Step-by-Step 4/14/14

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Sometimes we don’t need a whole lot of discussion. We just need a problem broken down step-by-step. This question comes from the GMAT Official Guide, 13th edition, Section 6.3 #110.

“Tom, Jane, and Sue each purchased a new house. The average (arithmetic mean) price of the three houses was $120,000. What was the median price of the three houses?

(1) The price of Tom’s house was $110,000.

(2) The price of Jane’s house was $120,000.”

Step 1: Identify your target and the given information- Here our goal is to find the median home price. We’re given the average price which tell us that the total of the three sales is $360,000.

Step 2: Find paths to sufficiency- in other words, figure out what you’d need to know in order to be able to find the median. When arranged in increasing order, the median is the middle term. In order to find the middle term we’ll either need to know the values of all three terms, or be guaranteed that one of our terms is the middle one.

Step 3: Assess the first statement- If Tom’s house was $110,000 we know that the two other houses sum to $250,000. If Jane’s house was $100,000 and Sue’s house was $150,000 the median would be $110,000. Alternatively if Jane’s house was $120,000 and Sue’s house was $130,000 the median would be $120,000. Since there is more than one possibility statement 1 is insufficient.

Step 4: Assess the second statement- If Jane’s house was $120,000 we know that the two other houses sum to $240,000. On the basis of this statement alone we can’t determine the exact prices of the other two houses. Many students will stop there, call this insufficient and get this question wrong. Remember, we could also have a sufficient amount of information if we could guarantee that $120,000 was the middle value. Since the other two houses sum to $240,000 we only have two options. The first option is that that one house is less than $120,000 and one is more. If that’s the case $120,000 is our median. The only other alternative is that all three houses cost $120,000. In that situation our median is still $120,000. Statement two is sufficient.

Follow all of these steps carefully and you’re well on your way to a great score. In data sufficiency questions you especially don’t want to jump ahead to combining both statements until you’ve fully evaluated each one.

Study hard!

 Concentrated Liquids

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If you’re looking for a technical, precise explanation on handling concentrated liquid problems, look elsewhere. For the rest of you, this may be helpful.

When I say concentrated liquid problems, I mean problems that take the following form:

5 gallons of a mixture of cranberry juice and water is 15 percent juice. How much water will have to be added in order to create a mixture that is 10 percent juice?

You see these problems in various forms, sometimes asking you to evaporate out some water, or sometimes to add some water. Sometimes you’re given the total volume of the desired mixture and asked to determine the concentration. Other times you’re given the desired concentration and asked to figure out the total volume in some form. However, despite the specific form that you’re given, I have some very eloquent advice that’s going to make tackling these problems much simpler: Find out how much stuff you’ve got.

Yup, that’s right “stuff”. Almost all of the problems of this type that I’ve encountered involve a mixture of water and something else. That something else is the “stuff” we’re interested in. Water will simply be a filler that will take up all of the space not occupied by “stuff”. Let’s take a look at the problem I mentioned earlier to illustrate this concept.

Here we have 5 gallons of mixture and we know that it’s 15% cranberry juice. That’s our stuff. We need to find out how much cranberry juice we have. The simple calculation is:

Once we know how much stuff we have, we just need to set that 0.75 gallons equal to 10 percent of x to find out the volume of the new mixture. This one is relatively straightforward, so we know that the total volume will equal 7.5 gallons. Since we started with 5 gallons, and all we’re adding is water to get to 7.5 gallons, we need to add 2.5 gallons of water and there’s your correct answer.

Regardless of the question type or form, when you see one of these concentrated liquids problems make sure you find out how much “stuff” you have! The rest will be water and you’ll be well on your way to a correct answer!

 Strategy Sunday: Skipping Questions

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There are many advantages to being able to take the GMAT on the computer. You get a much truer sense of your abilities because the computer can be adaptive while a paper test cannot. You eliminate the risk of smudged ink making a geometry question impossible to decipher. You get the flexibility to take the test almost any time that’s convenient for you without the test-makers having to worry about questions “getting out”. But it does make timing more challenging.

 

Unlike a paper-based test where you’re free to skip a difficult question so that you can come back and work on it later, once you’ve skipped a question on the computer-based GMAT it’s gone, never to be seen again. Well, maybe you’ll see it in those haunting dreams you have the night after the test. Because, you see, the vast majority of the questions you face are going to be things that you could answer… if you had enough time. The task being asked of you is not so much what can you do, but rather what can you do in 75 minutes.

 

The computer-adaptive nature of the test can seem cruel to test-takers of any ability. Whether you’re scoring in the 400s or the 700s there are going to be math questions that are at the limit of your abilities. If you continue to answer questions correctly, you’re eventually going to get to those questions. And that’s where the dilemma comes in: do you spend some extra time to get that really tough question, or do you take a guess and move on?

 

With few exceptions, the correct tactical decision is to guess and move on. Questions at the beginning of the test are no more important than the ones at the end of your test, and if you’re sacrificing the time you need to answer two questions correctly at the end in order to answer one question in the middle of your test, you’re losing that trade. Plus, even if you do manage to get that very difficult question correct, your reward will be an even tougher question! Go through this cycle a few more times and you’re faced with questions that are certainly beyond your reach and a time deficit that will be nearly impossible to overcome.

 

Take this strategy to heart: set time markers. Make sure you’re finished with 10 questions with 55 minutes to go. Make sure you’re finished with 20 questions with 35 minutes to go. Make sure you’re finished with 30 questions with 15 minutes to go. Hitting these markers might require making a quick guess on number 9 or taking a complete shot in the dark on number 20, but it’s worth it. The GMAT penalizes you heavily for failing to answer the last question(s) so it’s likely that any good you do yourself early in the test will be more than offset later.

 

Keep those time markers in mind. If you get a little ahead, you know you have some freedom to take an extra minute on that tough question. If you fall behind, take quick measures to get yourself back into a good position. Even if you realize after the test that the question you skipped and took a wild guess on was one you could have answered, if you follow this advice you’ll be too happy celebrating a great score to worry too much about that!

 Reduce Before You Multiply Fractions

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I often use this space to talk about broad topics that should be of interest to the takers of many tests in many situations.  Today I focus on something much more narrow that’s primarily of interest to GMAT test-takers who do not have the benefit of a calculator while working through the math portion of the test. The key is simple: reduce fractions before you multiply.

You often encounter probability problems where you end up needing to multiply two or more somewhat unruly fractions. The rule for multiplying fractions is, as always, multiply the numbers across the numerators to create the new numerator and multiply the numbers across the denominators to create the new denominator. Here that is in general form:

However, if you perform this operation before reducing those fractions you’re likely to end up with an unruly product that may be very difficult and time-consuming to reduce to its simplest form.  Take the following example:

If your answer to this problem is 504/9240, you’ve found the correct value, but you won’t find that choice in your answers because the answer choices will all be in their reduced form. Furthermore, reducing that fraction is going to be a bit of a mess.  It’s obvious to see that both the numerator and the denominator are multiples of 2, but beyond that it’s going to take some time to unwrap what’s going on.

Rather than save the reducing for the end, make sure that you reduce before you multiply to lessen the risk of a multiplication error and to speed up your process. Take a look at the problem I showed you earlier.

Notice that none of the individual fractions can be reduced. However, that isn’t going to slow us down. Any factor found on the top can be canceled when we find the same factor anywhere on the bottom. Doing this work on the front end produces a much neater result.

 

All of the color-coded cancelling is pretty simple. Plus, having cancelled everything that we could have on the front end, we can be guaranteed that the product will end up in reduced form. Keep this simple rule in mind to save time and reduce mistakes: when multiplying fractions, reduce before you multiply!