Advanced GMAT Quant
Data Sufficiency: Is ax - x(b + a) = 3 + bx?
Data Sufficiency: Is ax - x(b + a) = 3 + bx? First, I don't like the parentheses, so let's get rid of them: Is ax - bx - ax = 3 + bx? Oh, great, now I can combine the "ax" terms: Is -bx = 3 + bx? Ah, and now the "bx" terms: Is -3 = 2bx? And I know that algebra is all about trying to "isolate the variables," so I'm going to do that next: Is (-3/2) = bx? That's all they're asking me? Whether I can find one specific value for the expression bx? Great!
Data Sufficiency: Follow a consistent process
Do your work on paper, not in your head. Label everything and separate everything physically on your paper. (Facts from question stem, question, statement 1, statement 2.) Rephrase the question and statements whenever possible. (In particular, try to rephrase the question BEFORE you dive into the statements. This habit will no only save you time and effort; occasionally, it will make all the difference between getting the problem right and getting it wrong.) Evaluate the easier statement first. Physically separate the work that you do on the individual statements.
PS Strategy: Test answer choices
Doing so can often help you avoid demanding calculations or the need to set up and solve complicated algebraic expressions. A question might contain a phrase such as "Which of the following..." This is a great time to test answer choices.
Understand the Basics: Slow down on difficult problems. Make sure that you truly GET the problem.
- What do I know? - What don't I know? - What do I want to know? - What is the problem testing me on?
Understand, Plan, Solve Mathematicians know that the real math on hard problems is not Solve; the real math is Understand and Plan.
1) Understand the problem first. 2) Plan your attack by adapting known techniques in new ways. 3) Solve by executing your plan.
Pattern Recognition: When you see a string of numbers for a pattern, do the following:
1. Compute the first 5 to 8 terms and try to match them to a stock pattern that you already know. 2. Look for repeating cycles. 3. If you are stuck, look for patterns within DIFFERENCES BETWEEN TERMS or SUMS ACROSS TERMS. a) Look at the difference between consecutive terms. b) Also look at the CUMULATIVE SUM of all the terms up to that point. c) Some sums involve MATCHING PAIRS that sum to the same number (or even cancel each other out). 4. Look at the characteristics of the numbers: positive/negative, odd/even, integer/non-integer, etc. Once you have extended the pattern for several terms, these characteristics will generally repeat or alternate in some predictable way.
Solving hard inequalities problems:
1. Know the tricky details: manipulating variables can be dangerous when we don't know the signs; 2. If you know the theory inside and out, think it through theoretically; if not, test some numbers. Either way, on data sufficiency yes/no questions, your goal is to try to find a "yes" and a "no." If you can, then you know the info was not sufficient; if you keep getting only "yes" answers or only "no" answers, then it's probably the case that the info is sufficient. 3. When testing numbers, reuse your work as much as possible. If you're allowed to reuse the same numbers for both statements, do so (but be careful to evaluate each statement separately first!).
Pattern Recognition: The most basic patterns are...
1. The counting numbers (1, 2, 3, 4, ... ), also known as the POSITIVE INTEGERS. 2. A REPEATING CYCLE of numbers. For example: 4, 2, 6, 4, 2, 6, 4, 2, 6...
What is the prime factorization of 9¹⁰ - 3¹⁷?
3²⁰ - 3¹⁷ = 3¹⁷(3³ - 1) = 3¹⁷(27 - 1) = 3¹⁷(26) = 3¹⁷(13)(2)
Data Sufficiency: Work from facts to questions
Always ask, "If I start with the applicable facts and consider ALL possibilities, do I get a DEFINITIVE answer to the question? No matter what, when you are judging sufficiency, you should always proceed from the FACTS to the QUESTION. After you've rephrased, put the question on hold and work FROM the statements and any other given facts TO the question.
DS Strategy: Compute to completion
If you can't tell for certain whether the answer can be calculated in theory, keep going on the calculations ALL THE WAY. This is particularly common in the following situations: - Multiple equations involved - particularly if they are non-linear. - A complicated inequality expression is present. - Variables hidden with a Geometry problem are related. In a multiple scenario problem, be sure to compute each scenario all the way through to determine whether the end result for each scenario is ACTUALLY DIFFERENT.
If you get stuck...
If you get stuck, ask yourself (do, don't, want, testing me on) and WRITE EVERYTHING DOWN. 1. Look for patterns. 2. Draw a picture. Or visualize a word problem, put yourself into the situation. 3. Solve an easier problem (one without the given constraints). When you get stuck, try some real numbers to understand what's going on. Don't go over time; if you have to guess, do so and move on. Afterwards (if it's not the real test!), you can go back and try numbers until you figure out exactly how the problem works and why.
DS Strategy: Use a Scenario Chart
In problems with a limited number of potential outcomes, list those outcomes on paper and test them in a structured way. The Scenario Chart can help us organize our approach to these problems: 1. Each variable gets its own COLUMN. 2. Each scenario gets its own ROW. 3. Consider FACT CHECK COLUMNS (forces a check that the scenarios comply with more complicated constraints). 4. Create a QUESTION & ANSWER COLUMN (forces the generation of an answer for every allowable scenario). 5. MAKE SEPARATE CHARTS for each statement (avoids statement carryover). Work towards EFFICIENCY (e.g., evaluating only the scenarios that fit the given information) rather than EXHAUSTIVENESS (e.g., evaluating all possible scenarios and then crossing out the scenarios that violate the given information). This keeps you focused on the scenarios that remain possible, rather than wasting time on scenarios that are irrelevant. Also, be sure to CROSS OUT or mark as invalid any scenarios that do not fit your facts, as soon as you know that! You can't use these scenarios as "No" cases - they don't count at all.
DS Strategy: Extract the equation
Represent word problems with algebraic equations to avoid embedded tricks that can be difficult to spot otherwise. Be sure to translate all word problems into algebra so they can be properly evaluated.
PS Tactic: Look for answer pairs
Some Problem Solving questions have answer choices that pair with each other in someway. The right answer may be part of one of these pairs. Pairs of answers may: - Add up to 1 on a probability or fraction question - Add up to 100% on questions involving percents - Add up to 0 (be opposites of each other) - Multiply to 1 (be reciprocals of each other)
PS Tactic: Look at positive-negative
Some Problem Solving questions include both positive and negative answer choices. In such cases, look for clues as to the correct sign of the right answer.
PS Tactic: Apply cutoffs
Sometimes a back-of-the-envelope estimation can help you eliminate any answer choice above or below a certain cutoff. You have to imagine that the problem is slightly different (and easier) to come up witha threshold value, but once you do, you can often get rid of two or three answer choices. This strategy can sometimes be used in combination with an Answer Pairs Strategy, as pairs of answers are often composed of a high and a low value.
Inequalities: Squares (and any numbers raised to an even power) are also dangerous...
Squares (and any numbers raised to an even power) are also dangerous: because they hide the sign of the underlying number.
Hidden constraints: Sometimes, the test writers will simply tell us a piece of information. Other times, those keys will be hidden in the details of the problem. Problems in Data Sufficiency form often benefit from some upfront work on the question stem before you ever look at the statements.
Start looking for hidden constraints while you're studying. If you don't notice until after you're done with the problem, try it again, even if you got it right. Maybe noticing that hidden constraint at the beginning could have helped you spot a shortcut and answer the question more quickly. Do the details limit the possibilities in some way? Figure out what the limits are. Do we need to find one exact number? Sometimes yes, but often no; know when you do and when you don't. Take the time to understand what's going on. It's worth it.
When n is odd, is n² - 1 divisible by 8?
n² - 1 = (n - 1)(n + 1) Because they also talk about n, this is a CONSECUTIVE INTEGER problem: (n - 1), n, and (n + 1). Every other even integer, by definition, is a multiple of 4, not just a multiple of 2. 0, 2, 4, 6, 8, 10, 12... starting with zero, every other one is a multiple of 4. So, whenever I multiply two consecutive even integers, one of them has to be a multiple of 4. So, when n is odd, n² - 1 must be divisible by 8.
Consecutive integer problems:
(1) They can test you on a concept without naming that concept. Be able to recognize a consecutive integer problem in disguise. The form n² - 1 is probably the most common indicator and n³ - n is another very common indicator because n³ - n = n(n² - 1) = n(n+1)(n-1). (2) Also be able to recognize some of the common "useful pieces of information" that might be given on consecutive integer problems. Knowing something as seemingly simple as whether certain terms are even or odd can make a big difference, especially if the question deals with divisibility or remainders.
If you know you can Solve quickly, then you can take more time to comprehend the question, consider the given information, and select a strategy. To this end, make sure that you can rapidly complete calculations and manipulate algebraic expressions.
Avoid focusing too much on speed, especially in the early Understand and Plan stages of your problem-solving process. A little extra time invested upfront can pay off handsomely later.
Top-Down detective sets overall agenda (Plan)
Bottom-up bloodhound notices patterns (Understand)
DS Strategy: Use the constraints
Bring implicit and explicit constraints to the surface. These constraints will often be necessary to determine the correct answer. Make note of any additional information given to you in the question stem (for example, "x is positive" or "x is an integer"). When constraints are given, they are usually ESSENTIAL to the problem. Integer constraints in particular are very potent: they often limit the possible solutions for a problem to a small set. Sometimes this set is so small that it contains ONLY ONE ITEM. In some problems, constraints will be implicit (for example, "number of people" must be both positive and an integer).
Common terms (algebra): b/a...
Common terms (algebra): b/a is the reciprocal of a/b
Common terms (algebra): b² - a²...
Common terms (algebra): b² - a² is negative a² - b²
Common terms (exponents): Exponents can be manipulated when...
Common terms (exponents): Exponents can be manipulated when either bases or exponents are common. Also look for bases that have common factors, such as 3 and 12 (common factor of 3). You can often create a common base.
Common terms (factors and multiples): When many terms share a factors...
Common terms (factors and multiples): When many terms share a factors, pull that shared factor OUT TO THE SIDE. Factorials are particularly noteworthy, as they often have an abundance of shared factors. More generally, factorials are "super-multiples." Without ever computing their precise value, you can tell that they're divisible by all sorts of numbers.
Heavy Multiplication
Factor/eliminate common terms as you go.
DS Strategy: Test numbers systematically
For some one-variable Data Sufficiency problems, many possible values may exist, but underlying those values is some sort of pattern. Use SYSTEMATIC NUMBER TESTING when possible values cannot be listed consecutively, and use DISCRETE NUMBER LISTING when the values are "integer-like" (and thus can be listed consecutively). SYSTEMATIC NUMBER TESTING: We can often discover that pattern by intelligently picking numbers that cover a wide range of potential patterns. {-2, -3/2, -1, -1/2, 0, 1/2, 1, 3/2, 2} Remember this list as "every integer and half-integer between -2 and 2." Boundary principal: test values that are CLOSE TO BOUNDARIES given in the problem. (If the boundary is 1, test 0.9 and 1.1) For discrete numbers, TEST CONSECUTIVE VALUES that fit the criteria - it would be too easy to leave out the exception that proves insufficiency. Never skip numbers that fit the constraint.
Understand how scoring works:
Getting an easier question wrong hurts your score more than getting a harder question wrong. In fact, the easier the question, relative to your overall score at that point, the more damage to your score if you get the question wrong. (Note: it is still very possible to get the score you want even if you make mistakes on a few of the easier questions.)
DS Tactics: Spot one statement adding nothing IF one statement adds no information to the other...
IF one statement adds no information to the other... THEN eliminate C.
DS Tactics: Spot one statement inside the other IF one statement is "contained with" (i.e. is a subset of) the other...
IF one statement is "contained with" (i.e. is a subset of) the other... THEN eliminate C and "broader statement only."
DS Tactics: Spot clear sufficiency IF the two statements are clearly sufficient together...
IF the two statements are clearly sufficient together... THEN eliminate E.
DS Tactics: Spot identical statements IF two statements tell you exactly the same thing (after rephrasing)...
IF two statements tell you exactly the same thing (after rephrasing)... THEN the answer is either D or E.
Inequalities are dangerous when we aren't given information about the signs (positive or negative) of particular numbers...
Inequalities are dangerous when we aren't given information about the signs (positive or negative) of particular numbers: we can't, for example, multiply or divide by variables if we don't know the signs of those variables.
Inequalities are dangerous:
Inequalities are dangerous when we aren't given information about the signs (positive or negative) of particular numbers; we can't, for example, multiply or divide by variables if we don't know the signs of those variables. Squares (and any numbers raised to an even power) are also dangerous, because they hide the sign of the underlying number. Squares can also be tricky because different things can happen to different kinds of numbers; whole numbers (whether positive or negative) will generally get larger when squared, with the exception of zero and one, which stay the same; fractions between 0 and 1 will get smaller when squared.
Inequalities: Squares can also be tricky...
Inequalities: Squares can also be tricky: because different things can happen to different kinds of numbers; whole numbers (whether positive or negative) will generally get larger when squared, with the exception of zero and one, which stay the same; fractions between 0 and 1 will get smaller when squared.
PS Strategy: Avoid needless computation
Look for opportunities to avoid tedious computation by factoring, simplifying, or estimating. If it seems like calculating the answer is going to take a lot of work, there's a good chance that a shortcut exists. Keeping track not only of your current estimate, but also the degree to which you have overestimated or underestimated, can help you pinpoint the correct answer more confidently.
PS Tactic: Draw to scale
Many Geometry problems allow you to eliminate some answer choices using visual estimation, as long as you draw the diagram accurately enough.
Stealth constraints: Non-countable items (non-negative)
Many non-countable quantities must be NON-NEGATIVE NUMBERS, though not necessarily integers. Again, zero is only an option if the underlying object might not exist. If the problem clearly assumes the existence and typical definition of an object, these quantities must be POSITIVE. Examples: - The side of a triangle must have a positive length. - The weight of a shipment of products must be positive in any unit. - The height of a person must be positive in any unit.
Stealth constraints: Non-countable items (negative)
Many other non-countable quantities are allowed to take on negative values. Examples: - The profit of a company. - The growth rate of a population. - The change in value of essentially any variable.
Heavy Long Division
Move the decimals of both the numerator and denominator, so that you're dealing with integers.
Train your Top-Down brain to be:
ORGANIZED in an overall approach to difficult problems. FAST at executing mechanical steps FLEXIBLE enough to abandon unpromising leads and try new paths
On "MUST be true" questions:
On "MUST be true" questions, we have to find something that is always true, not just sometimes true. If possible, try to find one case in which the statement is false; then you know that you can eliminate that statement or answer. There will be traps revolving around things that are true sometimes but not all the time - watch out for those traps!
Stealth constraints: Countable items
The number of countable items must be a NON-NEGATIVE INTEGER. Note that zero is only a possibility if it is possible for the items not to exist at all - if the problem clearly assumes that the items exist, then the number of items must be POSITIVE. For example, - Number of people - Number of yachts - NUmber of books
Is y² less than y?
The question is really asking: is y between 0 and 1, or is 0 < y < 1? If you get a question or statement about a number getting smaller when it's squared, then you know we're talking about fractions between 0 and 1.
Data Sufficiency: Be a contrarian
To avoid statement carryover and gain insight into the nature of a problem, you should deliberately try to violate one statement as you evaluate the other statement. This will make it much harder for you to make a faulty assumption that leads to the wrong answer. When evaluating individual statement, ACTIVELY trying to violate the other statement can help you see the full pattern or trick in the problem. you will be less likely to fall victim to STATEMENT CARRYOVER.
PS Strategy: Plug in numbers
Try this when concepts are especially complex or when conditions are placed on key inputs that are otherwise unspecified (e.g., n is a prime number). Two common categories of problems almost always allow you to pick numbers: - VICs problems - Fractions and Percents problems without specified amounts Another problem type would be problems that put specific conditions on inputs but do not give you exact numbers.
DS Strategy: Beware of inequalities
Whenever a Data Sufficiency problem involves inequality symbols, you should be especially careful - the GMAT loves to trick people with inequalities.
Data Sufficiency: Assume nothing
You must avoid assuming constraints that aren't actually given in the problem - particular assumptions that seem natural to make. Do not assume integers. Do not assume that any unknown is positive UNLESS IT IS STATED AS SUCH in the information given (or if the unknown counts physical things or measures some other positive-only quantity).
If you're looking for a 650+ score, be aware that your primary task is NOT to do as many problems as possible - really!
Your task is to learn as much as you can from each problem you do such that you can apply this knowledge to other problems in future. Knowing the basics and inserting yourself into the situation is time-consuming, but this is absolutely how you learn to recognize what to do, strip off disguises, avoid traps, and so on. When you start to get to the higher levels of the GMAT, the task becomes so much harder because of the way in which these problems are written, not just because of the content being tested.