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For many students, GMAT number properties is one of the most daunting sections of the exam. But it doesn’t have to be that way. The GMAT number properties section is just like any other difficult section during a standardized test:it can be mastered. The key to mastering a standardized test is knowing how to take one. Remember the SAT or ACT back in High School? Learning how to the take those exams was crucial to a high score. Fortunately, those rules that you learned for those examinations still apply. Therefore, being prepared for the GMAT number properties portion is all about understanding and reviewing very basic math concepts in order to save time. Below is a review of very basic mathematical definitions:

Integers

Integers are numbers sans a fractional part such as 3, 2, 1. A number like 2.25, which is a decimal, is not an integer. Integers can also be negative, such as -3,-2,-1 but do not have a fractional part as well. Positive integers are defined as being whole numbers. The 0 is also an integer but is considered to be positive or negative.

Factors

Factors are considered to be numbers that divides equally into another number. For example the number 3 is a factor of 12 because 12/4=3. It is also a factor of 6 because 6/2=3 or 9 because 9/3=3.

Primes

Prime numbers are whole numbers that only have two divisors, the actual number itself and one. For example, the number 7 is a prime number because its only two divisors are 7 and 1.

Greatest Common Factor

The Greatest Common Factor or GCF for short is the largest number that divides two numbers evenly. In order to determine the Greatest Common Factor is by setting up a prime factorization of two numbers and comparing common factors. The largest common factor between the two numbers is the GCF.

Least Common Multiple

To find the least common multiple, you perform a prime factorization in the same manner as one would do for the GCF. However, the least common multiple is the smallest number of a multiple of two numbers.

Units Digits

Unit digits are the number to the right of the tens position. For example, the units digit for the number 364 is 4.

After reviewing basic topics such as the ones previously described, creating a study schedule with practice questions is a good way to see where your strengths and weaknesses are. Once, you know where your weaknesses are, study accordingly.

This post will introduce weighted average questions you’ll see on the GMAT.  There is one main formula you need to solve simple GMAT Average questions:

  • Average = SUM / # of observations

Note that this can be rearranged to read:

  • SUM = Average x (# of obs)
  • # obs = SUM / Average

So, if you are given ANY 2 of the 3 values, you should be able to find the 3rd. For example:

John drinks an average of 1.5 cups of water/day. After how many days has he drank 3 gallons of water? (1 gallon = 16 cups.)

In this case, we are looking for the number of days (or observations) such that we total 48 cups (3 gallons) of water.

# = SUM / Average

  • # days = 48 cups / 1.5 cups/day
  • # days = 32 days

NEVER AVERAGE AVERAGES!

Class A  has 15 students and an average height of 60”. Class B has 20 students. What is class B’s average height if the average height of both classes is 65”.

One might say:  (A + B) / 2 = 65”; A = 60”; so B must be 70”. However, keep in mind:

TOTAL AVERAGE = TOTAL SUM / TOTAL OBS

CLASS A + CLASS B = BOTH

  • 15 students + 20 students = 35 students
  • 60” average + 68.75” average = 65” average
  • 900” total in A + 1375” total in B = 2275” total in Both

The given information is in black. The necessary intermediate steps are in blue, and the red is your answer. Note that the average of the averages ≠ total average. We must calculate each average separately, and to do this we need the SUM and # of observations for each category. This brings us to the idea of WEIGHTED AVERAGES.

A WEIGHTED AVERAGE is needed when you are taking average of a large group in which there are subgroups with a different number of observations in each. Take a look at this generalized formula, assuming there are 3 groups A, B and C.

(Average of A x Obs in A) + (Average of B x Obs in B) + (Average of C x Obs in C)

(Obs in A) + (Obs in B) + (Obs in C)

Think of weighted averages like a tug of war between numbers. The “stronger” one side (dog) is, the more that weighted average (tennis ball) will be “pulled” in that direction.

In the previous question, we had:

CLASS A + CLASS B = BOTH

  • 15 students + 20 students = 35 students
  • 60” average + 68.75” average = 65” weighted average

Note that the weighted average is CLOSER to B’s average than it is to A’s. This is because there are 20 students in Class B compared to only 15 students in Class A.

Two More Examples

At a certain restaurant, the average (arithmetic mean) number of customers served for the past x days was 75. If the restaurant serves 120 customers today, raising the average to 90 customers per day, what is the value of x?

A. 2

B. 5

C. 9

D. 15

E. 30

WITHOUT using the formula, we can see that today the restaurant served 30 customers above the average. The total amount ABOVE the average must equal total amount BELOW the average. This additional 30 customers must offset the “deficit” below the average of 90 created on the x days the restaurant served only 75 customers per day.

30/15 = 2 days. Choice (A).

WITH the formula, we can set up the following:

  • 90 = (75x + 120)/(x + 1)
  • 90x + 90 = 75x + 120
  • 15x = 30

x = 2  Answer Choice (A)

Use whichever makes more sense to you!

Anita spent a total of $780 on 52 bottles of wine for her wedding. She then decided to buy 8 bottles of sparkling wine for the toasts, as well. Was the average (arithmetic mean) price per bottle of wine less than $20?

(1) Each bottle of sparkling wine cost more than $15.

(2) Each bottle of sparkling wine cost less than $40.

Take another look at what exactly the question is calling for: the TOTAL average price of all the wine at the wedding. We should look at the suggested average ($20) and use that as our threshold amount.

  • 60 bottles * $20/bottle = $1200 total
  • $1200 total – $780 (given) = $420 (left for sparkling wine)
  • $420 / 8 bottles = $52.50/bottle of sparkling wine (for the total average to equal $20)

Which of the answer choices are conclusively above or below $52.50/bottle of sparkling wine? Only (2). With (1), we can be below OR above the threshold, so (1) is not sufficient.

Answer Choice (B)

Now, you’re score will be above average! Please visit the Grockit forum or leave a comment here if you have more questions on weighted averages.

Good luck!

Three types of pencils, J,K, and L, cost $0.05, $0.10, and $0.25 each, respectively. If a box of 32 of these pencils costs a total of $3.40 and if there are twice as many K pencils as L pencils in the box, how many J pencils are in the box?

(A) 6
(B) 12
(C) 14
(D) 18
(E) 20

Highlight to see answer: C

Please post your explanations in the comments below!

For me, one of the most exciting parts of Kaplan’s new GMAT revision for 2010 is its emphasis on Identifying the Task in Quantitative Problem Solving questions.  Often, Problem Solving questions give a test-taker a large amount of information–sometimes as much as five or six equations’ worth–and when the pressure is on and the clock is ticking, it can be tough to get this information organized enough to answer the question, not to mention the amount of time that can be lost in doing so.

One of the most efficient ways of zeroing in on the answer in complicated questions is to use what I call a “master equation.”  For example, if I see the phrase “average (arithmetic mean)” in the first line of a Quantitative question, I immediately shoot down to my noteboard and jot down the average formula: “Average = (Sum of terms)/(Number of terms),” or my preferred abbreviated version, “Avg = sum/#.”

The logic behind it?  Well, if a question stem is mentioning an average in its first few words, I’m going to have to use the formula at some point, whether the question ends up needing the value of the average, the sum of the terms, or some other relationship.  With the formula already written before I even finish reading the question stem, I save myself the time and effort of recalling it later on, when I’ll be more concerned about the tricky, GMAT-ty aspects of solving the problem.  If I’m lucky–say, for example, the question from above gives us an average and a number of terms, and asks for the sum of the terms–I can even plug values directly into my master equation, solve algebraically, and get to the answer that much faster.  This can be very effective even complicated algebra questions where you’re not sure what the first step is–at least you can use the equations you know to do SOMETHING, instead of sitting there without knowing what to do.  When you keep your eventual goal (your “master equation”) prominent and proud at the very top of your scratch work, you’ll find it very difficult to lose focus!

In 1980 the government spent $12 billion for direct cash payments to single parents with dependent children. If this was 2,000 percent of the amount spent in 1956, what was the amount spent in 1956? (1 billion = 1,000,000,000)

(A) $6 million
(B) $24 million
(C) $60 million
(D) $240 million
(E) $600 million

Highlight to see answer: E

Please post your explanations in the comments below!

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Because I am not a native English speaker, I had lots of problems especially with GMAT English. But after getting this book, I feel that the English part became much easier! The math review is also very helpful. So if you are a foreign GMAT taker, this book will help you a lot. I recommend it any foreigners who have over 80 toefl score! :)
- Youngji Kim
Sungkyunkwan University