divides the original semicircle into two identical arcs, what is the measure of angle AOB in the folded cone?
Ratios: Box ‘Em Up (Or Just Pour A Drink)
On the GMAT, you may see a 3 to 5 ratio expressed in a variety of ways:
3:5
3 to 5
x/y = 3/5
5x = 3y (Yes, that’s the same as the other 3. Think about it.)
In the real world, we encounter ratios in drink recipes more often than anywhere else (3 parts vodka, 5 parts cranberry), perhaps explaining why–after drinks that strong–we forget how to handle them.
Keep in mind: ratios express a “part to part” relationship, whereas fractions and percentages express a “part to whole” relationship. So the fraction of the above drink is 3/8 vodka (or 37.5% of the whole). Either way, hold off on mixing that drink until after this post.
I like to set up ratios using a “ratio box.” The box is a variant on the “Unknown Multiplier” technique from page 65 of our FDPs book, but it’s a nice way to visually manage ratios without resorting to algebra.
Let’s take the beginning of a typical ratio question:
“The ratio of men to women in a class is 3:2…”
Instead of doing anything fancy with variables, I just set up a tracking chart:
Men | Women | Total | |
Ratio | 3 | 2 | 5 |
From this point alone, I have sufficient information to answer a bunch of questions.
-What fraction of the students are men? (3/5)
-What percent of the students are women? (40%)
-What is the probability of choosing a man? (3/5)
-etc.
However, I have nowhere near enough information to answer anything about the REAL numbers of students in this class. Suppose the GMAT were to add a little more information:
“The ratio of men to women in a class is 3:2. If there are 35 students in the class…”
Now we can calculate almost everything about the real numbers of people. First, make a bigger box with 3 lines. The unfilled box looks like this: