### Archives For Problem Solving

How did it go last time with the rate problem? I’ve got another story problem for you, but this time we’re going to cover a different math area.

Just a reminder: here’s a link to the first (and long ago) article in this series: making story problems real. When the test gives you a story problem, do what you would do in the real world if your boss asked you a similar question: a back-of-the-envelope calculation to get a “close enough” answer.

If you haven’t yet read the earlier articles, go do that first. Learn how to use this method, then come back here and test your new skills on the problem below.

This is a GMATPrep® problem from the free exams. Give yourself about 2 minutes. Go!

* “Jack and Mark both received hourly wage increases of 6 percent. After the wage increases, Jack’s hourly wage was how many dollars per hour more than Mark’s?

“(1) Before the wage increases, Jack’s hourly wage was \$5.00 per hour more than Mark’s.

“(2) Before the wage increases, the ratio of Jack’s hourly wage to Mark’s hourly wage was 4 to 3.”

Data sufficiency! On the one hand, awesome: we don’t have to do all the math. On the other hand, be careful: DS can get quite tricky.

Okay, you and your (colleague, friend, sister…pick a real person!) work together and you both just got hourly wage increases of 6%. (You’re Jack and your friend is Mark.) Now, the two of you are trying to figure out how much more you make.

Hmm. If you both made the same amount before, then a 6% increase would keep you both at the same level, so you’d make \$0 more. If you made \$100 an hour before, then you’d make \$106 now, and if your colleague (I’m going to use my co-worker Whit) made \$90 an hour before, then she’d be making…er, that calculation is annoying.

Actually, 6% is pretty annoying to calculate in general. Is there any way around that?

There are two broad ways; see whether you can figure either one out before you keep reading.

First, you could make sure to choose “easy” numbers. For example, if you choose \$100 for your wage and half of that, \$50 an hour, for Whit’s wage, the calculations become fairly easy. After you calculate the increase for you based on the easier number of \$100, you know that her increase is half of yours.

Oh, wait…read statement (1). That approach isn’t going to work, since this choice limits what you can choose, and that’s going to make calculating 6% annoying.

Second, you may be able to substitute in a different percentage. Depending on the details of the problem, the specific percentage may not matter, as long as both hourly wages are increased by the same percentage.

Does that apply in this case? First, the problem asks for a relative amount: the difference in the two wages. It’s not always necessary to know the exact numbers in order to figure out a difference.

Second, the two statements continue down this path: they give relative values but not absolute values. (Yes, \$5 is a real value, but it represents the difference in wages, not the actual level of wages.) As a result, you can use any percentage you want. How about 50%? That’s much easier to calculate.

Okay, back to the problem. The wages increase by 50%. They want to know the difference between your rate and Whit’s rate: Y – W = ?

“(1) Before the wage increases, Jack’s hourly wage was \$5.00 per hour more than Mark’s.”

Okay, test some real numbers.

Case #1: If your wage was \$10, then your new wage would be \$10 + \$5 = \$15. In this case, Whit’s original wage had to have been \$10 – \$5 = \$5 and so her new wage would be \$5 + \$2.50 = \$7.50. The difference between the two new wages is \$7.50.

Case #2: If your wage was \$25, then your new wage would be \$25 + \$12.50 = \$37.50. Whit’s original wage had to have been \$25 – \$5 = \$20, so her new wage would be \$20 + \$10 = \$30. The difference between the two new wages is…\$7.50!

Wait, seriously? I was expecting the answer to be different. How can they be the same?

At this point, you have two choices: you can try one more set of numbers to see what you get or you can try to figure out whether there really is some rule that would make the difference always \$7.50 no matter what.

If you try a third case, you will discover that the difference is once again \$7.50. It turns out that this statement is sufficient to answer the question. Can you articulate why it must always work?

The question asks for the difference between their new hourly wages. The statement gives you the difference between their old hourly wages. If you increase the two wages by the same percentage, then you are also increasing the difference between the two wages by that exact same percentage. Since the original difference was \$5, the new difference is going to be 50% greater: \$5 + \$2.50 = \$7.50.

(Note: this would work exactly the same way if you used the original 6% given in the problem. It would just be a little more annoying to do the math, that’s all.)

Okay, statement (1) is sufficient. Cross off answers BCE and check out statement (2):

“(2) Before the wage increases, the ratio of Jack’s hourly wage to Mark’s hourly wage was 4 to 3.”

Hmm. A ratio. Maybe this one will work, too, since it also gives us something about the difference? Test a couple of cases to see. (You can still use 50% here instead of 6% in order to make the math easier.)

Case #1: If your initial wage was \$4, then your new wage would be \$4 + \$2 = \$6. Whit’s initial wage would have been \$3, so her new wage would be \$3 + \$1.5 = \$4.50. The difference between the new wages is \$1.5.

Case #2: If your initial wage was \$8, then your new wage would be \$8 + \$4 = \$12. Whit’s initial wage would have been \$6, so her new wage would be \$6 + \$3 = \$9. The difference is now \$3!

Statement (2) is not sufficient. The correct answer is (A).

Now, look back over the work for both statements. Are there any takeaways that could get you there faster, without having to test so many cases?

In general, if you have this set-up:

- The starting numbers both increase or decrease by the same percentage, AND

- you know the numerical difference between those two starting numbers

? Then you know that the difference will change by that same percentage. If the numbers go up by 5% each, then the difference also goes up by 5%. If you’re only asked for the difference, that number can be calculated.

If, on the other hand, the starting difference can change, then the new difference will also change. Notice that in the cases for the second statement, the difference between the old wages went from \$1 in the first case to \$2 in the second. If that difference is not one consistent number, then the new difference also won’t be one consistent number.

## Key Takeaways: Make Stories Real

(1) Put yourself in the problem. Plug in some real numbers and test it out. Data Sufficiency problems that don’t offer real numbers for some key part of the problem are great candidates for this technique.

(2) In the problem above, the key to knowing you could test cases was the fact that they kept talking about the hourly wages but they never provided real numbers for those hourly wages. The only real number they provided represented a relative difference between the two numbers; that relative difference, however, didn’t establish what the actual wages were.

* GMATPrep® questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.

In the past, we’ve talked about making story problems real. In other words, when the test gives you a story problem, don’t start making tables and writing equations and figuring out the algebraic solution. Rather, do what you would do in the real world if someone asked you this question: a back-of-the-envelope calculation (involving some math, sure, but not multiple equations with variables).

If you haven’t yet read the article linked in the last paragraph, go do that first. Learn how to use this method, then come back here and test your new skills on the problem below.

This is a GMATPrep® problem from the free exams. Give yourself about 2 minutes. Go!

* “Machines X and Y work at their respective constant rates. How many more hours does it take machine Y, working alone, to fill a production order of a certain size than it takes machine X, working alone?

“(1) Machines X and Y, working together, fill a production order of this size in two-thirds the time that machine X, working alone, does.

“(2) Machine Y, working alone, fills a production order of this size in twice the time that machine X, working alone, does.”

You work in a factory. Your boss just came up to you and asked you this question. What do you do?

In the real world, you’d never whip out a piece of paper and start writing equations. Instead, you’d do something like this:

I need to figure out the difference between how long it takes X alone and how long it takes Y alone.

Okay, statement (1) gives me some info. Hmm, so if machine X takes 1 hour to do the job by itself, then the two machines together would take two-thirds…let’s see, that’s 40 minutes…

Wait, that number is annoying. Let’s say machine X takes 3 hours to do the job alone, so the two machines take 2 hours to do it together.

What next? Oh, right, how long does Y take? If they can do it together in 2 hours, and X takes 3 hours to do the job by itself, then X is doing 2/3 of the job in just 2 hours. So Y has to do the other 1/3 of the job in 2 hours. Continue Reading…

Stop! Before you dive in and start calculating on a math problem, reflect for a moment. How can you set up the work to minimize the number of annoying calculations?

Try the below Percent problem from the free question set that comes with your GMATPrep® software. The problem itself isn’t super hard but the calculations can become time-consuming. If you find the problem easy, don’t dismiss it. Instead, ask yourself: how can you get to the answer with an absolute minimum of annoying calculations?

 District Number of Votes Percent of Votes for Candidate P Percent of Votes for Candidate Q 1 800 60 40 2 1,000 50 50 3 1,500 50 50 4 1,800 40 60 5 1,200 30 70

* ” The table above shows the results of a recent school board election in which the candidate with the higher total number of votes from the five districts was declared the winner. Which district had the greatest number of votes for the winner?

“(A) 1

“(B) 2

“(C) 3

“(D) 4

“(E) 5”

Ugh. We have to figure out what they’re talking about in the first place!

The first sentence of the problem describes the table. It shows 5 different districts with a number of votes, a percentage of votes for one candidate and a percentage of votes for a different candidate.

Hmm. So there were two candidates, P and Q, and the one who won the election received the most votes overall. The problem doesn’t say who that was. I could calculate that from the given data, but I’m not going to do so now! I’m only going to do that if I have to.

Let’s see. The problem then asks which district had the greatest number of votes for the winner. Ugh. I am going to have to figure out whether P or Q won. Let your annoyance guide you: is there a way to tell who won without actually calculating all the votes?

We need to know a lot of different facts, rules, formulas, and techniques for the quant portion of the test, but there are 4 math strategies that can be used over and over again, across any type of math—algebra, geometry, word problems, and so on.

Do you know what they are?

Try this GMATPrep® problem and then we’ll talk about the first of the four strategies.

* ” If mv < pv < 0, is v > 0?

“(1) m < p

“(2) m < 0”

All set?

How did you do the problem? Most quant questions have more than one possible approach and this one is no exception—but I want to use this problem to talk about a particular technique called Testing Cases.

This question is called a “theory” question: there are just variables, no real numbers, and the answer depends on some characteristic of a category of numbers, not a specific number or set of numbers. When we have these kinds of questions, we can use theory to solve—but that can get very confusing very quickly. Instead, try testing real numbers to “prove” the theory to yourself.

(Note: I chose a particularly tough question for this exercise; testing cases can also be useful and fast on easier questions!)

This problem gives one inequality:

“mv < pv < 0″

The test writers are hoping that you’ll say, “Oh, let’s just divide by v to get rid of it, so the equation is really m < p < 0.” But that’s a trap! Why?

When you divide an inequality by a negative, you have to flip the signs. But you don’t know whether v is positive or negative, so you don’t know whether to flip the signs! Never divide an inequality by a variable if you don’t know the sign of the variable.

The question itself contains a clue (two, actually!) pointing to this trap. The given inequality asks about “< 0” and the question also asks whether v > 0? Less than zero and greater than zero are code for “I’m testing you on positives and negatives.”

In the first part of this article, we talked about how GMAT quant problems are often written to imply a certain approach or solution path that is not actually the best way to do the problem. We want to reorient our view in order to pick an easier, more efficient setup or solution (if at all possible).

We finished off with a homework assignment; here’s the problem I gave you (from the free problems that come with the GMATPrep software):

* ” If , then  =

“(A) -1/2

“(B) -1/3

“(C) 1/3

“(D) 1/2

“(E) 5/2   ”

The answers are fractions but they aren’t horrible fractions. They give me a value for x / y. The question is kind of annoying though, because the form doesn’t match x / y.

Or does it? Is there any way for me to rearrange that thing to make it look more like x / y?

Yes! Check it out:

Now, how did I know to do that? I’ve actually seen another problem with the same shortcut: split the numerator into two fractions. The first time I saw that other problem, though, the way I figured it out was that whole “Well, this is annoying, why did they give it to me that way!” And so I started looking at it differently and asking myself some questions:

“They gave me a value for x / y. But the question doesn’t give me x / y. Is there any way I can make x / y? There is an x on top and a y on the bottom; what if I put those two together?

“Oh, yeah, I see! It’s totally legal to split the numerator and get two separate fractions, so that would give me x / y for one of the fractions. Does that make my life any easier, though?

“The other fraction just turns into 1! That’s fantastic! I know what I’m doing now.”

Et voilà ! I know that  , so . Plug that in and get 1 – 1.5 = -0.5.

Note that it’s easier to add and subtract in decimal (or percent) form, so if fractions can be converted easily (as 3/2 can), then consider doing the subtraction in decimal form. You already know that it will be easy to convert back into the final answer because look at the answer options—they’re all easy fractions to convert.

Quick! Glance at the answer choices for the above problem. If you did no work at all and had 1 second to make a guess, which answer would you NOT pick? Continue Reading…

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! Continue Reading…

In honor of the final season of Breaking Bad, we decided to put together our ultimate Breaking Bad GMAT quiz. Those of you who fall in the overlapping section of the “Breaking Bad Fan” “GMAT student” Venn diagram should test your skills below… yo!

## 1. Data Sufficiency

Does x+4 = Walter White?

(1) x+4 is the danger
(2) x+4 is the one who knocks

A. Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient
B. Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient
C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
D. EACH statement ALONE is sufficient
E. Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed

## 2. Discrete Quant

The front portion of Walter White’s Roof is a 7 ‘ by 15’ rectangle. If the diameter of a pizza is 22”, what is the approximate area of the shaded region of this diagram?

A. 13,600 inches sq.
B. 14,740 inches sq.
C. 15,120 inches sq.
D. 15,500 inches sq.
E. 16,640 inches sq.

## 3. Critical Reasoning

Today, Walter White will cook 100 pounds of methamphetamine.

This argument is flawed primarily because:

A. Cooking methamphetamine presents a moral dilemma for Walter White.
B. Walter White has to prioritize the needs of his wife and children and be a better father.
C. Walter has already paid for his cancer treatment and no longer needs to cook methamphetamine.
D. There is a fly in the laboratory.
E. He was told not to cook that day and is obeying his instructions.

## 4. Critical Reasoning

Hank’s collection of rocks includes over 400 different items. Hank’s rock collection is clearly the most impressive in New Mexico.

This argument is flawed primarily because:

A. Rock collections are not judged by the total number of rocks but by the rarity of each item included.
B. Rock collections are not impressive to anyone.
C. Hank’s rock collection is a metaphor and therefore cannot be judged against other rock collections.
D. Hank’s wife stole most of the rocks and it is therefore ineligible for any superlatives.
E. They aren’t rocks, they are minerals.

## 5. Discrete Quant

Walter Junior eats 3 eggs for breakfast every morning. Given that Walter Junior never misses breakfast, how many eggs does Walter Junior consume in March?

A. 60
B. 74
C. 82
D. 93
E. 107