Most people don’t like weighted averages, and for good reason. The formula is complicated, and these often come in the form of story problems, which are hard to set up. We’re going to talk today about a couple of great little techniques to make these fast and easy well, easier anyway!

First, try this GMATPrep problem. Set your timer for 2 minutes. and GO!

*  A rabbit on a controlled diet is fed daily 300 grams of a mixture of two foods, food X and food Y. Food X contains 10 percent protein and food Y contains 15 percent protein. If the rabbit’s diet provides exactly 38 grams of protein daily, how many grams of food X are in the mixture?

 

(A) 100

(B) 140

(C) 150

(D) 160

(E) 200

Wow. I’m glad I don’t have to feed this rabbit. This sounds annoying. : )
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We invite you to test your GMAT knowledge for a chance to win! Each week, we will post a new Challenge Problem for you to attempt. If you submit the correct answer, you will be entered into that week’s drawing for a free Manhattan GMAT Prep item. Tell your friends to get out their scrap paper and start solving!
Here is this week’s problem:

The positive difference of the fourth powers of two consecutive positive integers must be divisible by…

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On October 13, 2012, one of the major sports books in Las Vegas said that there was a 108.8% chance of one of the four teams left in the baseball postseason would win the World Series. Of course it didn’t actually say there was a 108.8% chance of this happening, but the odds that they released to bettors did and helped ensure that over the long run, Vegas wins and we, as a whole, lose.

If you haven’t already, check out Part 1 for a review of AND vs OR probability. Now let’s imagine that instead of betting on outcomes, like we did in the previous article, you’ve wised up and decided to open your own sports book, gMATH. You decide to start simple and offer bettors a chance to bet on which number, 1-4, randomly rolls out of a bingo cage. You realize that the probability of each number being selected is 25%, but you need a way to translate this for paying bettors. In a scenario where four different people each put down $1 on each of the four numbers, one person would win $3 ($4 total – $1 they bet). So you place the very first odds at gMATH’s number guessing game at 3 to 1.

In the long run, gMATH’s inaugural betting event may attract a clientele of people who enjoy watching ping pong balls with painted numbers roll around, but it won’t be bringing you the fortunes that you passed up on business school for. You realize that you need a new betting game that will attract more than just the bingo-loving crowd, involves a small amount of luck, and allow you to make a profit no matter which team wins. As there are exactly four teams left in the postseason, you decide that baseball would make a perfect switch.
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Remember those times when you were sure you got the answer right, only to find out that you got it wrong? For a moment, you even think that there must be a mistake in the answer key. Then, you take a look at the problem again, you check your work, and you say, I can’t believe I did that! You knew exactly how to do this problem and you should have gotten it right, but you made a careless mistake.

What’s a Careless Error?

By definition, a careless mistake occurs when we did actually know all of the necessary info and we did actually possess all of the necessary skills, but we made a mistake anyway. We all make careless mistakes (yes, even the experts!); over 3.5 hours, it’s not reasonable to assume that we can completely avoid making careless mistakes. Our goal is to learn how to minimize careless mistakes as much as possible.

How Can We Minimize Careless Errors?

Isn’t the whole point of a careless error that we don’t know when we’re going to make them? They just happen randomly and we can’t control that!
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Are you ready for a challenge? Try to solve the following question in under two minutes:

How many different positive divisors does the number 147,000 have?

If you feel like two minutes are not nearly enough to solve the problem, you’re not alone. Even the most seasoned GMAT veterans might find the problem challenging, as it requires a deep level of understanding of two mathematical concepts: Divisibility and Combinatorics (just a fancy word for ˜counting’).

If I replaced the number 147,000 with the number 24, many more people would be able to come up with an answer:

You could just pair up the divisors (factors) and count them. Start with the extremes (1×24) and work your way in:

1×24

2×12

3×8

4×6

A quick count will show the number 24 has exactly 8 different positive divisors.

The number 147,000 will have many more positive divisors “ too many to count This is a strong indication that we will need to use combinatorics.

Divisibility: Any positive integer in the universe can be expressed as the product of prime numbers.
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We invite you to test your GMAT knowledge for a chance to win! Each week, we will post a new Challenge Problem for you to attempt. If you submit the correct answer, you will be entered into that week’s drawing for a free Manhattan GMAT Prep item. Tell your friends to get out their scrap paper and start solving!
Here is this week’s problem:

The function F(n) is defined as the product of all the consecutive positive integers between 1 and n², inclusive, whereas the function G(n) is defined as the product of the squares of all the consecutive positive integers between 1 and n, inclusive. The exponent on 2 in the prime factorization of F(3)/G(3) is

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Have you ever gotten a GMAT question wrong because you thought you were supposed to take a square root and get two different numbers but the answer key said only the positive root counted? Alternatively, have you ever gotten one wrong because you took the square root and wrote down just the positive root but the answer key said that, this time, both the positive and the negative root counted? What’s going on here?

There are a couple of rules we need to keep straight in terms of how standardized tests (including the GMAT) deal with square roots. The Official Guide does detail these rules, but enough students have questioned us about the OG explanation that we decided to write an article in hopes of clearing everything up. : )

I want to mention one thing before we dive in: the vast majority of the time, both roots do count, and it’s rare to miss an official question as long as you do take both roots. You could just decide that you’re not going to worry about it and you’re going to solve normally (always taking both square roots). Many students do still stress about this topic, though, so if you’re in that group, read on!

Doesn’t the OG say that we’re only supposed to take the positive root?

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Note: This is the first of a two-part series on Probability. The baseball odds used in this article were true on the morning of October 11 and are definitely no longer true. Using this gambling advice will likely cost you money in the long run, as Joe will explain in Part 2.

In terms of excitement, the World Series of Coin Flipping would rate right next to solving Data Sufficiency problems in the general public’s mind. But any Vegas oddsmaker worth his weight in comped beverages would be able to calculate the odds for every coin flipping contest in the time it would take the coin to land. In an 8-team bracket-style tournament, every squad would have 1 in 8 odds to win it all because only 1 team out of the 8 could outguess their way to the championship. But what if each matchup of two teams was a 3 game series? It wouldn’t affect the odds at the beginning of each series (still a 50% probability for each team to win), but once the first outcome was decided, those Vegas oddsmakers would require some knowledge of And Probability to keep the odds fair.

Imagine we have two teams competing in the first round of our Coin Flipping Playoffs- let’s call them Baltimore & New York. In our three game series, New York happened to have some late flipping heroics to go up 1-0. What are the odds that Baltimore comes back in this series and what are the odds that Baltimore comes back and then wins the whole tournament? The key to this type of question is understanding that many things have to go right in order for Baltimore to win it all- first they need to win Game 2, then win the winner-takes-all Game 3, then win their semi-final series, and then still have enough thumb strength to flip the World Series in their favor. Four unique events need to happen, and every single one of those events must happen for Baltimore to emerge victorious. Since the odds of each game and each series going Baltimore’s way would be 1/2, we can solve this by finding the odds that Baltimore wins Game 2 (1/2) AND Game 3 (1/2) AND the next series (1/2) AND the championship (1/2). In probability, whenever we want X AND Y to occur, we need to multiply the respective odds together. The odds of Baltimore winning Game 2 and 3 would be 1/2 x 1/2 = 1/4. The odds of winning all four events would be (1/2)⁴ = 1/16.
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In an earlier post, we tackled a medium-level GMATPrep weighted average question; click here to read that article before reading this one. This week, we’re trying 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 earlier, 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 week, we focused on understanding how weighted averages work via writing some equations. We’ll try to apply that understanding to our harder problem this week, 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 3^rd 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 week 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. Last week, 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.

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This week, we’re going to tackle a GMATPrep question from the quant side of things. We’ll tackle a medium-level question this week in order to learn how to master weighted average questions in general, and in the next article, we’ll try a very hard one “ just to see whether you learned the concept as well as you thought you did. : )

Before we begin, I want to mention that every weighted average problem I’ve seen 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. So we’re going to work on that conceptual understanding today and then we’ll discuss a neat calculation shortcut next week (built on the same principles!), just in case we do need to solve.

Let’s start with a sample problem. Set your timer for 2 minutes. and GO!
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