Tricky DS Please help

[RonPurewal]

#2:
What I think you're doing here is exploiting the idea that the trapezoid could be either ABCD (in which case it's the sum) or ABDC (in which case it's not).

You've totally nailed one key aspect of GMAT problems here: namely, that "obvious" answers are incorrect. (If you just assume that the trapezoid is ABCD, you'll think it's always the sum.)

On the other hand, the set-up is too "tricky" for this test.
GMAC has written a couple of number-line problems ilke this one (though, to my knowledge, none about quadrilaterals)"”and, in every case, the problem says something like "A, B, C, and D are points on the number line, not necessarily in that order."

They really, really, REALLY go out of their way NOT to be "tricky".
This may be a reflection of the test writers' philosophy (= making an honest, what-you-see-is-what-you-get test), or it may just be pure pragmatism ("tricky" questions wouldn't have the right correlation to testers' scores, and so would almost always be thrown out of the item pool). Or both.
Whatever. The point is that they don't write "trick questions". Really, they don't.

Many of my better quant students have the most trouble with problems that are about little more than following directions, doing routine arithmetic, and/or testing an exhaustive set of cases. They are often so bent on finding "cute shortcuts" or "lightbulb moments" that the thought of just grinding the problem simply never occurs to them (or is dismissed by them immediately).

[RonPurewal]

#3:
This problem has no definitive answer, because the word "during" creates ambiguity: Rat could slow down at any point during the return trip.
If the slip happens near the beginning of the return trip, then Rat's average will be close to 3 miles/hour (= the average speed of a trip that's 6 mi/hour in one direction and 2 mi/hour in the other direction). If it doesn't happen until near the finish line, though, then Rat's average is close to 6 mi/hr.
So we don't know.

(The question itself is also ambiguous, for what I suspect are language-barrier reasons.
I think you're just asking for an approximation of the difference between the two runners' average speeds for the whole race. The literal wording of the question, though, suggests that we're tracking the average speeds as the race evolves, and are looking for the single point at which they're closest together.)

Also, 3 mi/hr and 4 mi/hr are typical walking speeds, not running speeds. (GMAC also takes pains to write word problems with realistic, unexceptional quantities.)


But, yes, overall these are very cool. I like.

[NL]

Thanks Ron for your idea and your careful analysis. My brain actually prefers this learning style. It suddenly recognizes that it has to study materials again/look at stuff differently. Otherwise, I cannot reach my goal.

Please start a new thread with these.

Oh no, those are not problems (although they take a form as problems) that can be posted on the forum. They’re just learning experience responding to the original post and the opinions.
(like your problem posted here with 2m+n. It would not have the intended meaning if it were posted separately).

I wouldn't start with a general category (as you do here). Instead, I'd start from some very specific mathematical observation"”something that seems salient enough to be the basis of a good problem.
For instance... "Traditional algebra doesn't solve problems whose solutions are restricted to whole numbers."

From that, I'd make a problem like this one:
If m and n are positive integers, what is the value of 2m + n?
(1) 5m + 7n = 48
(2) 6m + 3n = 36

You actually start from hidden rules, the rules that if test takers know, they will solve problems much quicker. (They will go the same way that the makers went through)
And you start from this point because you’re an expert. You have lots of insight and discoveries, so you focus more on strategies. Maybe one day you may want/have to start from a general idea or a situation in life.

You can try that problem if you want. You have to watch your assumptions and refrain from jumping to conclusions, but this is not a "trick question".

Yes, I might jump to B. But it’s Ron’s question, let’s be careful!

(2) 6m + 3n =36 → 2m+n =12. Sufficient.
(1) 5m + 7n =48
--> only m=4 and n=4 satisfy this statement
--> 2m+n = 12. Sufficient.
OA: D

If I were feeling less lazy today, I'd make it a word problem; the math would ultimately be the same. The only advantage of a word problem is the ability to make "positive integers" emerge organically from whatever m and n actually represent in the problem.

A simple form may look like this:

Ron goes into supermarket and buys apples and oranges. How much does he have to pay for 2 apples and 1 orange.

(1) If Ron bought 6 apples and 3 oranges, he would pay 36 dollars
(2) If Ron bought 5 apples and 7 oranges, he would pay 48 dollars

(they must be super organic fruits, so they were paid with those prices)

[NL]

When I see "pretend to be true", I interpret that as "People don't think about the exceptions".

I meant test makers create a rule for a problem, not using actual math rules. E.g. If 1+3 = 2, then 2+3 =?

If that's what you mean, then, yes, this is one of the core principles of data sufficiency.
I.e., by minimizing calculations and staying focused on "sufficient"/"insufficient", the DS format probes this whole kind of thing more deeply than the multiple-choice probelms do.

Could you elaborate your point here. It seems important.

Well... no. Almost all official problems require some "grabbing the shovel and digging". And, even when there is a cute shortcut, brute force still tends to work pretty well"”as long as you don't spend too long staring at the problem first.

This is important. At the beginning of my preparation, I liked DS very much because I didn’t have to use my poor math skills, just using common sense or visualization. But when questions get harder, using reasoning doesn’t work well. I have to use case testing, algebra, organization, and so on.

But when I learn to make these questions, another thought pops out: I wonder whether the shortest way to solve a question is the way the maker "went through" to create the question? If yes, how could I recognize it in around 2mins while the maker might use hours/months to observe "rules" and create it?

E.g: here is a question from MGMAT CAT:
Is x > 0?
"¨"¨(1) |x + 3| = 4x - 3
"¨"¨(2) |x + 1| = 2x - 1

The "hidden rule" of this question is the same as that of the question 1, statement (1) here. If I know it, it just takes less than a minute to solve. Otherwise, it will take 3 minutes to go through all cases. Unfortunately, I don’t explore numbers/concepts/questions enough to know many this kind of rule.

[NL]

As a fashion-obsessed person, I am appalled by your casual equating of "sexy" and "revealing"... but let's not digress.

I’d like to invite this King to step out of the concept of sexy clothes for human beings for a while. "The concept" of sexy clothes for problems is a different one.

[NL]

Q1:Bad a-Good a
Is a >0?

(1) |2a| = a-8
(2) (a-1)^9 > -1

This is a non-problem, because statement 1 has no solutions at all. Every DS statement has just one solution (= sufficient) or else more than one (= not sufficient).

I agree. In term of gmat, statement (1) should be re-written.

But from my unintended error, I see a challenge to understand the underground of DS theory. - A little digression here"”just curious.
In the business world or life, a problem that doesn’t have any solution is still a problem. And if there is no solution found out at the present, the result is still sufficient (to throw it out or put it into a closet). Furthermore, many problems that have 2 solutions or more are still sufficient (still good for decision making). But yes, a yes-and-no solution is not a good solution in general.

Q2:Naughty triangles
A trapezoid has two bases that are AB and CD.
What is the relationship between the area of the trapezoid and the sum of two triangles ABC and ACD?

A. The area of the trapezoid is bigger the sum of the two triangles
B. The area of the trapezoid is smaller the sum of the two triangles
C. The area of the trapezoid is equal the sum of the two triangles
D. It cannot be determined.

What I think you're doing here is exploiting the idea that the trapezoid could be either ABCD (in which case it's the sum) or ABDC (in which case it's not).

I didn’t have much thought as you thought. It was just for fun. I saw a trapezoid and suddenly thought about triangles.

On the other hand, the set-up is too "tricky" for this test.
GMAC has written a couple of number-line problems ilke this one (though, to my knowledge, none about quadrilaterals)"”and, in every case, the problem says something like "A, B, C, and D are points on the number line, not necessarily in that order."

Actually, this question is not tricky, but requires us to think about different cases. That means 2 triangles can "jump" from left to right and vice versa, just keeping AB and CD as bases of the trapezoid. That’s why it was named naughty triangles :)
The 2 triangles located in either side, the result is the same.

Q3: Ron and Rat
In a laboratory, a race is organized for Ron and Rat. They are fed the same nutrients, bathed twice a day, and not allowed to meet girlfriends or some sort of. They run the same round trip. Rat runs at a constant rate of 6 miles per hour, but during the return, he slips on a banana’s peel (that Ron secretly throws out), so slows down to an average speed of 2 miles per hour. Ron’s average speeds are more stable than Rat’s, with a constant rate of the going-trip is 4 miles per hour and 3 miles per hour in average when return.

What is the closest difference between average speeds of Ron and Rat?
A. 0
B. ½
C. 1
D. 2 ½
E. 3


I've never seen any question that was as awful as this one. And I made a big mistake here. Constant speed vs. average speed. The intended words are "average speed".
You pointed out a big lesson for me: Words in math must be used precisely.

Also, 3 mi/hr and 4 mi/hr are typical walking speeds, not running speeds. (GMAC also takes pains to write word problems with realistic, unexceptional quantities.)

You assumed that Ron and Rat are humans, but there is no fact supporting it in the question.
But, ok, let make them human beings. So, I search "the fastest man in the world". Here we go: "Usain Bolt, the fastest human footspeed on record is 27.79 mph".
Ron and Rat have the legs’ length that is a half of Usain Bolt (although the length of their backs are comparable to that of Usain’s-so not a big deal!) . So let’s say, their average speeds are around one-fourth of Usain’s.
(It would be reasonable to make their speeds close to 14, but the numbers would be very ugly to calculate if we don’t know the hidden rule. Curious: how do you solve this problem: ugly numbers for gmat problems but close to real-world data)

All the questions are re-written below.

[NL]

Re-written questions:

E.g. 1: Bad a-Good a
(Changing statement 1)

Is a >0?
(1) |a| = 3a-8
(2) (a-1)^9 > -1


E.g. 2: Naughty triangles
(changing as Ron suggested)

A trapezoid ABCD (A, B, C, and D are points on the number line, not necessarily in that order) has two bases that are AB and CD.
What is the relationship between the area of the trapezoid and the sum of areas of two triangles ABC and ACD?

A. The area of the trapezoid is bigger the sum of the two triangles
B. The area of the trapezoid is smaller the sum of the two triangles
C. The area of the trapezoid is equal the sum of the two triangles
D. It cannot be determined.


E.g.3 Ron and Rat.
(Changing data and some words, the core is kept. Distracting information was cut. Please help me with the writing. I feel awkward when using words here)

A race is organized for Ron and Rat. They run the same round trip from A to B. Rat runs from A at an average speed of 10 miles per hour, but slows down to an average speed of 4 miles per hour when return. Ron’s speeds are more stable than Rat’s, with an average rate from A to B of 8 miles per hour and 6 miles per hour in average when return.

What is the closest difference between average speeds of Ron and Rat?
A. 0
B. ½
C. 1
D. 2 ½
E. 3

Answers: 1.D; 2.C; 3.C

[RonPurewal]

If that's what you mean, then, yes, this is one of the core principles of data sufficiency.
I.e., by minimizing calculations and staying focused on "sufficient"/"insufficient", the DS format probes this whole kind of thing more deeply than the multiple-choice probelms do.

Could you elaborate your point here. It seems important.

It's not a very deep point. Really two things.

1/
On many DS problems, you don't have to all (or even most) of the work. On a few, it's actually impossible to do so.
So, in relative terms, you spend less time doing routine work, and more time setting things up and exploring.

2/
DS also introduces the concept of "too much information""”especially in terms of C-trap answers, but in other contexts too.
For instance, if a DS problem asks for x + y, but you're trying to find x and y individually, you'll almost certainly get the problem wrong"”even if you find x and y.
E.g.,
What is x + y?
(1) 2x + 2y = 11
(2) x - y = -3

If this were a multiple-choice problem, with both equations given, and you wasted the time to solve the system, you'd still be fine. You'd smack yourself in the head when you realize you can just divide the first equation by 2"”but you wouldn't get the problem wrong.
In the DS format, this approach gives C, which is the wrong answer.

"Too much information" is impossible in multiple choice. Adding that consideration is, arguably, the most important role played by DS problems.

[RonPurewal]

But when I learn to make these questions, another thought pops out: I wonder whether the shortest way to solve a question is the way the maker "went through" to create the question? If yes, how could I recognize it in around 2mins while the maker might use hours/months to observe "rules" and create it?

If you see it, you see it. If you don't, you don't.

If you are sitting on a solution method that will actually work, you should NEVER sit there and wonder whether there is "a better way". Just do what you already have in mind.

E.g: here is a question from MGMAT CAT:
Is x > 0?
"¨"¨(1) |x + 3| = 4x - 3
"¨"¨(2) |x + 1| = 2x - 1

I would just solve these the normal way; to me, there's nothing immediately salient about either of them.

[RonPurewal]

The "hidden rule" of this question is the same as that of the question 1, statement (1) here. If I know it, it just takes less than a minute to solve. Otherwise, it will take 3 minutes to go through all cases.

Three minutes to solve four linear equations?
No way.

Most people HUGELY overestimate the time needed for this sort of thing. I bet you're doing the same.

If you have something that will work, don't talk yourself out of using it.

[RonPurewal]

But from my unintended error, I see a challenge to understand the underground of DS theory. - A little digression here"”just curious.
In the business world or life, a problem that doesn’t have any solution is still a problem. And if there is no solution found out at the present, the result is still sufficient (to throw it out or put it into a closet). Furthermore, many problems that have 2 solutions or more are still sufficient (still good for decision making). But yes, a yes-and-no solution is not a good solution in general.

This discussion has no relevance to the GMAT exam, and thus does not belong here.

[RonPurewal]

Actually, this question is not tricky, but requires us to think about different cases. That means 2 triangles can "jump" from left to right and vice versa, just keeping AB and CD as bases of the trapezoid. That’s why it was named naughty triangles :)

That wasn't a personal judgment; it was an objective statement about how the test does and doesn't operate.
If the test were to contain a question that did anything like this, there would be an explicit warning of "not necessarily in that order".

[NL]

E.g: here is a question from MGMAT CAT:
Is x > 0?
"¨"¨(1) |x + 3| = 4x - 3
"¨"¨(2) |x + 1| = 2x - 1

I would just solve these the normal way; to me, there's nothing immediately salient about either of them.

Wasn’t it created from a point that if we see the same variable in both sides of an equation with absolute value at one side, it’s likely that the equation will have 1 valid solution? If I’m short of time, I just pick D and move on?

If you have something that will work, don't talk yourself out of using it.

I saw you said this a couple of times.
But my biggest weakness is that my brain is not interested in mastering some fundamental skills such as calculation, organization. It always wants to run and seeks for something new. Dummy! It doesn’t like boring tasks. I don't know how to deal with it.

That wasn't a personal judgment; it was an objective statement about how the test does and doesn't operate.
If the test were to contain a question that did anything like this, there would be an explicit warning of "not necessarily in that order".

I know. I just try to protect my points until you break them or accept them. Either way, I get benefits -- sharpening the brain :)
Changed it! (The question 2)

[RonPurewal]

Wasn’t it created from a point that if we see the same variable in both sides of an equation with absolute value at one side, it’s likely that the equation will have 1 valid solution? If I’m short of time, I just pick D and move on?

* If you just shuttle the absolute-value bars to the other side of each of these equations, the resulting equations, x + 3 = |4x - 3| and x + 1 = |2x - 1|, have two solutions each.

* The answer to the question at the end of this paragraph is "You must be overestimating how long it will take to solve those equations."
How long will it realistically take to solve two linear equations?
Skipping the solving process is high-risk, low-return. Not the kind of investment I'd want.

Honestly, there's no point in trying to find this many shortcuts"”especially on data sufficiency, which is designed in large part to test the exceptions to "rules of thumb" rather than the rules themselves.

You have 75 minutes for the quant section. That's actually a lot of time, if you don't sit there and stare at problems when you're stuck.
If you had 30 minutes for the whole section, then there'd be much more of a point in looking for these kinds of shortcuts. But you have a lot more time than that.

[RonPurewal]

If you have something that will work, don't talk yourself out of using it.

I saw you said this a couple of times.
But my biggest weakness is that my brain is not interested in mastering some fundamental skills such as calculation, organization. It always wants to run and seeks for something new. Dummy! It doesn’t like boring tasks. I don't know how to deal with it.

Sometimes I make up games, to try to make boring or off-putting tasks more fun.

If my brain is acting like a petulant child, though, I'll just as often tell it to shut up, pick up the shovel, and dig.