For every positive EVEN integer n, the function h(n) is defined to be the product of all the even integers from 2 to n, inclusive. If p is the smallest prime factor of h(100) + 1, then p is

(A) between 2 and 10

(B) between 10 and 20

(C) between 20 and 30

(D) between 30 and 40

(E) greater than 40

So, where do you start?

It is difficult to figure out how to approach a question like this one, since it’s a mix of so many different concepts: properties of even numbers, prime numbers, factors, and functions. No one wants to see a complex question like this on the GMAT, especially when you’re struggling to keep up your pace and finish the section on time.

The best advice for tackling complicated questions like this? Don’t waste too much time thinking about the perfect way to start. **Just start writing!**

Start by writing out what you know:

n is EVEN

Function h(n) = product of even integers from 2 to n

**h(100) + 1**

p is the smallest factor of h(100) + 1

Once you take the important elements out of the word problem, the problems should become clearer. The most important value in the problem is h(100) + 1, so let’s try to find out what that is.

h(n) = product of even integers from 2 to n.

So h(100) = product of even integers from 2 to 100.

h (100) = 2 x 4 x 6 x 8 … x 100

Hmm… now what? We don’t have time to calculate the value of h(100), so let’s see what we can do with the expression. Since the question is asking about prime factors of h(100) + 1, it will probably be helpful to find the prime factors of h(100). Let’s start there.

We know that h(100) is the product of only **even** integers, so let’s factor out a 2 from each number.

h(100) = 2 x 4 x 6 x 8 …. x 100 = 2(1) x 2(2) x 2(3) x 2(4) … 2(50)

Factoring out all the 2s (and there are 50 of them!) gives us:

2^50(1 x 2 x 3 x 4 … x 50)

We just found all the factors of h(100)! The factors are 2, and every integer from 1 to 50 inclusive. But the question is asking us about h(100) + 1. Now what?

Okay, let’s think about it. This is a good place to test the function with simpler numbers. If I know 3 and 4 are factors of 12, are they also factors of 12 + 1? No. 12 + 1 is 13, which is a prime number. So essentially, if x is a factor of n, then x will not be a factor of n+1, since n+1 cannot be a multiple of x. Test it out with other values if you’re not sure.

And since the integers from 1 to 50 are all factors of h(100), none of them can be a factor of h(100) + 1! Ah ha!

If p is the smallest prime factor of h(100) + 1, then p must be greater than 50. So the answer must be **E**, greater than 40.

Whew! In this problem, like in many others, just getting started allowed us to see more clearly what we were dealing with. If we had spent all our time trying to figure out how to start, we would have been out of time before we knew it.

Takeaway: When in doubt about how to solve a GMAT math problem, start writing, and then look for the most important value to find or solve.

Start practicing thinking with your pen!

]]>*Kyle Hausmann is a Content Developer at Knewton. *

If the background sound of your dreams over the last month has been the endless blowing of vuvuzelas, then you either have a really annoying neighbor or you are caught up in World Cup fever. At Knewton, we thought we could tie some of the football fanfare to probability. (Because that’s what we do here.)

So, here is a data sufficiency example:

If the probability is 0.7 that Uruguay will win any given match, and if the two semi-final matches are Uruguay vs. the Netherlands and Germany vs. Spain, what is the probability that Germany will win the final?

- The probability that Germany beats another European team in the semi-finals or in the final is 0.6.
- If they make it to the final, the probability that the Netherlands wins the Cup is 0.4.

The question assumes that the outcome of each game is independent from the others. Since we have independent events, we can multiply the probabilities of each event together.

First, we consider Statement 1 alone. With Statement 1, we know the probability (0.6) that Germany will beat Spain, a European team, and make it into the final. And from the prompt, we know the probability that the other team in the final would be Uruguay (0.7) or the Netherlands (0.3). Thus, the probability that the final match is between Germany and Uruguay is: 0.6 Ã— 0.7 = 0.42. And the probability that the final match is between Germany and the Netherlands is: 0.6 Ã— 0.3 = 0.18. (These add up to 0.6, which makes sense, because that is the probability that Germany makes it to the final.)

Now, we have to determine the probability that Germany wins each of those to possible games — we know that the probability that Uruguay wins is 0.7, so the probability that Germany would win is 0.3. That means the probability that the Germany vs. Uruguay final game takes place AND that Germany wins is 0.42 Ã— 0.3 = 0.126. And we know the probability that Germany would beat the Netherlands, a European team, is 0.6, so the probability that the Germany vs. the Netherlands game takes place AND that Germany wins is 0.18 Ã— 0.6 = 0.108.

By adding those two probabilities together, we cover all the possible outcomes in which Germany takes home the cup. So, the probability that Germany wins the final is 0.126 + 0.108 = 0.234. Since we know this from Statement 1 alone, the statement is Sufficient.

Now we look at Statement 2 alone. It tells us the probability that the Netherlands would win in the final match, were they to make it that far, is 0.4. But this tells us nothing about Germany — maybe the probability Germany makes it to the final match is 0 (sorry, fans). In that case, they certainly are not winning the whole thing; the probability of winning the final is thus 0, too. But maybe Germany has a great chance of making it. We just do not know from the information we have, so Statement 2 is insufficient.

(This would be answer choice A.)

]]>Answer the 2 brutally hard challenges in comments below and then sign up for the Monday night webinar to get the answers.Â Are YOU up to the challenge?

Challenging the hypothesis that tuberculosis could only be transmitted when someone inhaled air that had been exhaled by an infected individual, an article published in Nature magazine in 1872 reported that when examined, monkeys that had been kept in confinement and had consequently never breathed contaminated air still displayed tubercles in their lungs and presented with the same symptoms that individuals infected with tuberculosis did, and that they did not always transmit the disease upon coming in contact with an uninfected monkey.

A) and had consequently never breathed contaminated air still displayed tubercles in their lungs and presented with the same symptoms that individuals infected with tuberculosis did, and that

B) consequently never breathed contaminated air, still displaying tubercles in their lungs and presenting with the same symptoms as individuals who had been infected with tuberculosis did, and that

C) and consequently had never breathed contaminated air, still displayed tubercles in their lungs and had presented the same symptoms as individuals infected with tuberculosis, and

D) had consequently never breathed contaminated air, but still they displayed tubercles in their lungs and presented with the same symptoms as individuals infected with tuberculosis, and that

E) and consequently they had never breathed contaminated air still displayed tubercles in their lungs and presented with the same symptoms that individuals infected with tuberculosis, and

]]>In a roomful of spies, some spies carry Argentinian passports, some carry Bolivian passports, and some carry Chilean passports. Every spy carries at least one passport, no spy carries more than one passport from a given country, and only one spy carries passports from all three countries. The number of spies who carry Bolivian passports is two more than the number of spies who carry Argentinian passports, two less than the number of spies who carry Chilean passports, twice the number of spies who carry both Bolivian and Chilean passports, and half the total number of spies in the room. The number of spies who carry both Argentinian and Bolivian passports is two more than the number of spies who carry both Argentinian and Chilean passports and two less than the number of spies who carry both Bolivian and Chilean passports. If a spy is selected at random, what is the probability that he will carry an Argentinian passport?

A) 2/9

B) 13/57

C) 6/23

D) 3/7

E) 1/2

In an earlier post, I discussed Dan Meyer’s visionary talk about the future of math education. Halfway through the talk, Meyer mentions a classic problem in which students must determine how long it takes to fill a tank with water.

A typical textbook would give students all (or most of) the necessary pieces and then ask them to construct the puzzle — that is, plug the numbers into a formula. Meyer, however, advocates doing away with all the information and simply posing the question: “How long will it take to fill the tank with water?” Students then have to figure out what they need to answer the question. This approach forces them to think patiently and creatively.

By no small coincidence, we at Knewton teach the exact same question in the Data Sufficiency portion of our GMAT course:

An empty rectangular tank has uniform depth. How long will it take to fill the tank with water?

- Water will be pumped at the rate of 480 gallons per hour (1 cubic foot = 7.5 gallons).
- The tank is 100 feet deep and 30 feet wide.

- (A) Statement (1) ALONE is sufficient, but statement (2) alone isÂ 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.

Okay, so the question isn’t *exactly* the same. Meyer’s tank is an octagonal prism; ours is a rectangular solid. Details aside, both questions require the same level of critical reasoning.

Imagine that the prompt was pared down to simply, “How long will it take to fill the tank with water?” Give it a moment’s thought. What do you need to know to answer this question? This is, at its core, what the Data Sufficiency section tests. You never have to compute the final answer on a DS question; you just have to know what it would take to find the answer.

Here, you need to know how quickly the water is flowing into the tank — that is, you need a rate. Next, you need to know how big the tank is — you need a volume. The prompt tells you that the tank is rectangular, with uniform depth, so its volume will be a product of its length, its width, and its depth. In total, then, you need four quantities to answer the question: (1) rate, (2) length, (3) width, (4) depth.

Statement 1 gives you quantity (1), the rate, but nothing more; it can’t be sufficient. Statement 2 gives you quantities (3) and (4), the width and depth, but since it doesn’t give you the length or the rate, it can’t be sufficient, either. When you put the statements together, you’ve got quantities (1), (3), and (4), but you still don’t have quantity (2), the length of the tank. Even together, the statements are not sufficient.

If you approach this question critically, you can polish it off in a matter of seconds. “I need four quantities. I see only three quantities. Answer **choice E** is correct.” If you approach it passively, though, you get lost in the numbers and are far more likely to get it wrong. Nearly 25% of students do.

To be sure, questions like this are rare. In general, standardized math tests and creative problem solving do not go well together. As we change the way we teach students by drawing them further into the conversation, we will also need to change the way we assess them. The sooner, the better.

]]>In Part I of this series, I talked about approaching wordy GMAT questions as a businessperson would–by carefully reading these questions the first time around in order to absorb all information. The following GMAT problem has inspired me to expand this approach questions to include the actual process of decision-making:

A square countertop has a square tile inlay in the center, leaving an untiled strip of uniform width around the tile. If the ratio of the tiled area to the untiled area is 25 to 39, which of the following could be the width, in inches, of the strip?

I. 1

II. 3

III. 4a. I only

b. II only

c. I and II only

d. I and III only

e. I, II, and III

In case you haven’t figured out the answer, the strip could be any width. Answer choice **E is correct**. How do we arrive at this answer?

The problem-solving version of this question is taken from p. 85 of the GMAT Quantitative Review, 1st Edition. The solution given is computational; the basic steps are:

Let [pmath]t^2[/pmath] = area of the square tile inlay

Let [pmath]s^2[/pmath] = area of the entire countertop

Then [pmath](t^2)/(s^2 – t^2) = 25/39[/pmath], so that [pmath](t^2)/(s^2) = 25/(25 + 39) = 25/64[/pmath].

Taking the square root of both sides, we see that t/s = 5/8, which means that the length of the side of the square tile inlay is 5/8 the length of the side of the entire square countertop. At this point, the solution in the QR proceeds to write an expression for the width of the strip, w, in terms of these two variables: w = (s – t)/2, and then substitute an expression for t in terms of s to obtain an expression for w in terms of one variable, s:

[pmath]t/s = 5/8[/pmath] –> [pmath]t = (5/8)s[/pmath]. Then [pmath]w = (s – (5/8)s)/2 = (3/16)s[/pmath].

Presumably, you need to be able to write w = ks for some constant k, in order to see that w can take on any positive value. This is a lot of computation, and it will surely put you beyond the scant average of 2 minutes per question that you should allow. Is there a better way?

Imagine that this problem was posed as a data sufficiency question. It could look something like this:

A square countertop has a square tile inlay in the center, leaving an untiled strip of uniform width around the tile. What is the width, in inches, of the strip?

1. The ratio of the tiled area to the untiled area is 25 to 39.

2. The ratio of the tiled area to the total area of the countertop is 25 to 64.

Granted, this version of the question is slightly easier, because statement II gives you a clue to what your first step should be. But the same insight is necessary to solve this, and I maintain that it does not require computation. From either statement, you can reach the point reached in the solution given above: t = (5/8)s. Instead of plowing ahead with more computation, take a step back and use your imagination. The only constraint given by the problem is a proportion relating the length of a side of the tiled area to the length of a side of the table. FURTHERMORE, the question asks for width of the strip IN INCHES. This is slightly peculiar, given that the only other information about the table (a ratio of areas) does not mention ANY UNITS OF MEASUREMENT. What if the question had said “in feet.” Or “in millimeters.” Or “in miles.” Any one of these is possible given the information in the statements. The fact is that there is no way to determine absolute length measurements in terms of a specific unit if only proportions are known. A simpler question might look like this:

If a red spherical balloon contains twice the volume of a blue spherical balloon, which of the following can be the surface area of the red balloon, in square meters?

1. 1

2. 2

3. 3ANY of these numbers could be the surface area.

GMAT questions force the test-taker to make decisions about quantitative matters as quickly and as accurately as possible–very often without having to make calculations. The key behind these questions is to realize that computation is not necessary–a sense of what CAN be computed, however is indispensable.It makes sense that questions like these are on the GMAT. Very often in world of business, folks with an MBA are faced with the following decision: Should we spend all this time to do this computational task?

In the world of management, understanding when to ask this question can often be the difference between money wasted and money well-spent. On the GMAT, understanding this question will raise your score.

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