Data Sufficiency
Tips and Tricks
&
The 2-Paths Strategy
Data Sufficiency: Course Goals
This course will help you to:
1. Realize that 'sufficiency' is everything !!!
2. Recognize the DS question 'type'
3. Take control of DS question stems
4. Understand the 'A/D – B/C/E Paths'
5. Recognize some of the most common traps
6. Work through some non-advanced examples
7. Gain confidence through further practice.
Data Sufficiency: Sufficiency is Everything!
The Quantitative section of GMAT is not really a 'math' test (so don't be afraid of GMAT math topics) but is a key part of the 'reasoning' assessment that business schools etc. want you to go through. This fact is particularly applicable to DS questions.
DS questions always have 4 parts:
1. The question stem (q-stem)
2. Statement (i)
3. Statement (ii)
4. 5 answer choices.
In the next slide, you will see that the 5 answer choices in DS questions are always the same.
Data Sufficiency: Sufficiency is Everything!
GOOD NEWS: You do not need to memorize or even use the information in these 5 answer choices.
You will now learn a simple system that is used by all our GMAT students.
Data Sufficiency:�� The Answer Choice A
As you can see, the answer choices talk a lot about sufficiency:
A. Statement (i) is sufficient alone to answer the question but statement (ii) is not.
Data Sufficiency:�� The Answer Choice B
As you can see, the answer choices talk a lot about sufficiency:
B. Statement (ii) is sufficient alone to answer the question but statement (i) is not.
Data Sufficiency:�� The Answer Choice C
As you can see, the answer choices talk a lot about sufficiency:
C. Both statements (i) and (ii) are required to answer the question.
Data Sufficiency:�� The Answer Choice D
As you can see, the answer choices talk a lot about sufficiency:
D. Both statements (i) and (ii) alone are sufficient to answer the question.
Data Sufficiency:�� The Answer Choice E
As you can see, the answer choices talk a lot about sufficiency:
E. The question cannot be answered either with statements (i) and (ii) alone or together.
Data Sufficiency�v.s.�Problem Solving
Unlike Problem Solving questions, DS questions are not really designed to test your ability to actually find the one correct 'solution' to a problem. Rather, your task is to identify whether enough information has been provided to answer the question.
There is a subtle difference between these two tasks, and we will now show you how to safely approach sufficiency problems.
Data Sufficiency:��The Two Paths
Almost every GMAT book and school will teach this system in one form or another.
There are various names for it, but we call it ‘The A/D or B/C/E Paths.
Start by taking a small piece of paper and write A/D on one side and B/C/E on the other.
You will understand shortly why you only work with one side, depending on the sufficiency of statement (i).
Data Sufficiency:��The 6 Steps of The Two Paths
Here are the 6 steps:
Step 1: Identify whether the q-stem is 'yes/no' or 'value’.
Step 2: Identify or introduce 'the variables’ and note their ‘limits’ if given.
Step 3: 'Take control' of the question.
Step 4: Analyze information from the q-stem and only statement (i).
Step 5: Analyze information from the q-stem and only statement (ii).
Step 6: IFF (if and only if) necessary, analyze information from the q-stem and both statements (i & ii) together.
Data Sufficiency:�� Step 1 of The Two Paths
Step 1: Identify whether the q-stem is 'yes/no' or 'value’.
GOOD NEWS: The question stem clearly indicates which type of question you face, so just look at the wording:
EG Value: “what”, “how much”, “which” etc.
EG Yes/No: “is”, “can”, “will” etc.
Data Sufficiency:�� Steps 2&3 of The Two Paths
Step 2: Identify or introduce 'the variables’ and note their ‘limits’ if given.
Step 3: 'Take control' of the question.
This is such an important early stage (Steps 2 & 3) which will help you avoid many mistakes and traps.
Yes, GMAT is full of traps and throughout this course we will show you how to recognize and avoid them.
Data Sufficiency:�� Steps 2&3 of The Two Paths
Step 2: Identify or introduce 'the variables’ and note their ‘limits’ if given.
Write down any variables and their limits given in the q-stem, and/or introduce variables in word problems; for example, let the original positive value = x
Step 3: 'Take control' of the question.
Do NOT look at the additional information yet but read the q-stem carefully and decide which area of math is involved, which formulas are needed, and which approach to use.
Data Sufficiency:��A/D or B/C/E��Step 4 of The Two Paths
Use statement (i) alone to choose either the 'A/D' path or the 'B/C/E' path.
You can never walk down both paths!
If the information from the q-stem and additional statement (i) is enough to answer the question, we are on the A/D path.
If not, we are on the B/C/E path.
Data Sufficiency:��A/D or B/C/E��Step 5 of The Two Paths
Use statement (ii) alone to progress along either the 'A/D' path or the 'B/C/E' path.
If statement (i) from Step 4 was sufficient, we are choosing between A and D.
Now, if statement (ii) is also sufficient alone, we choose D and move on to the next question.
However, if statement (ii) is not sufficient alone, we choose A and move on to the next question.
Data Sufficiency:��A/D or B/C/E��Step 5 of The Two Paths
Use statement (ii) alone to progress along either the 'A/D' path or the 'B/C/E' path.
If statement (i) from Step 4 was NOT sufficient, we are choosing among B, C, and E.
Now, if statement (ii) is sufficient alone, we choose B and move on to the next question.
However, if statement (ii) is not sufficient alone, we need to move on to Step 6.
Data Sufficiency:��Warning About Step 6!
A very common mistake is to combine the information from the 2 additional statements during earlier steps or after you choose answer A, D, or B.
Only do Step 6 if after Step 5 on the B/C/E path, neither statement alone was sufficient to answer the question.
We will show you soon how this trap is sometimes quite sneaky!
Data Sufficiency:��C or E?��Step 6
Combine and use the information from both statements (i) & (ii) to choose between C and E.
If neither statements (i) nor (ii) from Steps 4 & 5 were sufficient alone, we choose either C or E.
If statements (i) & (ii) combined are sufficient together, we choose C and move on to the next question.
If statements (i) & (ii) combined are NOT sufficient together, we choose E and move on to the next question.
Data Sufficiency:��‘Decision Points’
In the following sets of slides, we will walk you through some sets of worked examples grouped by the main math topics tested on GMAT.
You will see key 'decision points' that occur in each question, and you will realize that success in GMAT DS questions is all about making good decisions.
Your first good decision is to follow this process, step-by-step. Do not look for shortcuts yet. They will come later in the course.
Data Sufficiency:��Learn from Your Mistakes!
TIP: Any time you make a bad decision, analyze it, keep a record of it in your log, and move forward with confidence.
You will find this record of questions that caused you problems a great resource later in your GMAT journey.
Data Sufficiency:��Yes/No��Questions
Yes/No questions ask you to decide whether the answer to the q-stem is either 'yes' or 'no’. The word ‘either’ is vital to focus on.
Sufficiency in��Yes/No��Questions
Sufficiency in Type 1 DS questions is defined as: there is enough information provided to establish whether the answer is either 'yes' or 'no' to the problem presented in the q-stem.
Notice the 'or' in this definition! Both 'yes' and 'no’ is not an option for sufficiency, and no official GMAT DS question will provide a 'yes' answer for one statement and a 'no' for the other statement. All will become clear soon.
Insufficiency in��Yes/No��Questions
Insufficiency in Type 1 DS questions is defined as: there is not enough information provided to establish whether the answer is either 'yes' or 'no' to the problem presented in the q-stem.
TIP: The quickest way to reach the correct answer in yes/no questions is to look for both a 'yes' and a ‘no’.
Hunt as quickly as possible for both a ‘yes’ and a ‘no’…if you cannot find both, then the information is sufficient.
Data Sufficiency:��Value��Questions
Value questions ask you to establish whether there is enough information provided to find the value of a variable (or combination of variables) in the q-stem.
Sufficiency in��Value��Questions
Sufficiency in Type 2 questions is defined as: there is enough information provided to find the value or set of values (and only one possible value or set of values) of a variable (or combination of variables) in the problem presented in the q-stem.
Notice the 'and only one possible value or set of values' in this definition. No official GMAT DS question will provide different ‘correct’ solutions from each statement. All will become clear soon.
Insufficiency in��Value��Questions
Insufficiency in Type 2 DS questions is defined as: there is not enough information provided to find the value or set of values (and only one possible value or set of values) of a variable (or combination of variables) in the problem presented in the q-stem.
Data Sufficiency:�� Yes/No Example Question
We will now work through a basic yes/no question:
Is a > b ?
(i) a > 6
(ii) b > 5
TIP: It doesn't matter whether you are doing a low-level or a 700+ level DS question, the basic strategy for yes/no questions is always the same.
Data Sufficiency:��Value �Example Question
We will now work through a basic value question:
What is the value of integer x?
(i) x > 6
(ii) 5 < x < 8
TIP: It doesn't matter whether you are doing a low-level or a 700+ level DS question, the basic strategy for yes/no questions is always the same.
Data Sufficiency:��Traps & Time Pressure
The time limit adds pressure to your decision making, so traps become even more important to identify and avoid.
We will look at a few of the most common traps found in DS questions, and some common errors.
You can learn a lot from your mistakes, so remember to keep your log.
Data Sufficiency:��Difficulty Levels
No matter how good you are at math, GMAT will sometimes get the better of you.
In fact, the algorithm adapts the difficulty level in both the Verbal and Quantitative sections according to how well or badly you are doing.
The better you do, the harder and more valuable the questions become.
The worse you do, the easier and less valuable the questions become.
Data Sufficiency:�� The 'IFF' Error
You should only consider both statements together 'if and only if' neither statement is sufficient by itself.
Avoid the temptation to dive straight into DS questions and get all the information straight away.
Let's look at a simple example:
Is integer y a perfect square?
(i) √y = |√x| = √16
(ii) y > 0
Data Sufficiency:�� The 'IFF' Error
TIP: Unfortunately, this mistake can be made by the best of us if we are not concentrating properly.
So, as we tell all our students, do not take any shortcuts with DS questions.
It can be tempting to quickly look at both statements together, especially as you improve and feel more confident, but the test writers can structure information in cunning ways.
Data Sufficiency:�� The 'Forget-me-not' Error
Sometimes the information provided in statement (i) can be so beautiful, impressive or obvious that you will be unable to 'forget' it when you move to Step 5!
Let's look at a simple example:
Is z a prime number?
(i) z is positive
(ii) z4 = 16
Data Sufficiency:�� The �'Forget-me-not' Error
TIP: This trap is almost psychological because the nature of the human brain is to categorize information so as to remember it.
The secret is to have the utmost discipline in your approach to each DS question, taking no mind of the beguiling nature of what secrets you have learned from statement (i) and if necessary, cover statement (i) with your finger as you move down to statement (ii) during Step 5.
Data Sufficiency:��The 'Hidden-in-plain-sight' Trap
Very often in DS questions, the q-stem will disguise vital information, hiding it either within the math, or in plain sight. So, we push students to carefully analyze the q-stem.
Here is a not-so-subtle example:
What is the value of positive integer y?
(i) y2 + 3y – 28 = 0
(ii) y2 – 16 = 0
Data Sufficiency:��The 'Hidden-in-plain-sight' Trap
TIP: Always do Step 2 carefully, and do not rush it.
In this example question, remember that the q-stem gives a limit for the variable: y must be a positive integer – not a fraction or zero.
It is easy to rush over such information when you are under time pressure, so even though both statements indicate more than one value for y, only the positive solutions are valid.
Data Sufficiency:�� The 'False Assumption' Error
The reasoning involved in GMAT is particularly applicable to DS and Critical Reasoning.
One of the most common false assumption made on DS questions is that variables are odd or even integers. Unless you are explicitly given this information, the variable could be odd, even, a fraction, a decimal etc.
Example:
If x is even, is x + y odd?
(i) x ÷ y is even
(ii) 8y is even
Data Sufficiency:�� The 'False Assumption' Error
TIP: Again, the key to avoiding such an error is to maintain vigilance at all times.
As explained in the video, I have a set of numbers that I write down for such 'open-variable' questions. This forces me to consider that a variable could be positive, negative, zero, integer or fraction.
As more information is provided, the 'limits' of the variable may become clearer.
Data Sufficiency:��The 'Equivalence' Trap
Some DS questions tempt you to misuse a rule of thumb called the 'number of variables vs. number of equations' rule. As you will see in the following video, this rule can sometimes lead to a very annoying mistake.
If a and b are positive integers, how much greater is a than b?
(i) 7b – 11a = -1
(ii) 27.5a = 17.5b + 2.5
Data Sufficiency:��The 'Equivalence' Trap
Wow, such a frustrating mistake to make. I can tell you honestly, almost all GMAT takers have fallen into this trap under time pressure.
GOOD NEWS: As shown in the video, discipline yourself to always write any linear equation underneath another, with the same sequence of variables, and you should avoid this trap.
Data Sufficiency:��Number Properties
The first 3 steps in our DS system are closely connected.
As seen in the examples we've already covered, you often need to look at a problem from various angles (no pun intended).
GMAT expects you to know basic number theory (aka number properties), such as even/odd; positive/negative; prime numbers; remainder theory; divisibility rules; and inequalities involving absolute values.
Data Sufficiency:��Number Properties
We've already seen a few easy algebra examples, so let's dive into some DS examples involving strategies to cope with number properties.
GOOD NEWS: Although you do need to learn quite a few rules, there are approaches that make attacking these problems a bit easier and often much quicker.
Data Sufficiency:�� Number Properties �&� Limits of Variables
It is absolutely essential to spend time analyzing the information in the q-stem, because as the difficulty level increases, the 'way' the limits are given often becomes more difficult to identify, or easy to miss.
We will now look at 3 example questions, of increasing difficulty, to illustrate this point.
GOOD NEWS: Because number properties are so common in different types of GMAT Quantitative problems, you will have lots of practice at identifying the limits of variables.
Data Sufficiency:��Number Properties �&�Limits of Variables
Example 1: Is x a prime number?
(i) x has 3 distinct prime factors
(ii) 70 < x < 74
Data Sufficiency:��Number Properties �&�Limits of Variables
Example 2: If the integers p, q and r are even, is r > p?
(i) q > r
(ii) p2 < q2
Data Sufficiency:��Number Properties �&�Limits of Variables
Example 3: Is 2x2y + 3xz2 positive?
(i) y and z are negative integers
(ii) x = 1
Data Sufficiency:��Algebra
In DS problems that directly involve algebra, you will be expected to apply knowledge of standard topics such as commutative & associative properties; isolating a variable; solving simultaneous equations; factoring quadratic expressions; and the inevitable 'Classic Quadratics'. We love Classic Quadratics!.
Data Sufficiency:��Algebra
GOOD NEWS:
None of the algebra that comes up on the GMAT is advanced, and all the problems you face can be solved without a calculator.
We will explain some nice tricks, highlight common patterns, and walk you through a few alternative approaches.
Data Sufficiency:��Algebra ��Example 1
What is the value of (x + y)?
(i) (x + y)2 = 36
(ii) x and y are both positive integers
Data Sufficiency:��Algebra ��Example 2
For all real numbers p and q, the operation ж is defined by
p ж q = (p + 3)(q – z).
How many possible values does this operation have?
(i) p, q and z are all non-zero integers
(ii) p = 2q = 2z
Data Sufficiency:��Algebra ��Example 3
Does (abd + acd) ÷ 2 = 1
(i) b = c
(ii) abd = 1
Data Sufficiency:��Arithmetic
As with algebra, arithmetic is used by the writers of GMAT across most parts of the Quantitative section.
We will work through examples of each subtopic, such as fractions, inequalities, radicals and absolute values, in both the Data Sufficiency and Problem Solving Courses.
Data Sufficiency:��Picking �Numbers
Sometimes, the quickest and most efficient method of approaching 'sufficiency' problems is to pick numbers to plug in for the given variables.
As with the problems involving number properties, how you decide which values to use depends on the limits given in the q-stem.
We will practice how to decide whether to use prime numbers, negative integers, multiples of 100 etc.
Data Sufficiency:��Arithmetic��Example 1
If the product of s and t is non-zero,
and r = s ÷ t,
is ‘r’ a rational number?
(i) s = 21/2
(ii) t2 = 200
Data Sufficiency:��Arithmetic ��Example 2
If w is an even prime number, what is the value of r – s?
(i) s = (w + 3)2
(ii) r = (w - 3)2
Data Sufficiency:��Arithmetic ��Example 3
Is |a| - |b| > |b| - a ?
(i) a < 0
(ii) a > b
Data Sufficiency:��Number Properties’ Rules
The math topics associated with Number Properties are some of the most important rules you must learn as you prepare for GMAT Quantitative questions, especially DS.
You have already seen a few of these rules used in both the algebra and arithmetic examples, and each rule helps you better understand the world of math.
GMAT was designed to be a 'reasoning test’, and the rules of number properties are often used in the reasoning steps in DS problems.
Data Sufficiency:��Reasoning & Number Properties
Deciding whether we have enough information to make a decision is a fundamental form of reasoning.
To quote Sherlock Holmes, “In solving a problem of this sort, the grand thing is to be able to reason backwards”.
As mentioned previously, if we know the limits of variables, we can pick possible values according to the given properties.
Data Sufficiency:��Flashcards of Number Properties
TIP: Make flashcards, stick them to various walls in your favorite locations at home or work and walk from spot to spot - maybe stretch as you read them.
The change of poise and simple movement can improve your mood and helps the learning process. Don’t just sit for hours in one spot.
Data Sufficiency:��Number Properties��Example 1
If the remainder when ‘a’ is divided by 2 is 1, and the remainder when ‘a’ is divided by 5 is 3, what is the value of positive integer ‘a’?
(i) ‘a’ is a 2-digit number
(ii) ‘a’ is a multiple of 11
Data Sufficiency:��Number Properties��Example 2
If x and y are even integers, is z a prime number?
(i) z + 1 is odd
(ii) (x + z) ÷ y is a prime number
Data Sufficiency: Number��Properties��Example 3
Are the values of ‘a’ and ‘b’ equal?
(i) 2(a ÷ b) is negative
(ii) 3(a + b) is positive
Data Sufficiency:��Ratios & Proportions
GOOD NEWS: In terms of sufficiency, we can consider ratios, proportions, percentages, fractions, decimals and probabilities, to be pretty much the same thing.
We could in fact word the ratio ‘x to y’ as:
So we can create a relationship between variables x & y in any of these formats and then do the math as a fraction or decimal.
Data Sufficiency:��Ratios & Proportions
TIP: In the following 3 examples, you will notice that I switch to fractions asap.
I love fractions.
I like how I can simplify them, and how prime factorization makes this process really easy.
Data Sufficiency:��Ratios & Proportions��Example 1
If each member of the Good Old Father Time Over 40's Cricket Club is either a left-handed or right-handed batsman, and none is both, how many members are left-handed batsmen?
(i) The ratio of left- to right-handed batsmen is 1:9.
(ii) The cricket club has 30 members, but if 3 more left-handed only batsmen joined, the ratio of left- to right-handed batsmen would be 2:9.
Data Sufficiency:��Ratios & Proportions��Example 2
If c and d are prime, such that c < d, what is the value of the ratio c1/2 : d2 ?
(i) c+1 ÷ d-2 = 1 ÷ 3
(ii) c is even.
Data Sufficiency:��Ratios & Proportions��Example 3
Charlie usually proof reads a medical publication of ‘p’ standard pages at a total cost of ‘d’ dollars. At this usual price, how many dollars will he charge per page for a publication of ‘1.5 p’ standard pages, assuming no discount for the increase in number of standard pages?
(i) He charged (c + x) dollars per page for a medical publication of p + 15 pages at a total cost of (d + y) dollars.
(ii) He recently charged a repeat customer a rate two-thirds that of his usual rate, at a total cost of (d - y) = 3,720 dollars for a 372-page medical publication.
Data Sufficiency:��Statistics – Mean, Median, Mode, & Range.
Before we look at probability, let’s remind ourselves of a little bit of basic statistics:
The mean of a set is its average.
The median of a set is the 'middle' term or terms.
The mode of a set is the 'most seen' term.
The range of a set is the largest term minus the smallest term.
Data Sufficiency:��Statistics -��Variance & �Standard Deviation
The variance and standard deviation of a set are closely related and communicate the ‘spread’ of each element in the set from the mean.
GOOD NEWS: GMAT only rarely moves into the slightly more complicated worlds of variance and standard deviation, and unless you are faced with 700+ level questions, you will only be expected to know the general connection between them.
Data Sufficiency:��Statistics��Example 1
If x < y, and x < z, what is the median of x, y and z?
(i) y < v
(ii) z > v
Data Sufficiency:��Statistics��Example 2
Is the median of the consecutive even integers ‘m’ through ‘n’ greater than the mean of the consecutive odd integers ‘u’ through ‘v’?
(i) m < u
(ii) n > v
Data Sufficiency:��Statistics��Example 3
What is the average of the set of consecutive integers ‘u’ through ‘v’, if u < v?
(i) The sum of the middle 5 integers in the set is 105.
(ii) u and v are both prime, such that 40 < u + v < 44, and 18 < v – u < 21.
Data Sufficiency:��Probability
In terms of probability rules, GMAT expects you to know a few and to apply them.
You will come across the topic in questions focused on almost any of the other math topics.
Common examples include:
Data Sufficiency:��Probability
There are many ways to approach the topic, but GMAT rarely uses standard math notation or terms, such as U to represent 'sample space’, etc.
If you are mathematically minded, there is no harm at all in reviewing this aspect of set theory, but only for a higher level of understanding of the context that GMAT problems can involve.
Data Sufficiency:�� Probability Rules 1 & 2
The best approach is to integrate the information in the q-stem and statements i & ii into these 4 rules:
1. The probability of a 'desired event' = the number of ways the 'event' can occur ÷ the total number of possible 'events'
2. The probability of an 'event' occurring + the probability of the 'event' not occurring = 1
Data Sufficiency:�� Probability Rules 3 & 4
The best approach is to integrate the information in the q-stem and statements i & ii into these 4 rules:
3. The probability of an event AND another event both occurring = the product of their respective probabilities
NOTE: This assumes that the first event has already occurred.
4. The probability of one event OR another event occurring = the sum of the probabilities of each event occurring individually – the probability of both events occurring
Data Sufficiency: ��Probability ��Example 1
On Mondays, a restaurant offers a deal on any selected 3-course dinner, made up of a starter, a main course, and a dessert. What is the probability of a particular meal deal being chosen?
(i) There are twice as many main courses as starters.
(ii) There are 3 times as many main courses as desserts.
Data Sufficiency: ��Probability ��Example 2
The probability of teen sensation The Plastic Hero scoring a top 10 hit from his new album is y%. The probability of his favorite soccer team winning the league this season is x%. What is the probability of at least 1 of these events happening?
(i) x = y
(ii) 1 – x = 1 – y = 11%
Data Sufficiency: ��Probability ��Example 3
In a game of darts, if Alesia can hit the bull's-eye, which is worth 50 points, on average once in 5 throws, did she hit more than 2 bull's-eyes in her '1st to 100 points' game against Federico?
(i) Both players threw 7 darts each, and the bull's-eye was hit at least once.
(ii) The actual probability of Alesia hitting a bull's-eye in this game was between 0.2 and 0.4.
Data Sufficiency: ��Combinations & Permutations
Over the past few years, questions involving this area of math have become increasingly common in below advanced-level GMAT questions.
We find many students, especially those without a solid math background, are intimidated by such questions, and the wording can be deliberately confusing.
Data Sufficiency: ��Combinations & Permutations
There are various approaches to taking control of such questions, but basically you will want to decide whether to logically ‘count through’ the events outlined in the question, or to use a step-by-step approach to decide which formula to apply.
Data Sufficiency: ��Combinations & Permutations ��Example 1
In how many ways can the kitchen staff of Billy Bob's Posh Restaurant be positioned in the annual Christmas photo if the head chef, Billy, has to be sat in the middle seat of the single row of seats, with a female member of kitchen staff sat on either side of him.
(i) There is one more female kitchen staff than male, and 11 in total.
(ii) There are 5 male kitchen staff.
Data Sufficiency: ��Combinations & Permutations ��Example 2
A shop offers many types of beer, including craft beer. A mini-crate of 6 craft beers is to be selected as a gift for Uncle John. In how many ways can the selection be made if there must be at least 2 of Uncle John's favorite Blue Merman Smile craft beer included?
Assume each bottle is considered unique.
(i) There are 4 types of craft beer in the shop, and at least 2 bottles of each still available.
(ii) The Blue Merman Smile beer is the least available, with 4 left in stock.
Data Sufficiency: ��Combinations & Permutations ��Example 3
For a lady's 5-a-side soccer match, 5 players can be organized to wear one of the team's shirts with one different number from 11 through 15 on each.
How many different possible team arrangements can be made from a group of ‘x’ ladies if each number represents a specific position on the pitch?
(i) If there were x – 2 ladies to choose from, there could be 2,520 different team arrangements.
(ii) If the number on each shirt did not represent a specific position on the pitch, and any number could play anywhere on the pitch, there would be 126 possible team selections.
Data Sufficiency: ��Sequences
This is a fairly common sub-topic in GMAT DS and involves remembering the formulas for arithmetic and geometric sequences and some basic number properties.
In the following 3 examples, notice how the wording is sometimes designed to make the problems appear a bit more complex than they really are.
Data Sufficiency: ��Sequences
Many students experience a 'fear factor’ when reading the wording in some Quantitative questions. Guess what, this is part of the way the test writers try to keep your score around average.
GOOD NEWS: With familiarity comes confidence, and if English is not your native language, your vocabulary is going to be boosted as you prepare for your GMAT.
Data Sufficiency: ��Sequences & Series ��Example 1
In arithmetic sequence S, the 5th term is x, and the 4th term is y. What is the 1st term in the sequence?
(i) The 5th term in the sequence is 27½.
(ii) x – y = 7
Data Sufficiency: ��Sequences & Series ��Example 2
M is an infinite, geometric sequence, and term My is an integer > x.
If set L is comprised only of the elements of sequence M with a positive value below 100, is My part of set L?
(i) x is both a perfect square and a perfect cube, and My = x + 21.
(ii) 80 < My < 90
Data Sufficiency: ��Sequences & Series ��Example 3
Every time Lu shares a new blog post, it is shared to 'a' unique followers in her network. As soon as each of these followers receives the post, it is immediately shared with 'r' of each of these recipients' unique followers, who each shares the post with 'r' further unique followers. This process continues for a long time! If the 4th round of shares reaches 6,000 unique followers, none of whom has received the post more than once, how much greater is 'r’ than 'a'?
(i) After the 5th round of shares, the total number of unique followers reached was 60,000.
(ii) The 6th round of shares reached 600,000 unique followers.
Data Sufficiency: ��Word Problems
GMAT will offer you many Quantitative questions that use words to tell a 'story'. Such questions vary in length and are structured around most Quantitative topics. These stories can usually be broken down into manageable parts.
The trick is to extract the variables and identify not only the math required, but also the fastest and most effective approach to solve the problem.
Data Sufficiency:��Word Problems��DiRT Journey Example
We sub-categorize ‘DiRT journey’ questions into:
Round trips (to a destination and back)
Two objects travelling toward each other
Two objects travelling away from each other
Two objects travelling in same direction
Compass headings (N, S, E, W etc.)
Circular motion
Average speed.
These are covered in detail in the related Problem Solving examples, but here is an example to illustrate the DS approach.
Data Sufficiency:��Word Problems��DiRT Journey Example
Train A is traveling through a mountain tunnel from one direction while train B is traveling in the reverse direction on a parallel track. If at one moment the distance of track separating them is 1.5km, how fast is train B travelling, assuming no change in speed occurs before they pass?
(i) The rate of train B is 2/3 that of train A.
(ii) Train A travels for 24.5 seconds before passing train B.
Data Sufficiency:��Word Problem��Combined Work Rates
The key rule for such questions is the formula: Work = Rate x Time
The extension to this formula is shown above and considers the work done by more than one object, be they people or machines.
By replacing ‘Work’ with the unit 1, and dividing both sides by ‘Time’, we can then calculate the combined ‘Rate’ to complete 1 ‘job’.
Data Sufficiency:��Word Problem��Combined Work Rates
Example: In 1 hour, Scott can assemble x identical shelving units and Tim can assemble y of such units, where x and y are integers. If Tim takes x hours break during a 6-hour period, and Scott takes no break, will they have completely assembled at least 6y units during that time?
(i) 3 < x < y < 6
(ii) The average combined rate of assembly for Scott and Tim is 6 min 40 seconds.
Data Sufficiency:��Word Problem��Mixtures
Mixture problems are far easier to tackle in DS than in Problem Solving. Remember the principle of ‘number of variables vs number of equations’ and focus on ‘sufficiency’.
GOOD NEWS: An equivalent Problem Solving question would require more effort and time than does this DS question, giving more opportunities for mistakes. Sufficiency is everything!
Data Sufficiency:��Word Problem��Mixtures
Example: Anna made a fruit juice mix for herself and another for her guest. Her mix and her guest’s contained 30% and 50% grapefruit juice, respectively.
However, neither Anna nor her guest had time to drink any of their mixes before leaving, so Anna poured both mixes into one container and stored the resulting mix in her fridge.
If grapefruit juice constituted 42% of the mix in the fridge, how many ml of fruit juice mix did Anna make for herself?
(i) The mix in the fridge was a total of 300ml.
(I) Grapefruit juice constituted 126ml of her guest’s mix.
Data Sufficiency:��Word Problem��Ratios
Example: On a scale model of a certain passenger jet, 1 inch on the model’s surface represents 6 feet on the actual passenger jet’s surface. What is the approximate wingspan of the actual passenger jet?
(i) The wingspan of the model is 1/72nd the wingspan of the actual passenger jet.
(ii) The wingspan of a model that is 100% larger is approximately 67 inches.
Data Sufficiency:��Word Problem��Proportions
Example: The time it took for Bran and Freya, 2 window cleaners, to complete their task on a particular size of building was halved when a 3rd colleague, Hercules, joined them when they started to clean the windows on an identical, neighboring building. In how many hours could Hercules have done the whole task by himself, assuming all 3 work at a constant rate and all only stop twice for a 30 minute and 20 minute break before the task is complete?
(i) Bran can complete such a task by himself in 2/3rd the time it would take Freya by herself.
(ii) Working together, Hercules and Freya could complete the task in 4 hours, 33 minutes and 16 seconds, without the aid of Bran.
Data Sufficiency:��Word Problem��Percentages
A luxury car dealer keeps 12% of the gross profit margin made by the manufacturers of a certain luxury car. What amount, in dollars, does the dealer keep from the actual sale price of the car?
(i) The manufacturer received 95% of the actual sale price of the car.
(ii) The manufacturer’s suggested retail price was $85,000, and the actual sale price was 100% of this figure.
Data Sufficiency:��Word Problem��Number Properties
Number Properties is a fascinating area of math. Here are the fundamentals upon which many such GMAT questions are based:
Data Sufficiency:��Word Problem��Number Properties
Example: Is the product of integers x, y and z even?
(i) x & y are both evenly divisible by 8
(ii) z = 0
Data Sufficiency:��Word Problem��Coins
Example: Elvira and Gina have been discarding their loose change for a week in two separate jars. Elvira has discarded a total of 22 coins that are a mix of cents, nickels, dimes and quarters. Gina’s jar contains no nickels but has one more in total of the others 3 types. If they pool the contents of both jars, will they be able to pay for the $7.99, 2-for-1 pizza delivery they have ordered?
Gina has 3 times as many quarters as she does cents, and Elvira has the same number of cents as Gina has dimes.
Gina has half as many dimes as does Elvira, whose 12 quarters number twice her own dimes and four times her own cents.
Data Sufficiency:��Word Problem��Coins
GOOD NEWS: You will only be asked to convert between American and non-American weights and measures with an appropriate conversion rate provided in the question.
Data Sufficiency:��Word Problem��Relative Ages
Example: Mr Loopez was born on the same day but 41 years before Mr Santany, and today is their birthday.
How many years ago was Mr Loopez 60? Assume Mr Santany has only 1 wife.
(i) In 1 year, Mr Santany’s wife, Lana, will be half of Mr Loopez’s current age.
(ii) In 5 years Mr Santany will be his wife’s current age.
Data Sufficiency:��Word Problem��Relative Ages
Do not rush through such problems but carefully build algebraic equations and solve step by step.
TIP: Notice how we build the equations with a focus on the ‘=‘ sign…build on the balance that must exist by definition either side of the ‘=‘ and you will avoid the trap of adding to, subtracting from, dividing or multiplying the years of the wrong variable.
Data Sufficiency:��Word Problem��Interest Rates
Simple and compound interests are 2 topics that you need to know the formulas for. As you can see below, the compound version is a bit more complicated.
Data Sufficiency:��Word Problem��Interest Rates
Example: Star & Whippet's investment portfolio guarantees 6% interest compounded biannually on investments of, or over, $20,000, but offers only 3% on investments below that limit. Did Roger invest over $20,000?
Data Sufficiency:��Word Problem��Growth / Decay
On the GMAT, growth/decay questions relate to such issues as changes in a population, a specific rate of decay in science, and interest rates – the key to this example is to identify how a sequence develops.
Data Sufficiency:��Word Problem��Growth / Decay
Example: A certain giant branching coral grows at a constant rate every year. Assuming a constant rate of growth for the past 10 years, how fast does the coral grow?
(i) 4 years ago, the coral was n2 cm smaller than it is today.
(ii) 8 years ago, the coral was 2n2 cm smaller than it is today.
Data Sufficiency:��Word Problem��Averages
Example: A new team member joins MuskyRide’s marketing team to help promote the recently completed hyperloop linking Antarctica and Tierra del Fuego.
By how many dollars did the average wage increase due to the new team member, if the total monthly wage cost of the team was $444,000 prior to her joining?
(i) Before the new member joined, the team had 111 members, each paid exactly the same wage of $4,000 per month, due to the company’s on-going equal-pay-for-all policy.
(ii) The new team member increases the average pay by exactly 0%.
Data Sufficiency:��Word Problem��Basic Set Theory
Example: In a given year, the percentage of total applicants accepted to the most selective business school in Europe, in Asia, and in Australia, was 20%, 9% and 37%, respectively.
How many applicants applied to at least 2 of these business schools?
(i) 11% of those applicants who applied to 2 of these business schools also applied to the other.
(ii) Only 39 applicants applied to all three of these business schools, representing 2% of the total who applied to 2 or more of them.
Data Sufficiency:��Word Problem��Basic Set Theory
We recommend using either a Venn diagram or a standard table to better visualize the information contained in such word problems.
Data Sufficiency:��Word Problem��Probability
Example: A survey is taken of a group of MBA students in a particular faculty. The question asked how many smart devices they carry with them to class. If each student carries at least 1 such device, what is the probability that a student chosen at random carries only 1 such device?
(i) 68% of the students carry exactly 2 smart devices to class.
(ii) 32% of students carry either exactly 1 or more than 2 smart devices to class.
Data Sufficiency:��Word Problem��Permutations
Example: In a game of 9-ball pool, which utilizes only the first 9 numbered balls from the set, a certain trick shot is to be set up using the first ‘n’ of the numbered balls. If the order of the balls set up in the trick shot matters, and no ball can be used more than once, how many different ‘versions’ of the trick shot can be set up?
(i) If the pool player used ‘n+1’ balls for his trick shot, there would be an even number of possible versions.
(ii) If the pool player used ‘n’ balls for his first trick shot, but then 1 fewer ball for his second trick shot, the number of versions would suddenly not be a multiple of 10.
Data Sufficiency:��Word Problem��Combinations
Example: From a group of 12 actors, ‘r’ are chosen to perform a skit designed to inform the audience on gender rights. How many possible selections of ‘r’ actors can be made, assuming r >1?
(i) The resulting number of possible versions is even and a multiple of 3.
(ii) If r were 1 smaller, then r! would equal the sum of each r, (r-1) and (r-2).
Data Sufficiency:��Word Problem��Max / Min
Example: A moving company had 4 sizes of trucks that it’s customers reserved in the first quarter of a particular year, and each size was reserved at some point during that quarter. Of 153 individual customers of the moving company, 1/9th reserved a 22’ truck, while reserving a 26’ truck was the least common choice. What was the maximum possible number of reservations for the most-reserved size of truck/trucks?
(i) The 12’ & 16’ trucks were the most common choice during the quarter, with an equal number of customers reserving each.
(ii) The 26’ trucks were reserved an even number of times during the quarter.
Data Sufficiency:��Word Problem��Max / Min
‘Max/min’ questions may seem complicated, but the secret is to identify the variables and how they affect each other. Often, as one variable increases, there is the possibility that another will decrease, or not, depending on the desired outcome.
TIP: Many such questions involve set theory, so in such cases sketch a Venn diagram or a table to visualize the big picture and try values for the variables at the extreme ends of the possible range of limits.
Data Sufficiency:��Word Problem��Remainders
Example: The professor of a certain MBA course likes to have her students discuss case studies in different group sizes as follows:
CASE STUDY 1 – groups of 3
CASE STUDY 2 – groups of 4
CASE STUDY 3 – groups of 5
In each session, in order to maintain the desired group size, any left-over students are allocated to a separate ‘excess group’ led by the professor.
Assuming all ‘n’ of her students attend each of these case study sessions, how many possible values of ‘n’ exist, assuming ‘n’ must be less than 50?
(i) The session with groups of 5 had twice as many MBA students in the excess group as had the one with groups of 3.
(ii) The session with groups of 4 MBA students was the only one in which the professor constituted 50% of the excess group.