Tackle the GMAT’s toughest quant problems with the world’s leading GMAT Prep company. To get into top business programs, you need top GMAT scores, and GMAT Advanced Quant is designed to get you there. Written for students striving for 800, this book focuses on building your high-level quantitative skills by combining elite techniques for problem solving and data sufficiency with intense practice. Learn the tactics most often used by top-scorers and train your instincts as you work through over 250 of the toughest practice problems, then test your skills online with access to two additional online practice sets! Written by Manhattan Prep’s 99th percentile GMAT instructors, GMAT Advanced Quant will help you get over that last hurdle on your path to 800.
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Chapters 1 and 2 of this book focus on the more fundamental of the two types of GMAT math questions: Problem Solving (PS). Some of the content applies to any kind of math problem, including Data Sufficiency (DS). However, Chapters 3 and 4 deal specifically with DS issues.
This chapter outlines broad principles for solving advanced PS problems. You've already seen very basic versions of the first three principles in the Introduction, in the dialogues between the Top-Down and the Bottom-Up brain.
As mentioned earlier, these principles draw on the work of George Polya, who was a brilliant mathematician and teacher of mathematics. Polya was teaching future mathematicians, not GMAT test-takers, but what he said still applies. His little book How To Solve It has never been out of print since 1945—it's worth getting a copy.
In the meantime, keep reading!
Principle #1: Understand the Basics
Slow down on difficult problems. Keep three broad activities in mind while trying to Understand the problem:
Glance Read Jot
Glance at the entire problem: is it PS or DS? If it's PS, glance at the answer choices to see what form they're in. If it's DS, glance at the statements: is most of the complexity in the question stem or in the statements?
Polya recommended that you ask yourself a few simple questions as you Read a problem. We whole-heartedly agree. Here are some great Polya-style questions that can help you Understand:
What exactly is the problem asking for?
What are the quantities I care about? These are often the unknowns.
What do I know? This could be about certain quantities or about the situation more generally.
Whatdon'tI know?
Sometimes you care about something you don't know. This could be an intermediate unknown quantity that you didn't think of earlier.
Other times, you don't know something, and you don't care. For instance, if a problem includes the quantity 11! (11 factorial), you will practically never need to know the exact value of that quantity.
What is this problem testing? In other words, why is this problem on the GMAT? What aspect of math are they testing? What kind of reasoning do they want me to demonstrate?
Don't forget to Jot down any given numbers or formulas on your scrap paper.
The Understand the Basics principle applies later on as well. If you get stuck, go back to basics. Re-read the problem and ask yourself these questions again (within a reasonable amount of time, of course).
Start the following problem by trying to Understand what's going on.
Try-It #1-1
x = 910 – 317 and
is an integer. If n is a positive integer that has exactly two factors, how many different values for n are possible?
(A) One
(B) Two
(C) Three
(D) Four
(E) Five
Glance. This is a PS problem. The answers are numbers but in written form; this format is reserved for questions that ask for the number of numbers or number of possibilities for something. The numbers are small.
Read. Dive into the text. Here are some possible answers to the Polya questions.
What exactly is the problem asking for?
The number of possible values for n. This means that n might have multiple possible values. In fact, it probably can take on more than one value. I may not need these actual values. I just need to count them.
What are the quantities I care about?
I'm given x and n as variables. These are the quantities I care about.
What do I know?
x = 910 – 317
That is, x = a specific large integer, expressed in terms of powers of 9 and 3.
is an integer.
That is, x is divisible by n, or n is a factor of x.
Finally, n is a positive integer that has exactly two factors. Prime numbers have exactly two factors. So I can rephrase the information: n is prime. (Primes are always positive.).
Whatdon'tI know?
Here's something I don't know: I don't know the value of x as a series of digits. Using a calculator or Excel, I could find out that x equals 3,357,644,238. But I don't know this number at the outset. Moreover, because this calculation is far too cumbersome, it must be the case that I don't need to find this number.
