Posted by Clear Admit on May 19, 2010, at 8:00 pm
Posted in:  GMAT - Quantitative , GMAT Practice Problem , GMAT Tips Today’s GMAT challenge question comes from our friends at ManhattanGMAT. To help you with your GMAT studying, try to solve the problem on your own, and then read on for the explanation of its solution:
Problem
For which of the following functions f(x) is the relation f(f(x)) = f(f(f(f(x)))) NOT true for at least some values of x not equal to zero?
(A) f(x) = -|x|
(B) f(x) = 2 – x
(C) f(x) = 3x
(D) f(x) = 4/x
(E) f(x) = 5
Solution
We are asked which function does NOT obey the rule f(f(x)) = f(f(f(f(x)))). This rule looks intimidating, but all it means on the left side is that we put some number x into the function, get the output, and then put that output back into the function and see what we get out. We do the same thing (feeding the function its own output) two more times on the right side, then compare the two sides.
Now let’s look at the functions.
(A) f(x) = -|x|
This function takes the absolute value of x, then puts a negative sign on. For instance, if x = 7, then f(x) = -|7| = -7. Likewise, if x = -8, then f(x) = -|-8| = -8. In words, f(x) turns any number negative (it’s the “negative absolute value.”) Applying this process twice gives you the same number as applying it 4 times. INCORRECT.
(B) f(x) = 2 – x
Let’s see what happens when we try to calculate f(f(x)). Work your way from the inside out:
f(f(x)) = f(2 – x) = 2 – (2 – x) = 2 – 2 + x = x. In other words, f(f(x)) just gives us x back. Therefore, applying two MORE f‘s to get f(f(f(f(x)))) will give us x again as well. INCORRECT.
(C) f(x) = 3x
If f(x) = 3x, then f(f(x)) = f(3x) = 3(3x) = 9x. Applying two MORE f‘s to get f(f(f(f(x)))) will give us 3(3(9x) = 81x. 9x does NOT equal 81x for any nonzero x, in fact. CORRECT.
(D) f(x) = 4/x
We should finish out the list, just to make sure.
f(f(x)) = f(4/x) = 4/(4/x) = x. As with the function in (B), this function brings us back to x if we apply it twice. Thus, if we apply it 4 times, we also get back to x. INCORRECT.
(E) f(x) = 5
f(f(x)) = f(5) = 5. This function may seem tricky, but it’s actually very simple: it gives you back a 5 no matter what you feed into it. If you give it a 5, in particular, you still get a 5 back, no matter how many times you go through that cycle. INCORRECT.
The correct answer is C.
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Posted by Clear Admit on May 12, 2010, at 8:00 pm
Posted in: GMAT - Quantitative , GMAT Practice Problem , GMAT Tips Today’s GMAT challenge question comes from our friends at ManhattanGMAT. To help you with your GMAT studying, try to solve the problem on your own, and then read on for the explanation of its solution:
Problem
At the beginning of year 1, an investor puts p dollars into an investment whose value increases at a variable rate of xn% per year, where n is an integer ranging from 1 to 3 indicating the year. If 85 < xn < 110 for all n between 1 and 3, inclusive, then at the end of 3 years, the value of the investment must be between
(A) $p and $2p
(B) $2p and $5p
(C) $5p and $10p
(D) $10p and $25p
(E) $25p and $75p
Solution
The quick solution to this problem is to pick a convenient number in the allowed range of growth rates. If xn is always between 85 and 110, then the most convenient growth rate to pick is 100. An annual growth rate of 100% is exactly equivalent to doubling one’s money. Doubling one’s money three times is equivalent to multiplying one’s investment by a factor of 8 (= 23). The only range that includes $8p is the third range ($5p to $10p).
Computing the exact outer limits of the allowed range is much more cumbersome. We would have to cube 1.85 for the lower limit and 2.1 for the upper limit. The cube of 1.85 is 6.331625, and the cube of 2.1 is 9.261. These values fall between 5 and 10.
The correct answer is (C).
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Posted by Clear Admit on May 5, 2010, at 8:00 pm
Posted in: GMAT - Quantitative , GMAT Practice Problem , GMAT Tips Today’s GMAT tip comes from test prep firm ManhattanGMAT. In this article, ManhattanGMAT instructor Stacey Koprince explains how to work through a rate problem:
This week, we’re going to tackle a challenging GMATPrep® problem solving question from the topic of Rates & Work.
Let’s start with the problem. Set your timer for 2 minutes…. and… GO!
* “Circular gears P and Q start rotating at the same time at constant speeds. Gear P makes 10 revolutions per minute, and gear Q makes 40 revolutions per minute. How many seconds after the gears start rotating will gear Q have made exactly 6 more revolutions than gear P?
“(A) 6
“(B) 8
“(C) 10
“(D) 12
“(E) 15”
Given info about two different gears, P and Q, we have to figure out something about how quickly they move relative to each other. In particular, we’re supposed to figure out when this is true: (# of Gear Q revolutions) = (# of Gear P revolutions) + 6.
Quick Quiz:
Name something critical that you MUST do on a problem solving question after reading the question but before starting to write down information or solve the problem.
Brainstorm a little bit. There are multiple good answers to the quiz, but I’m looking for one in particular, so come up with a few ideas to see if you can hit the one that I want.
Here’s the answer: it’s critical to look at the answer choices before starting to solve. Why? Because the answer choices can actually help you decide the best way to solve the problem. In this problem, when I glanced at the answers, I immediately noticed that they were small, whole numbers. Great! I could actually test some of the answers to help me solve. I might start with the lowest number or I might start with one of the middle numbers – I’m not sure about that until I get a bit further into the problem.
Back to the problem. Before we dive in, a word of advice: I strongly recommend drawing out what is happening. Draw two little circles for gears, label them and write the rates in. Draw a timeline that shows, with tick marks, when each one completes a full revolution, and so on. Make rate or work problems as visual and concrete as you can.
Okay, now that you’ve done that, you also noted that the first sentence gives us the rates in revolutions per minute, while the second sentence asks us to calculate something in seconds, right? We’re going to have to do some conversions in order to solve, so let’s start there.
The answer choices are represented in seconds, so we could convert those to minutes; that would involve converting five numbers and those numbers would all be fractions. That’s kind of annoying. Alternatively, the problem contains two rates that are given in terms of revolutions per minute. Converting two is easier than converting five, so let’s convert the numbers in the problem.
Next, what do we want to convert to? Do we need to know the rate in terms of revolutions per second? Or do we want to know how many seconds it takes for each gear to rotate once?
Remember that timeline we drew earlier? We want something that we could just add to that timeline; that is, we could start drawing tick marks for Gear P and say, okay, after x seconds, it has rotated once, and then after 2x seconds, it has rotated twice, and so on. It’s often a lot easier to do rate problems this way, especially because it’s so easy to make a mistake with the algebra. So I’m going to start with figuring out how many seconds it takes for each gear to rotate once.
Gear Q’s rate is 40 revolutions per 60 seconds. 60 seconds / 40 revolutions = 6/4 = 3/2 = 1.5 seconds for one revolution. After 1.5 seconds, Gear Q has rotated once. After 3 seconds, Gear Q has rotated twice.
Gear P’s rate is 10 revolutions per minute. There are 60 seconds in a minute, so it rotates 10 times in 60 seconds. 60 seconds /10 revolutions = 6 seconds for one revolution. Okay, so after 6 seconds, Gear P has rotated once. After 12 seconds, it has rotated twice.
From here, I’m going to give you two different ways to finish this problem off. The long way is up first:
Hmm. Those two numbers, 6 and 12, are jogging my memory. Those are two of the answer choices! Earlier, I noted that it would be easy to try the answer choices in the problem because they’re small integers. Let’s try 6 and 12.
I already know that Gear P will have rotated once after 6 seconds and twice after 12. What about Gear Q? After 6 seconds, Gear Q will have rotated 6/1.5 = 4 times. Okay, so once and four times… does that fit the question? (When will Gear Q have rotated 6 more times than Gear P?) Nope. Cross off A.
After 12 seconds, Gear Q will have rotated 12/1.5 = 8 times. Gear P will have rotated twice at this point. Does this fit the question? Yes! 8 = 2+6! Gear Q has rotated 6 more times than Gear P. We didn’t even have to use algebra or write any complicated equations (and, yes, this was the long way).
Here’s the short way: if you did draw out a timeline as I suggested earlier, start placing tick marks on your timeline, every 6 seconds for Gear P and every 1.5 seconds for Gear Q. Wherever they overlap (12, in this case!) is our answer. (Note that you already know you won’t have to extend the timeline very far to find the overlap because the largest number in the answer choices is 15. If the largest number had been 145, then the longer way, above, might be a better bet.)
The correct answer is D.
Key Takeaways for Problem Solving Rate Problems:
(1) Determine that you have a rate (or work) problem: this occurs when some type of rate is discussed, possibly a rate for the movement of some object or possibly a rate for an amount of work that is completed.
(2) Before you start solving, glance at the answer choices for all problem solving problems. Does the format give you any clues about possible solution methods? If the answers consist of small, whole numbers, you can probably use them to help you solve. (If the numbers are very spread out, you can probably estimate. If there are variable expressions in the choices, you can try your own numbers. And so on!)
(3) Imagine you’re in the actual situation and draw it out. Make a number line: a timeline or a representation of distance (say, a 100-mile train track) and, if there’s more than one thing moving or doing work, put them both on that same number line. Draw tick marks and “step” through the problem: after 1 hour, this is where each train is on the 100-mile track; after two hours, this is where each one is; and so on.
(4) As always, write out all of your work and, when studying, analyze your work. Even when you answer correctly, there may still be an easier or more efficient way to do the problem!
* GMATPrep® questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.
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Posted by Clear Admit on April 3, 2010, at 9:00 am
Posted in: GMAT - Quantitative , GMAT Practice Problem , GMAT Tips Today’s GMAT challenge question comes from our friends at ManhattanGMAT. To help you with your GMAT studying, try to solve the problem on your own, and then read on for the explanation of its solution:
Problem
x is replaced by 1 – x everywhere in the expression 1/x – 1/(1 – x), with x ≠ 0 and x ≠ 1. If the result is then multiplied by x2 – x, the outcome equals
(A) x + 1
(B) x – 1
(C) 1 – x2
(D) 2x – 1
(E) 1 – 2x
Solution
We can attack this problem by doing Direct Algebra. First, carry out the replacement. That is, literally replace every x in the expression with 1 – x, putting parentheses around the 1 – x in order to preserve proper order of operations:
Original: 1/x – 1/(1 – x)
Replacement:
1/(1 – x) – 1/(1 – (1 – x))
Now simplify the second denominator: (1 – (1 – x)) = (1 – 1 + x) = x
So the replacement expression becomes this:
1/(1 – x) – 1/x
This should make sense. If we replace x by 1 – x, then it turns out that we are also replacing 1 – x by x (since 1 – (1 – x) = x). Thus, the denominators of the original expression are simply swapped.
Now we can either combine these fractions first (by finding a common denominator) or go ahead & multiply by x2 – x, as we are instructed to. Let’s take the latter approach.
[1/(1 – x) – 1/x] (x2 – x)
Instead of FOILing this product right away, we should factor the expression x2 – x first. If we do so, we will be able to cancel denominators quickly.
x2 – x factors into (x – 1)x. We can now rewrite the product:
[1/(1 – x) – 1/x] (x – 1)x
= (x – 1)x/(1 – x) – (x – 1)x/x
The second term, (x – 1)x/x, becomes just x – 1 after we cancel the x’s.
Since (x – 1) = –(1 – x), we can rewrite the first term as –(1 – x)x/(1 – x) and then cancel the (1 – x)’s, leaving –x.
So, the final result is
–x – (x – 1) = –x – x + 1 = 1 – 2x
This is the answer.
Separately, since this is a Variables In Choices problem, we could instead pick a number and calculate a target. Since 0 and 1 are disallowed, let’s pick x = 2. We are told that x should be replaced by 1 – x, so we calculate 1 – x = –1 and put in –1 wherever x is in the original expression.
1/x – 1/(1 – x) = 1/(–1) – 1/(1 – (–1))
= –1 – ½
= –3/2
Now multiply this number by x2 – x = 22 – 2 = 2. We get –3 as our target number.
Finally, we plug x = 2 into the answer choices and look for –3:
(A) x + 1 = 2 + 1 = 3
(B) x – 1 = 2 – 1 = 1
(C) 1 – x2 = 1 – 22 = –3
(D) 2x – 1 = 2(2) – 1 = 3
(E) 1 – 2x = 1 – 2(2) = –3
We can eliminate choices A, B, and D, but to choose between C and E, we would need to pick another number. For instance, if we pick x = 3, we get a target of –5. Only E fits this target.
The correct answer is (E).
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Posted by Clear Admit on March 27, 2010, at 9:00 am
Posted in: GMAT - Quantitative , GMAT Practice Problem , GMAT Tips Today’s GMAT challenge question comes from our friends at ManhattanGMAT. To help you with your GMAT studying, try to solve the problem on your own, and then read on for the explanation of its solution:
Problem
The tens digit of 617 is
(A) 1
(B) 3
(C) 5
(D) 7
(E) 9
Solution
We know that there must be a pattern, since we can’t be expected to expand 617 out to all its digits. In other words, we must be able to spot a repeating cycle of digits.
The only way forward is to compute tens digits for powers of 6, starting with 61, and see what we get. To go up, multiply the previous result by 6 and drop any higher digits than the tens, but we have to keep the units digit (which, as we’ll see, will be 6 every time).
61 = 6 (no tens digit)
62 = 6 × 61 = 36 (tens digit = 3)
63 = 6 × 62 = ..16 (tens digit = 1)
64 = 6 × 63 = ..96 (tens digit = 9)
65 = 6 × 64 = ..76 (tens digit = 7)
66 = 6 × 65 = ..56 (tens digit = 5)
67 = 6 × 66 = ..36 (tens digit = 3)
Whew – the numbers finally started repeating! The cycle is 3, 1, 9, 7, 5 – which is 5 terms long. Every power will have the same tens digit as the 5th larger power, so 62, 67, 612, and most importantly 617 will all have 3 as their tens digit.
Notice that the pattern didn’t start until 62. 61 doesn’t have a tens digit (or has a tens digit of 0, but this digit is never repeated later in the cycle).
The correct answer is (B).
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Posted by Clear Admit on March 20, 2010, at 9:00 am
Posted in: GMAT - Quantitative , GMAT Practice Problem , GMAT Tips Today, our friends at Manhattan Review share their advice on tackling permutation problems on the GMAT:
Permutation problems are very common on the Quantitative section of the GMAT. In order to do well, it is very important to understand how to solve these types of problems efficiently.
Luckily, there are two very easy methods which you can use to understand and quickly find a solution to these problems.
Let’s look at an example problem:
4 Couples wish to stand in a row for a group photo. How many arrangements of the 8 people are possible if each person must stand next to his or her partner.
(a) 324 (b) 352) (C) 384 (D) 426 (E) 512
It’s easiest to think of the four groups of couples as
C1 C2 C3 C4
Now how many different ways can we arrange these couples?
Since these couples are a factorial of 4 we simply multiply them out. 4 x 3 x 2 x 1.
And we are left with 24 permutations of the couples.
But we are not done yet! We still have to arrange the couples and each couple has the option of deciding who will stand on the right side. Since each couple has two ways to make this decision we can understand that as:
24 x 2 x 2 x 2 x 2 = 384 possible arrangements!
You’ve just solved this problem one way. But there is another, which you may or may not prefer.
Consider that we have 8 positions.
_ _ _ _ _ _ _ _
How many people can stand in the left position if there are 8 positions? Anyone!
8 _ _ _ _ _ _ _
So there are 8 possible options for the left position
But since only that person’s partner can stand next to him or her, there is only 1 possible option for the next position.
8 1 _ _ _ _ _ _
Now there will only be 6 possible people left for the next position.
8 1 6 _ _ _ _ _
Again, this person’s partner can only stand next to him or her, so once again there is only 1 possibility for this option.
8 1 6 1 _ _ _ _
Now there are only 4 people left.
8 1 6 1 4 1 _ _
We’re down to the last two people, which gives us only 2 options for this next position…and only one person remaining for the last option!
8 1 6 1 4 1 2 1
Now to arrive at the final answer, simply multiply these out.
This will be 8x6x4x2 = 384
It is important to understand both ways of doing permutations in order to succeed at the various problems one will face on the GMAT.
Manhattan Review focuses on GMAT preparation and prides itself in its highly experienced instructors and small class sizes. It has proven student success with top GMAT test scores and acceptance by top business schools and employers.
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Posted by Clear Admit on March 17, 2010, at 8:00 pm
Posted in: GMAT - Quantitative , GMAT Practice Problem , GMAT Tips Today’s GMAT challenge question comes from our friends at ManhattanGMAT. To help you with your GMAT studying, try to solve the problem on your own, and then read on for the explanation of its solution:
Problem
x8 – y8 =
(A) (x4 – y4)2
(B) (x4 + y4)(x2 + y2)(x + y)(x – y)
(C) (x6 + y2)(x2 – y6)
(D) (x4 – y4)(x2 – y2)(x – y)(x + y)
(E) (x2 – y2)4
Solution
You need to recognize the expression as a difference of squares. Like all other even powers, x8 is a square, equal to (x4)2, so we use the template a2 – b2 = (a + b)(a – b), with a = x4 and b = y4:
x8 – y8 = (x4)2 – (y4)2 = (x4 + y4)(x4 – y4)
We continue breaking down the second part of the resulting expression, which is also a difference of squares.
x8 – y8 = (x4)2 – (y4)2 = (x4 + y4)(x2 + y2)(x2 – y2)
And we’re not done yet, because the last expression is of course also a difference of squares!
x8 – y8 = (x4)2 – (y4)2 = (x4 + y4)(x2 + y2)(x2 – y2) = (x4 + y4)(x2 + y2)(x + y)(x – y)
This final product matches the expression in choice (B), so the correct answer is (B).
Plugging numbers is probably too time-consuming in this case. Among positive integers, only 0 and 1 are easy to compute the eighth power of (unless you’ve memorized that 28 = 256). Moreover, several of the answer choices are designed to give you 0 if you choose x = y = 1.
If you did plug in x = 2 and y = 1, then you would get the following for choice (B):
256 – 1 = 255 = (16 + 1)(4 + 1)(2 + 1)(2 – 1) = (17)(5)(3).
If you happen to know already that 28 = 256, then you could get 255 as your target number relatively quickly. Since 255 ends in 5, it must be divisible by 5. No choice besides (B) gives you 5 as a factor if you plug in x = 2 and y = 1, so you wouldn’t need to compute the actual values of every choice. However, it’s still the case that the best way to do this problem is to recognize the original expression as a difference of squares, and then factor.
Again, the correct answer is (B).
If you’ve been following our Challenge Problems, you might remember “Difference of Zenzicubes” from last October. At the end of the explanation, we mentioned that a “zenzizenzizenzic” is the square of a square of a square, or ((x2)2)2 = x8. This name was coined by Robert Recorde, who also introduced something far more useful: the equals sign (=).
http://en.wikipedia.org/wiki/Zenzizenzizenzic
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Posted by Clear Admit on March 3, 2010, at 8:00 pm
Posted in: GMAT - Quantitative , GMAT Practice Problem , GMAT Tips Today’s GMAT challenge question comes from our friends at ManhattanGMAT. To help you with your GMAT studying, try to solve the problem on your own, and then read on for the explanation of its solution:
Problem
Skier Lindsey Vonn completes a straight 300-meter downhill run in t seconds and at an average speed of (x + 10) meters per second. She then rides a chairlift back up the mountain the same distance at an average speed of (x–8) meters per second. If the ride up the mountain took 135 seconds longer than her run down the mountain, what was her average speed, in meters per second, during her downhill run?
(A) 10
(B) 15
(C) 20
(D) 25
(E) 30
Solution
First, we set up two RT = D equations, one for the downhill run and one for the ride back up the mountain.
Downhill run: (x + 10)t = 300
Ride back up: (x – 8)(t + 135) = 300
Technically, we just have to do some algebra & arithmetic from here on out. However, these equations are very difficult to solve in their current state. The tipoff for you is that the variable x does not represent, on its own, either the downhill or the uphill speed. Thus, the equations wind up being thorny (although still solvable).
However, we can reduce the complexity by creating a new variable, say r, that represents the speed on the ride back up. In other words, r = x – 8. We can rewrite this equation as r + 8 = x, and thus the downhill speed, x + 10, can be re-expressed as r + 18. As you’ll see, this simplifies the algebra. In this sort of situation, when a variable such as x does not represent any real speed in the scenario, our instinct should be to replace x with another variable that does represent a real speed.
Downhill run: (r + 18)t = 300
Ride back up: r(t + 135) = 300
Now we can set the expressions on the left side equal to each other, since they both equal 300:
(r + 18)t = r(t + 135)
rt + 18t = rt + 135r
18t = 135r
2t = 15r
t = (15/2)r
Finally, we substitute back into either equation (we’ll just pick the first). Since the numbers get large and we can see we’re going to get a quadratic, we might want to leave certain numbers factored as we go.
(r + 18)(15/2)r = 300
(15/2)r2 + 135r = 300
(15/2)r2 + 135r – 300 = 0
Now divide by 15 and multiply throughout by 2.
r2 + 18r – 40 = 0
(r + 20)(r – 2) = 0
Since r must be positive (it represents a speed), r must be 2. Thus, Lindsey’s downhill speed, in meters per second, is r + 18 = 20.
The correct answer is (C).
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Posted by Clear Admit on February 6, 2010, at 9:00 am
Posted in: GMAT - Verbal , GMAT Practice Problem , GMAT Tips Today’s GMAT tip comes from our friends at test prep firm ManhattanGMAT. In this article, ManhattanGMAT instructor Stacey Koprince offers advice on Critical Reasoning problems:
This week, we’re going to discuss how to analyze and master a particular GMATPrep® Critical Reasoning problem.
First, set your timer for 2 minutes and try this GMATPrep® problem:
“Because no employee wants to be associated with bad news in the eyes of a superior, information about serious problems at lower levels is progressively softened and distorted as it goes up each step in the management hierarchy. The chief executive is, therefore, less well informed about problems at lower levels than are his or her subordinates at those levels.
“The conclusion drawn above is based on the assumption that
“(A) problems should be solved at the level in the management hierarchy at which they occur
“(B) employees should be rewarded for accurately reporting problems to their superiors
“(C) problem-solving ability is more important at higher levels than it is at lower levels of the management hierarchy
“(D) chief executives obtain information about problems at lower levels from no source other than their subordinates
“(E) some employees are more concerned about truth than about the way they are perceived by their superiors”
After trying the problem, checking the answer, and reading and understanding the solution (if available), I try to answer these questions:
1. Did I know WHAT they were trying to test?
- Was I able to CATEGORIZE this question by topic and subtopic? By process / technique? If I had to look something up in my books, would I know exactly where to go?
→The question is a “Find the Assumption” CR question because the word “assumption” appears in the question stem. If I don’t remember how to do “Find the Assumption” questions, I’d go look in my book right now. If I had previously studied wrong answer choice types for Assumption questions, I’d also note which wrong answer choice types (if any) I recognized in this problem.
- Did I COMPREHEND the symbols, text, questions, statements, and answer choices? Can I comprehend it all now, when I have lots of time to think about it? What do I need to do to make sure that I do comprehend everything here? How am I going to remember whatever I’ve just learned for future?
→The first sentence of the argument is a premise – it’s designed to support what the author wants to claim. The second sentence is the conclusion – what the author is actually claiming. The author didn’t give any opposing information in this argument.
- Did I understand the actual CONTENT (facts, knowledge) being tested?
→CR questions don’t test particular facts, but they do test my knowledge of what I’m supposed to do on this type of CR question. On “Find the Assumption” questions, I need to find the answer that the author believes to be true, and that answer must also be something that is necessary in order for the author to draw his conclusion. If the author doesn’t have to believe the answer, then that answer isn’t necessary in order for the author to draw the conclusion.
2. How well did I HANDLE what they were trying to test?
- Did I choose the best APPROACH? Or is there a better way to do the problem? (There’s almost always a better way!) What is that better way? How am I going to remember this better approach the next time I see a similar problem?
→I forgot to read the question first, as I should have done – I read the argument first and only then read the question. Reading the question first allows me to categorize the argument immediately and have a better idea of what is important as I read through the argument. In some cases, reading the question first also tells me what the conclusion is (though not in this case). I need to make a note to read the question first every time and practice till it becomes a habit. Also, I don’t think I diagrammed (took notes) in the best way that I could have (see “careless mistakes” below).
- Did I have the SKILLS to follow through? Or did I fall short on anything?
→I didn’t actually remember that the correct answer would have to be necessary in order for the author to draw his conclusion. Because of that, I think I fell into a trap. I should also spend a bit more time studying the characteristics of wrong answers (see below).
- Did I make any careless mistakes? If so, WHY did I make each mistake? What habits could I make or break to minimize the chances of repeating that careless mistake in future?
→I didn’t immediately note that the first sentence gave a cause-and-effect scenario. That messed me up later because I didn’t note that the sequence of the argument was X → Y → Z, not just “a bunch of stuff leads to Z.” The word “because” at the beginning of the argument should have been my clue that even the premise was cause and effect. First, I’m going to re-write the notes the way they should have been done, then I’m going to make a list of all of the words that I can think of that signal cause-and-effect, and then I’m going to scan through some old CRs I’ve already done to try to spot cause-effect premises. (And, of course, I will keep an eye on this issue when I do future problems.)
For verbal, the following two questions can be combined:
- Am I comfortable with OTHER STRATEGIES that would have worked, at least partially? How should I have made an educated guess?
- Do I understand every TRAP & TRICK that the writer built into the question, including wrong answers?
→Answer A is tempting to choose because it seems like a pretty good assumption to make in the real world; Answer A is wrong, though, because how the problems “should” be solved doesn’t tell me anything about how well-informed the chief executive is about those problems.
→Answer B is tempting to choose because it sounds like a good way to resolve the problem described in the argument. Answer B is wrong, though, because we weren’t asked to resolve the problem; we were asked to articulate a belief (an assumption) of the author who is pointing out the problem.
→Answer C is tempting to choose because it sounds like a pretty good assumption to make in the real world. Answer C is wrong, though, because the ability to solve a problem still doesn’t tell me anything about how well-informed the chief executive is about those problems.
→Answer choice D is tempting to eliminate because it sounds like a pretty bad assumption to make in the real world; it’s probably not true that a CEO only gets info from subordinates. Answer D is right, though, because this is exactly the (bad!) assumption that the author makes to draw his conclusion. If CEOs really can’t get info from anyone other than their subordinates, and if those subordinates don’t want to tell them any bad news, then those CEOs are not going to be well-informed about problems.
→Answer choice E is tempting to choose because it is undoubtedly true in the real world –some people will tell their bosses the complete truth about problems. Answer choice E is wrong, though, because it weakens the argument: if some subordinates are speaking up, then the bosses aren’t less well-informed. We were asked to find an assumption, and an assumption is something the author must believe to be true in order to draw that conclusion. If the answer choice actually weakens the conclusion, then that answer can’t be a valid assumption (and now I know that’s true for all future Assumption questions!).
3. How well did I or could I RECOGNIZE what was going on?
- Did I make a CONNECTION to previous experience? If so, what problem(s) did this remind me of and what, precisely, was similar? Or did I have to do it all from scratch? If so, see the next bullet.
→Yes. I recognized that this was an assumption question because I’d studied how assumption questions are typically worded. I should have recognized more though (see below).
- Can I make any CONNECTIONS now, while I’m analyzing the problem? What have I done in the past that is similar to this one? How are they similar? How could that recognition have helped me to do this problem more efficiently or effectively? (This may involve looking up some past problem and making comparisons between the two!)
→I could have done better with recognizing the X → Y → Z setup more quickly so that I could have taken more clear notes. I also fell into a “sounds good in the real world” trap that caused me to pick the wrong answer, as well as a “sounds bad in the real world” trap that caused me to eliminate the right answer. In the future, I will know that how it sounds in the real world is not a good reason to pick or eliminate an answer.
- HOW will I recognize similar problems in the future? What can I do now to maximize the chances that I will remember and be able to use lessons learned from this problem the next time I see a new problem that tests something similar?
→I need to do everything I already described in my notes above. I’m also going to go back and look through some old Assumption problems that I’ve already done. I’ll identify why each answer is tempting (to choose or eliminate, as appropriate) and why it’s actually right or wrong, looking to see if I can recognize the kinds of traps I identified on this problem (especially the “sounds wrong in the real world” right answer!).
And that’s it! Note that, of course, the details above are specific to each individual person – such a write-up would be different for every single one of you, depending upon your particular strengths, weaknesses, and mistakes. Hopefully, though, this gives you a better idea of the way to analyze a problem. This framework also gives you a valuable way to discuss problems with fellow online students or in study groups – this is the kind of discussion that really helps to maximize scores.
* GMATPrep® question courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.
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Today’s GMAT challenge question comes from our friends at ManhattanGMAT. To help you with your GMAT studying, try to solve the problem on your own, and then read on for the explanation of its solution:
Problem
Two different primes may be said to “rhyme” around an integer if they are the same distance from the integer on the number line. For instance, 3 and 7 rhyme around 5. What integer between 1 and 20, inclusive, has the greatest number of distinct rhyming primes around it?
(A) 12
(B) 15
(C) 17
(D) 18
(E) 20
Solution
First, make sure that you understand the new concept that the problem presents: “rhyming primes,” which are the same distance on the number line from a central number. You are given an example: 3 and 7 rhyme around 5, since both are 2 units away from 5 on the number line. Don’t let the new terminology confuse you. Instead, try to rephrase the concept into something you’re more familiar with. Ideally, you recognize that “rhyming” is just another way to say “average (arithmetic mean)” – saying “3 and 7 rhyme around 5” is the same thing as saying “the average of 3 and 7 is 5.” So, rhyming primes rhyme around their average. Alternatively, we can say that the sum of two rhyming primes (e.g., 3 and 7) is twice the central number (2×5 = 10). Sums are quick operations, so it might be good to rephrase our question in terms of taking sums of two primes.
We are asked which integer between 1 and 20, inclusive, has the greatest number of rhyming primes around it. So we should list out the primes up to 40, since the larger number in any pair of rhyming primes that average to 20 would have to be below 40 (primes are restricted to positive integers).
Here are the primes less than 40:
2
3
5
7
11
13
17
19
23
29
31
37
Rephrasing the question in terms of sums, we can ask: what number between 1 and 20, when multiplied by 2, can be expressed as a sum of two different primes from this list in the greatest number of different ways?
We should now start from the answer choices, rather than test all 20 theoretical possibilities. Unfortunately, there is no shortcut; you actually have to check the possibilities. Primes are unevenly distributed, so there’s no way to intuit the answer.
We should start by checking the highest number, because we will probably be able to construct more valid pairs around larger numbers than around smaller numbers. Construct the pairs by inspecting your list of primes. Since you know the smaller primes better than larger primes, and since the larger primes are more spread out, put the larger prime first in the potential sum, then look for the smaller prime in the second position.
(E) 20×2 = 40
37 + 3 = 40
29 + 11 = 40
23 + 17 = 40
20 has 3 rhyming pairs of primes, or 6 rhyming primes.
(D) 18×2 = 36
31 + 5 = 36
29 + 7 = 36
23 + 13 = 36
19 + 17 = 36
18 has 4 rhyming pairs of primes, or 8 rhyming primes. If we had to pick right now, because of time pressure, we would pick D.
(C) 17×2 = 34
31 + 3 = 34
29 + 5 = 34
23 + 11 = 34
19 + 15 doesn’t work
Also, 17 + 17 doesn’t work, because the definition of “rhyming” indicates that the primes must be different.
17 has 3 rhyming pairs of primes, or 6 rhyming primes. D is still our tentative answer.
(B) 15×2 = 30
29 + 1 doesn’t work, because 1 isn’t prime.
23 + 7 = 30
19 + 11 = 30
17 + 13 = 30
15 has 3 rhyming pairs of primes, or 6 rhyming primes. D is looking better and better.
(A) 12×2 = 24
19 + 5 = 24
17 + 7 = 24
13 + 11 = 24
12 has 3 rhyming pairs of primes, or 6 rhyming primes.
Answer choice (D), 18, is the correct answer.
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Today’s GMAT challenge question comes from our friends at ManhattanGMAT. To help you with your GMAT studying, try to solve the problem on your own, and then read on for the explanation of its solution:
Problem
The consumer price index in Zeropia in 2009 relative to the year 2000 was 1.75, meaning that for every Zeropian dollar spent on consumer goods in 2000, $1.75 on average had to be spent in 2009. In Zeropian dollars, what was the increase in the price of Brand Z running shoes from 2000 to 2009, if these shoes’ price increased precisely according to the consumer price index?
(1) The price of Brand Z running shoes was $91 in 2009.
(2) The ratio of the dollar increase in the price of Brand Z running shoes to the price of the shoes in 2009 was 3:7.
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: Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.
Solution
The consumer price index gives us a ratio between prices in 2000 and prices in 2009. We are told that “for every Zeropian dollar spent on consumer goods in 2000, $1.75 on average had to be spent in 2009.” In other words, if something cost X dollars in 2000, it cost 1.75×X dollars in 2009 (as long as the price increased exactly according to the index, which is just an average). In dollar terms, the increase in price would then be 1.75×X – X = 0.75×X dollars.
We are asked for this dollar price increase for Brand Z running shoes. Representing the price of these shoes in 2000 as X, as we already have, we can rephrase the question as “What is 0.75×X?” We can further rephrase this question to “What is X?”
(1) SUFFICIENT. We are told that the price of the shoes in 2009 is $91. We have represented the 2009 price as 1.75×X dollars, staying consistent with our variable naming throughout the problem (never change variable designations midstream unless you’re starting over completely). So we can write an equation:
1.75×X = 91
We know we can solve for X, so we can answer the question. (Incidentally, if we had to solve for this X on a Problem-Solving problem, one fast way would be to convert 1.75 to a fraction. 1.75 = 7/4, so we can quickly write that X = 91×4/7. Since 91/7 = 13, we get X = 13×4 = 52.)
(2) INSUFFICIENT. We are told that the price increase in dollar terms, divided by the price of the shoes in 2009, is 3/7. However, this information is already completely implied by the stem. If the index is 1.75, then any good’s price increase was 75%, or 75 cents for every 2000 dollar. Since the 2009 price is $1.75 for every 2000 dollar, the ratio of the price increase ($0.75) to the 2009 price ($1.75) will always be 0.75/1.75, or 3/7. This holds true no matter what the original 2000 price is, so we cannot determine X through this bit of redundant information.
The correct answer is (A): Statement 1 by itself is sufficient to answer the question, but Statement 2 is not sufficient.
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Today’s GMAT challenge question comes from our friends at ManhattanGMAT. To help you with your GMAT studying, try to solve the problem on your own, and then read on for the explanation of its solution:
Problem
How many factors does x have, if x is a positive integer?
(1) x = pn, where p is a prime number.
(2) nn = n + n, where n is a positive integer.
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: Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.
Solution
We cannot easily rephrase the question. Note that we may not need to know x in order to know how many factors it has.
Statement (1): INSUFFICIENT. Without knowing the value of n, we cannot determine the number of factors x has.
Statement (2): INSUFFICIENT. This statement by itself is unconnected to the question, because the statement involves only the variable n, whereas the question only involves the variable x.
Statements (1) and (2) TOGETHER: SUFFICIENT. First, we should analyze the second statement further, to see whether we can find a unique value of n.
Since n is a positive integer, we can test simple positive integers in an organized fashion, checking for equality of the two sides of the equation.
11 = 1 + 1? No.
22 = 2 + 2? Yes.
33 = 3 + 3? No.
44 = 4 + 4? No.
Notice that the left side of the equation is growing at a much faster rate than the right side, so the equation will not be true for any higher possible values of n. Thus, we can determine that the value of n is 2.
Now, we do not know the value of p, nor of x, but we do now know that x = p2, with p as a prime number. Since a prime number has no factors other than 1 and itself, we can see that x has no factors other than 1, p, and p2. Thus, x has exactly 3 factors, and we can answer the question definitively.
The correct answer is C: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
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Today’s GMAT challenge question comes from our friends at Test Prep New York. To help you with your GMAT studying, try to solve the problem on your own first, and then read on for the explanation of its solution:
Television sitcom writers get no opportunities to craft scripts that are truly “out of the box.” One contributing factor is the pressure from the network to follow a formula that has a proven ability to deliver high ratings. At the same time, there is pressure from advertisers to avoid edgy or controversial material that might offend the audience. These factors taken together make it impossible for television sitcom writers to create scripts that break new ground.
Which of the following is an assumption that is required to draw the conclusion above?
(A) If advertisers believed that edgy material helped sell their products, television sitcom writers would have opportunities to write what they want.
(B) The formulas that sitcoms follow did not start out as edgy or innovative.
(C) The formulas that networks prefer are not always as safe and uncontroversial as the advertisers would like.
(D) Television sitcom writers do not engage in scriptwriting outside of that required for their jobs.
(E) The formulas, which guarantee high ratings, are also the formulas most likely to be approved by advertisers.
How should you approach this problem? First, read the set-up carefully and see whether anything seems suspicious. The set-up makes very strong claims: sitcom writers get “no opportunities” to write scripts that are out of the box; it’s “impossible” for them to create groundbreaking scripts. Remember that on the GMAT, these claims have to be taken literally: if the set-up says it’s impossible for the person to do something, it’s impossible, period. That’s an extremely strong claim, so look to see whether the argument has made an airtight case that it’s impossible.
What evidence does the set-up give to convince you that it’s impossible? Well, it tells you that writers are under pressure from both networks and advertisers. Does that prove that it’s impossible for them ever to write groundbreaking scripts? Look for loopholes. You can probably find a few. Scriptwriters could write groundbreaking scripts that never get produced because the network executives insist on approving all scripts before filming – but even if the episodes are never filmed, the writers would still have written groundbreaking scripts. Or they could write groundbreaking scripts that get filmed and aired, after which the advertisers and network executives complain, and the scriptwriters lose their jobs – but they still would have written the scripts. Or maybe the scriptwriters write scripts on the weekends that have nothing to do with the writing they do for their jobs (for all we know, maybe they want to do something different in hopes of landing new jobs). You may come up with other ways that the scriptwriters could write groundbreaking scripts, despite the pressure from the networks and the advertisers.
But now let’s look at the answer choices.
(A) is irrelevant — it says that there might be circumstances under which one of the premises would not hold. That has nothing to do with filling in an assumption that would make the argument stronger.
(B) is also irrelevant. Whether or not the formulas were innovative in the beginning has nothing to do with whether scriptwriters are forced to follow formulas now. It might seem tempting because it seems to make one of the premises stronger (i.e. not only are the formulas safe and boring, but they have always been that way), but it has nothing to do with the current situation.
(C) is also irrelevant – whether or not the two sources of pressure disagree with each other has nothing to do with the claim that is made.
(D) closes one of the loopholes we identified. It says that the scriptwriters don’t write any scripts outside of those they write for their jobs. That still leaves a couple of loopholes not addressed, but this is one of the assumptions that must be made to make the argument hold together, so this is the right answer.
(E) Is more or less the reverse of (C) – it might seem tempting to say that the two sources of pressure on scriptwriters have to agree with each other, because otherwise you might think there’d be wiggle room for scriptwriters to do something innovative. So it might seem like this is a necessary assumption. But in fact, it isn’t necessary. It could be the case that the two different groups (network executives and advertisers) both have extensive lists of demands, and the scriptwriters just have to combine the two lists and write only the scripts that are judged “permissible” under both sets of criteria. So we don’t have to assume that the two groups agree on their demands to make the argument hold, so the answer is not (E).
Notice how the sample GMAT question tries to trick you:
By labeling the people in question “television sitcom writers,” the question tries to trick you into thinking of these people only in terms of their jobs. You need to think outside the box by thinking more broadly – don’t just think of them as people who do nothing but write scripts that will appear in TV sitcoms, but imagine that they are actual people who might do all kinds of things, including writing scripts that won’t help their jobs at all. Thinking outside the box that the question sets up helps you to find the flaws in the argument, and thus the correct answer.
(This question was written by a Test Prep New York/TPNY, “Content Manager”, you can see a very similar trick played by the official GMAC/GMAT writers in question #32 of the 12th edition GMAT Official Guide.)
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Today’s GMAT tip comes from ManhattanGMAT. This article is a follow-up to an earlier one on weighted average problems. This time, ManhattanGMAT instructor Stacey Koprince offers advice on a more difficult weighted average problem:
Earlier, we tackled a medium-level GMATPrep® weighted average question. This time, we’ll try a harder GMATPrep® weighted average question in order to test whether you learned the concept as well as you thought you did.
As we discussed, every weighted average problem I’ve seen (so far!) on GMATPrep is a Data Sufficiency question. This doesn’t mean that they’ll never give us a Problem Solving weighted average problem, but it does seem to be the case that the test-writers are more concerned with whether we understand how weighted averages work than with whether we can actually do the calculations. Last time, we focused on understanding how weighted averages work via writing some equations. We’ll try to apply that understanding to our harder problem this time, along with a more efficient solution method.
Let’s start with a sample problem. Set your timer for 2 minutes…. and… GO!
* “A contractor combined x tons of gravel mixture that contained 10 percent gravel G, by weight, with y tons of a mixture that contained 2 percent gravel G, by weight, to produce z tons of a mixture that was 5 percent gravel G, by weight. What is the value of x?
“(1) y = 10
“(2) z = 16
There are two kinds of gravel: “10% gravel” and “2% gravel.” These are our two “sub-groups.” When the two are combined (in some unknown – for now! – amounts), we get a 3rd kind: “5% gravel.” The number of tons of “10% gravel” (x) and the number of tons of “2% gravel” (y) will add up to the number of tons of “5% gravel” (z), or x + y = z. We need to find the number of tons of “10% gravel” used in the mixture.
The problem this time throws in a new wrinkle: we’re not just trying to calculate a ratio this time. We have to have enough info to calculate the actual amount of “10% gravel” used. In the other problem, we never had to worry about the actual number of employees. We’ll have to keep that in mind to see how things might change.
This problem never mentions the word average. That’s annoying. – how are we supposed to tell that this is a weighted average problem? Basically, the problem should talk about 2 sub-groups that are combined in some way to make a 3rd overall group, or mixture of the original 2 sub-groups. The problem will often discuss these groups in terms of percentages (as this problem does) or ratios (as the other problem did). That starts to tell us that some kind of averaging is happening.
Next, we know that we’ll either have an unweighted (“normal”) average or a weighted average, so check to see whether this is a normal average. We start with 10% gravel and 2% gravel. If we mixed exactly equal amounts of each (a normal average), what would the resulting mixture be?
(10+2)/2 = 6
Does the problem say we end up with 6% gravel? No. The amount of “10% gravel” is not equal to the amount of “2% gravel.” Therefore, this is not a “normal” average; now we know we have a weighted average problem.
Let’s go back to our test of “10%” and “2%” as an unweighted average. Instead of calculating with a formula (as we did up above and last time), let’s draw it out visually. Draw a straight, horizontal line on a piece of paper. Label the left end “2” and the right end “10” to represent our two different sub-groups of gravel:
2————————–10
An average of any two numbers will always appear between those two numbers. If the average is unweighted (50% of each is used), then the average will be exactly halfway between the two:
2————6————10
If the average is weighted, then it will NOT appear exactly halfway between. It will appear closer to one end or the other, depending upon the weighting. For example, if we have all “2% gravel” and zero “10% gravel,” what’s the average? A 100% weighting of “2% gravel” will give us an “average” of 2.
2————————–10
In this problem, they tell us that the average of the mixture is 5, so the average is closer to the “2” end of the line than to the “10” end of the line. (Imagine a tug-of-war between 2 and 10.) Now we know that there’s more “2% gravel” in the mixture than “10% gravel” because the “2” end of the line has “pulled” the average closer to its end.
2———5—————10
To find the weighting: calculate the difference between the two ends of the line. In this case, the difference is 10 – 2 = 8. Determine how far the dominant end (2, in this case) has “pulled” the average: calculate how far away the weighted average appears from the other end of the line. In this case, the weighted average 5 is 5 units away from the 10 sub-group.
The “dominant” end (2) therefore has the weighting 5/8, because it has “pulled” the weighted average 5/8 of the way towards its end of the line.
The “non-dominant” end (10) is just the remaining amount of the “adds up to 1” figure (from our article last time): 3/8. The “non-dominant” end (10) has “pulled” the weighted average only 3/8 of the way toward its end of the line.
Now we know that, of the total mixture, 5/8 of it will be “2% gravel” and 3/8 of it will be “10% gravel.”
Conceptually, we want to realize that if a problem tells us the two starting points (the ends of the line) and the weighted average (the middle number), then we know we will be able to calculate the relative weightings of the two sub-groups. (This is the exact same concept that we learned on the last problem!)
Let’s examine the statements to see whether this knowledge might be useful.
Statement 1 says “y = 10,” which tells us that there are 10 tons of the “2% gravel.” If I know I have 10 tons of the “2% gravel,” and I also know that 5/8 of all of the gravel will be this “2% gravel,” then can I calculate x, the amount of “10% gravel?” Yes! 10 tons = 5/8 of the total. Divide each side by 5: 2 tons = 1/8 of the total. Multiply by 3: 6 tons = 3/8 of the total. (I don’t actually need to do this calculation on this problem because this is Data Sufficiency, but I would if I saw this on Problem Solving.) Statement 1 is sufficient; eliminate choices B, C, and E.
Statement 2 says “z = 16,” which tells us that there are 16 tons of the “5% gravel.” I also know that z represents the sum of x and y, or 8/8. If I know that 16 is 100%, or 8/8, of the amount, then I can also calculate x: 16 tons = 8/8 of the total. Divide by 8: 2 tons= 1/8 of the total. (Is this starting to look familiar?) Multiply by 3: 6 tons = 3/8 of the total. Statement 2 is also sufficient.
The correct answer is D.
We can simplify this further (for future data sufficiency questions) by saying: if we’re told the two “ends of the line” for calculating the average, as well as the overall weighted average, then we can calculate the relative weightings, or ratio, of the sub-groups (just as we did last time). If we’re also given one of the three actual amounts, then we can calculate all of the actual amounts in the problem. In this case, if we’d realized that before examining the statements, we could have asked ourselves, “Does the statement give me one of the three real amounts?” and quickly recognized that each statement is sufficient by itself.
Key Takeaways for Data Sufficiency Weighted Average Problems:
(1) Determine that you have a weighted average problem: this occurs when an average is described (even if the word “average” is not in the problem!), but that average is not a standard 1:1 or equally weighted average.
(2) Carefully write down what you were asked to solve, then determine what you know, what you don’t know, and what you would need to know in order to solve (before you look at the statements).
(3) Check the given statements to see whether you can find a “match” (that is, a statement tells you what you had already decided you would need to know in order to solve).
* GMATPrep® questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.
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