Data Sufficiency Questions for CAT

this article is from totalgadha.com

This article comes from our TathaGat Pune team. Yes, TG is in Pune. How many of you MBA aspirants knew that TathaGat- TotalGadha's classroom program- has a center in Pune also besides Delhi? Well, we do. And it is headed by an excellent team of exprienced CAT trainers who are MBAs themselves. This article comes from Rakesh Vishwakarma, our Quant wizard at TG Pune. An engineering graduate from NIT, an MBA in marketing from PUMBA, and an IIM-I call-getter from the last year CAT, Rakesh has always been passionate about teaching. Prior to teaching CAT aspirants, he was teaching engineering aspirants for regional and national level engineering entrance exams.

Do solve the exercise given at the end. You might be surprised by some of the answers but don’t worry, we shall discuss each and every question for the benefit of you guys. – Total Gadha

DATA Sufficiency has been a critical area of testing the problem analysing skills. Yes! you read it right. It is about analysing the problem NOT Solving it.

In GMAT, Data sufficiency questions account for about 1/3, i.e. 12 to 13, of the questions in the quantitative section. These questions are about working out what is the necessary and sufficient data to be able to answer a question- which is very different from the usual problems solving questions you have faced.

Let’s demystify the myth, mystery and enigma of DATA Sufficiency.

DATA Sufficiency Question kaisa hota hai?

DS questions consist of two parts:

1. Question Statement

2. Data statements

After going through the problem and the two statements, we are presented with some options:

1) Statement (A) ALONE is sufficient, but statement (B) alone is not sufficient.

2) Statement (B) ALONE is sufficient, but statement (A) alone is not sufficient.

3) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

4) EACH statement ALONE is sufficient.

5) Statement (A) and (B) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Meaning of the options:

Option 1: If the data/information of statement A is sufficient to give a UNIQUE solution of the question AND data/information of statement B is not sufficient to give a UNIQUE solution, then we will mark option 1.

Option 2: If the data/information of statement B is sufficient to give a UNIQUE solution of the question AND data/information of statement A is not sufficient to give a UNIQUE solution, then we will mark option 2.

Option 3: If the data/information of statement A alone is not sufficient to give a UNIQUE solution of the question, nor is statement B able to do so, but if we combine the data of both the statements together and get a UNIQUE solution, then this means that the statements alone are insufficient but together are sufficient. Hence we will mark option 3.

Option 4: If from statement A alone we are getting a unique solution and similarly we are getting a unique solution independently from statement B (while using data from statement B alone, we do not have to assume/use any data/information from statement A), we will mark option 4.

Option 5: If we are not getting a unique solution even after combining the data of both the statements, then it means that the data statements given are insufficient and more information is needed to get a unique solution. We will mark option 5.

Let us understand with an example:

What is the value of X, given X and Y are both real numbers in base 10?

(A) X + Y = 12

(B) X - Y = 5

Here we have been asked about the value of x. We have to find whether the statements A and B are sufficient to give the answer. From statement A alone we can’t find the unique value of x, nor with statement B alone. But if we combine both the statements we get a unique value of x by solving the equations. Hence here both the statements are necessary to give the answer.

Option 3.

Quest for a Unique Answer

The objective of a DS question is to find a unique answer i.e. the data statement/s is sufficient only if we are getting unique answers. Let us understand this with the help of another example

What is the value of x, given x is a real number?

(A) x² - 3x + 2 = 0

(B) x³ - 27 = 0

Answer: From statement A, by solving the quadratic equation, we get x = 1 and 2. Still statement A is NOT sufficient because it does not give a unique answer. On the other hand, from statement B alone we are getting x = 3 which is the only real solution of this equation. Hence from statement B alone we are getting UNIQUE solution therefore we will mark

option 2.

NOTE: For a statement to be sufficient it must give a unique answer, else it is not sufficient.

Approach for DS questions

Most of the students struggle in DS not because they lack the concepts but because of a faulty approach. The trick here is that we do not have to find the answer of the question but to select the right options. So even you get the answer there is no guaranty that you will mark the right option. If we follow the right approach we can increase the chances of getting it right. Below are the steps which must be strictly followed for every DS questions.

STEP 1: Always read the statement A first and check whether this statement alone is sufficient to provide a UNIQUE solution.

STEP2: Read statement B alone. While using data from statement B, do not consider any data/information from statement A.

STEP3: If you are not getting solution from either of the statements then combine the data of both the statements and analyse for the sufficiency of data statements for a UNIQUE solution.

Golden Rules for Solving DS Questions

1. ‘NO’ is as good as ‘YES’:

Many a question like 'is x > y?', 'is x an integer?', 'Is triangle ABC right-angled?' etc can have either a definite YES or a definite NO answer and both are acceptable i.e. sufficiently providing a Unique solution. Let us understand with an example.

Is x prime?

(A) X + Y = 20

(B) X = 12

Answer: Following the golden steps, statement A does not provide a unique answer. Hence statement A is ruled out. Now let’s analyse statement B. Here x = 12, therefore x is composite but still we are getting a unique and definite answer of the question that x is NOT prime. Hence, statement B alone is sufficient and we will mark option 2.

2. Different but unique answer from both the statements alone is acceptable:

This is in sync with objective of DS (quest for a UNIQUE answer) that one should examine each statement separately at first. It can happen that both the statements are giving unique answers to the question although the two answers might be different.

Three idiots Farhan, Raju and Rancho together have Rs. 15000. How many rupees Rancho alone has?

(A) Farhan and Raju together have Rs. 10000

(B) Rancho has 40% of the total amount

Answer: From statement A, It is clear that rancho has Rs. 5000. Hence statement A alone is sufficient.

From statement B, Rancho has 40% of 15000 = Rs. 6000 with him. Here also we are getting unique solution from statement B alone although the answers from both the statements are different but since we are getting unique answers from both the statement alone hence both the statement alone are sufficient to answer the question. We will still mark option 4 in such cases. Let’s have an example:

Is x prime?

(A) X = 17

(B) X = 12

It can also happen that in a 'yes/no' sort of question one statement is giving a definite 'no' while the other statement is giving a definite 'yes'. That means both the statements are able to answer the question alone and we will mark option 4 as our choice.

3. Making assumptions is prohibited:

Do not use your own information or make assumptions to answer the question. Do not assume any information about the properties of numbers, geometrical figures etc. unless it is given in the statements. This is the most obvious reason for getting DS question wrong. Remember that DS is about checking the sufficiency of data statements NOT about getting the solution of the question statement. Many data sufficiency questions are made with an intention to lead students into a trap of assumption.

Manish and Rakesh are running on a circular track of 1200 meters. Both started together from the same position on a track with different but uniform speeds. When will they meet for the first time after they start?

(A) Speed of Rakesh is 20m/s

(B) Speed of Manish is 30m/s

Answer: It is clear that both the statements alone can’t give the answer. If you combine both the statements and get a unique answer, you are making an assumption (either both running in same direction or in opposite direction). In the question statement nothing has been specified about the direction in which both are running. Hence even after combining the answer we are NOT getting a unique solution (taking both the directions separately we are getting different answers)

We will mark option 5.

Don’t believe in geometrical figures: In many of the geometrical questions in DS the figure is given along with the question statement, often the figure is not drawn on scale. It’s a trap to push you to make extra assumptions.

Example:

What is the length of BD (in the adjacent figure), if

(A) AD = 10

(B) DC = 10

Answer:  From the figure it appears that ABCD is either a rectangle or square. From statements A and B two sides are equal. Also angle D = 90 degrees.

Now there is the highest possibility that from the data above and the figure we are tempted to assume ABCD is a square. But since nothing has been specified in either of the question statement nor we can conclude from the data statement about what type of quadrilateral ABCD is, the data statements are insufficient to give conclusive a answer. Hence we will mark option 5.

4. Unless very sure, always try to solve the question till the end:

Although it is a common belief that one need not solve a data sufficiency question completely, it is not a good strategy to assume prematurely that a statement will give an answer; we should solve the question till the end and check. In case of equations, logarithms, geometry etc. we should especially solve and check if we're getting a unique answer.

Few problems with application of above rules:

Let p(x) = x² + 40. Then for any two positive integers i and j where i > j, is p(i) + p(j) a composite number?

(A) p(i) – p(j) is not a composite number

(B) p(2i) + p(2j) is a composite number

From statement A: p(i) – p(j) is not a composite number

--> i² - j² is a prime as i, j are positive integers and i > j, (i² - j²) can’t be 1

--> (i + j)(i - j) is prime so i – j = 1

Let p be the prime so i = (p+1)/2 and j = (p-1)/2

clearly p is not 2 hence all p is odd

p(i) + p(j) = 80 + (p² + 1)/2

now p² = 6k + 1, therefore p(i) + p(j) = 80 + (p²+1)/2 becomes

80 + (6k + 2)/2 = 81 + 3k = 3(27+k) so not a prime. Hence question can be answered from statement A alone.

From statement B: p(2i) + p(2j) is a composite number

--> 4(i² + j² + 20) is composite, now i and j can be anything hence can’t make any conclusions from B alone.

Option 1.

What is the value of x?   (CAT 2001)

(A) X and Y are unequal positive even numbers, less than 10 and x/y is an odd integer.

(B) X and Y are positive even numbers, each less than 10, and product of x and y is 12.

From statement A:

We are getting x = 6 and y = 2. Hence statement A alone is sufficient.  

From statement B alone:

12 = 1 × 12

   = 2 × 6

   = 3 × 4

Since both x and y are even and less than 10, the only values satisfying are (6, 2) and (2, 6)

Therefore x is either 2 or 6. No unique solution from statement B.

 Option 1

'n' is a natural number. State whether n (n² - 1) is divisible by 24.

(A) 3 divides 'n' completely without leaving any remainder.

(B) 'n' is odd.

From Statement A: n is a multiple of 3.

Now, say if we take n = 3, the expression is divisible, but in case, we put n = 6 or 12, then the expression is not divisible by 24. Hence statement A alone is insufficient.

From statement B: n is odd.

Now, if we put any odd value in place of n, we find that the expression is divisible by 24. Hence option B alone is sufficient.

Option 2

Kamal, Manish, Rajat, Rakesh and Sanjeev ran a 1000 metres race. Who won the race?

(A) Rakesh finished after Rajat and Sanjeev, but before Kamal and Manish.

(B) There are two person between Sanjeev and Kamal.

From statement A: We cannot find unique answer. The winner is either Rajat or SAnjeev.

From statement B: clear that no conclusion can be made.

Even by combining both the statements no conclusive solution hence

Option 5

Is x² – y² even? (both x and y are positive integers)

(A) x + y is odd

(B) x – y is odd

From statement A: If x + y is odd and given that x & y are positive integers, we can say that for x + y to be odd one of x or y is odd and other is even. In either case x² – y² will be odd. Hence statement A alone is sufficient.

For statement B same logic applies, hence statement B alone is sufficient

Option 4

What is the value of 2^2a where a is a positive integer less than 1000.

(A) a + 1 is a cube

(B) a - 1 is a square

It is clear that from both the statements alone we can’t find the unique solution, let’s combine both the statements; we have

a - 1 < a < a + 1. Where a - 1 is square and a + 1 is cube. The perfect squares less than 1000 are:

1, 4, 9, 16, 25, 36, …, 961

And perfect cubes are:

1, 8, 27, 64, 125, 216, 343, 512, 729.

From close observation to both the data we can easily find that the only value of a is 26. (25 is perfect square and 27 is perfect cube)

Option 3

If 5x + 2y + 3z = 27 and x, y, z are natural number. What is the value of x + y + z?

(A) 12z + 20x = 76

(B) 15x + 6y = 54

From statement A: 12z + 20x = 76

--> 5x + 3z = 19 putting in equation given in question statement

--> 2y = 27-19

--> y = 4. Also from 5x + 3z = 19 substituting x as 1, 2, 3 we get

for x = 1, z = 14/3

for x = 2, z = 3

for x = 3, z = 4/3

Since x, y, z are natural number therefore only possibly value is x = 2 and z = 3

Hence x + y + z = 2 + 4 + 3 = 9.

Therefore statement A alone is sufficient.

From statement B: 15x + 6y = 54

--> 5x + 2y = 18 putting in equation given in question statement

--> z = 3. Also from 5x + 2y = 18 substituting x as 1,2,3 we get

for x = 1, y = 13/2

for x = 2, y = 4

for x = 3, y = 3/2

Again as in statement A the only possible value of x = 2 and y =4.

Therefore x + y + z = 2 + 4 + 3 = 9.

Hence statement B alone is also sufficient to give answer.

Option 4

Last Tips for DS questions:

1. Be thorough with basics of Number Theory and geometry. This is the favourite area for DS.

2. Always follow the steps to solve DS. Never try to combine the statements at the first instance, you may get the answer BUT will mark wring option .

3. Read the directions given before solving the question. It may happen that the order of marking option is different.

4. Always keep in mind that we have to get UNIQUE solution.

Of anchor questions, leaks and Prometric’s plan for a glitch-free CAT 2010

There will be little change in the base process used behind the creation and evaluation of the Common Admissions Test (CAT) 2010 compared to CAT 2009. Prometric, the testing vendor commissioned by the Indian Institutes of Management (IIMs) for conducting the test will continue to use the “psychometric process to ensure that the CAT is valid, reliable and fair.”

Speaking to mediapersons in New Delhi last week, Prometric’s Vice President of Test Development Services Stephen Williams said that he was happy with the content creation and evaluation of CAT 2009. “It went exactly as we would have liked it to be,” he said.

Prometric India’s Managing Director Soumitra Roy was quick to clarify that Mr Williams was speaking only about the academic content and evaluation aspect of CAT 2009 (and not the hardware failures and virus attacks).



Much of what Mr Williams told journalists about the processes and standards used in creation and evaluation of the CAT question papers has been shared in the public domain before. I will summarize the highlights and some points of interest captured during the interaction.



In summary, during the test development phase, “Prometric works closely with IIM professors along with specially trained subject matter experts from other well-regarded Indian universities. Each exam question is written, edited and reviewed in an iterative process. All modifications and approvals are tracked electronically in an audit trail.”



“After the tests are administered, the raw scores are calculated on the basis of a +3 for a correct answer, a -1 for a wrong answer while un-attempted questions are ignored. After equating the scores (read below), they are linearly scaled to a 0-450 range before being presented to the candidates as test results.”



Anchor questions, Cloned questions and question leakages



“Each question paper has four ‘anchor questions’, which are used to adjust differences in difficulty between different question papers. Two of these questions would have appeared in the question paper of a previous slot, and the other two would appear in a consequent slot. All question papers are thus linked together in a sequential chain, the two anchor questions forming the links.” (Mr Williams refused to divulge whether the anchor questions were shared between question papers of consecutive slots.)



The performance of test-takers in these anchor questions is measured to establish a common metric of difficulty between two question papers and then adjust the raw scores accordingly.



So in CAT 2009, which was held over 30 slots (3 slots per day for 10 days), there would have been 60 anchor questions in total.



Mr Williams said that a few more questions similar in nature but different in form – called ‘cloned questions’ – were repeated across slots. A simplistic example is an algebraic problem in multiple versions, each version having a different set of substitution values. In reality, the differences between the versions would be a lot less apparent, said Mr Williams.



Anchor questions and cloned questions have a lot to do with the controversy over the ‘leakage’ of CAT 2009 questions on anonymous blogs, Orkut communities and coaching institute channels. For any candidate to gain an advantage in a future test slot, it is important that the leaked questions fall in the anchor or cloned categories.



Mr Williams played down the leakages. According to him, people tend to have nervous mindsets during a test and their ability to remember questions and reproduce them accurately a few hours later is overrated. “We went through the blogs and Orkut communities that were accused of sharing questions and found that very few of them had gotten it right. There was only a perception that the questions on the blogs were the same as those in the test, but they were not,” he said.



Not all candidates are exactly ‘nervous’, though. Test-preparation institutes, for example, were regularly sending ‘proxy candidates’ (teachers, content developers) to CAT 2009 slots with the intentional purpose of memorizing questions and sharing them with their students. These questions were being shared clandestinely in classrooms and secret email lists with test-takers in later slots.



Mr Williams’ defended this with the explanation that such concerted activity had not worked because “if that were to happen, then statistically we would have noticed strange patterns with the answering of cloned questions which we didn’t.”



Mathematically, the extent of the advantage gained by being privy to an anchor question is an interesting probability problem (anyone?).



CAT 2010 will be held in 32 cities and over fewer centers



The IIMs will release the CAT 2010 advertisement in the end of August, said Mr Roy. Prometric plans to use fewer testing centers this year after its disastrous tryst with India’s hardware infrastructure during CAT 2009.



“We will only choose test centers that adhere to the norms created by us for ensuring infrastructure quality,” he said.



As the testing window will also be longer (more like a month), there will clearly be more number of slots, more number of distinct question papers and therefore a larger pool of questions.

The Graduate Management Admission Test (GMAT), used for MBA admissions by a vast majority of business schools across the world will add a new section that will test advanced reasoning skills, starting June 2012.

The new section, known as ‘Integrated Reasoning’, will feature questions that will test skills of data analysis and interpretation, the ability to form relationships between information from multiple sources and form judgments.



Even though the ‘next generation’ GMAT will add another section, the total duration of the exam will remain the same at 3.5 hours (excluding the breaks). However, the Analytical Writing section will have only one essay instead of the two in the present avatar of the exam.



The GMAT exam’s verbal and quantitative sections however, will not change. As a result, when the new section is introduced in June 2012, tests will be scored on the same 200–800 scale used today. Test takers will receive a separate score for the essay — as they do now — and another distinct score on the new integrated reasoning section. The new score would thus look like something on the lines of 800/6/X, where X would be the score of the new Integrated Reasoning section.



“The new test will be launched on June 4, 2012 at all locations worldwide and registrations will begin six months prior, as they do at present,” Ashok Sarathy , Vice President of the GMAT Program told PaGaLGuY over telephone from the US. He added that there will be no change in the price of appearing for the GMAT as a result of the test revamp.



Those joining the classes of 2013-15 at b-schools worldwide would thus be the first batch to receive admits based on the new GMAT.



The Integrated Reasoning section



According to Mr Sarathy, the new section was added after a survey of business school faculty across the world including India about the additional skills they would like tested in the GMAT.

The section will feature questions that would require test-takers to use information from multiple sources, such as charts, graphs and spreadsheets to analyze information, draw conclusions and discern relationships between data points, just as they must do in business school, he added.

A sample Integrated Reasoning question released by GMAC consists of an interactive spreadsheet listing traffic data at 21 airports across the world. Test-takers can sort columns of the spreadsheet to interpret the data from different viewpoints. Questions that follow ask the test-taker to examine a few statements based on the data and determine which ones are true. Questions may also have more than one true statements. Other types of questions would test the ability to assess the reason for, or the likelihood of certain outcomes. Watch the video below or here for a demo released by GMAC

 

Mr Sarathy said that there would also be audio questions in the section, which will test the ability to listen to complex spoken material and form judgments, strengthen and weaken an argument or separate and convert data between arguments.

According to Dave Wilson, president and CEO of GMAC, “The new integrated reasoning section of the GMAT will be a microcosm of today’s b-school classroom. These questions will provide critical intelligence to schools about the ability of prospective students to make sound decisions by evaluating, assimilating or extrapolating data.”

Score scale, number of questions



GMAC has as yet not decided the number of questions in or the scale it will use to score the Integrated Reasoning section. “We are currently in pilot mode and will test the questions with 3,000 more students to determine how many to include in the section and how much of a question bank to create,” Mr Sarathy told PaGaLGuY.



On the score scale for the new section he said, “Once we finalize the questions, we will go to admission directors and ask them to define the scale. Since they have to use the scores, they will know best what scale to use for differentiating the candidates.”



Effect on the Analytical Writing section



In the present format, the Analytical Writing section requires the test-taker to write two essays, one on the analysis of an issue and the other on the analysis of an argument. Post-revamped launch, the section would shrink to just one essay, which could either be on the analysis of an issue or of an argument.

“We have been researching on the scores of these essays and admissions directors have said that performance in the two essays is correlated,” said Mr Sarathy, “so we decided to do away with one.”

Managing the launch



In the 2012 admission season, b-schools would thus receive two types of scores — from the tests taken before the new GMAT launch and those from the new format. “The verbal, quantitative and analytical writing scores would be common to both the old and new GMAT scores. After June 2012, schools will have one additional score component. It is upto them to decide how they will compare the two types of scores,” said Mr Sarathy.



In all likelihood, there would be a rush to take the GMAT in its old format in the months preceding June 2012. GMAC on its part will make the new GMAT sample test material available on its website. “We will also be working closely with test preparation companies around the world to make sure that students are as prepared as possible to take the new format,” said Mr Sarathy.

GMAT vs GRE

In recent years, the GMAT exam has had a growing rival in the Graduate Record Examination (GRE), traditionally used for applying to programs in arts and sciences in the US. The upgrade received by GMAT will help it in strengthening its competitive edge against the GRE.

GRE, though is giving a good fight. It has in the past couple of years convinced some big name b-schools such as Tuck-Dartmouth, Stanford, Wharton and others to accept the GRE score for their MBA admissions. With a revised b-school-focused version of GRE set to launch in August 2011, coupled with their ‘Personal Potential Index’ offering, GRE is aggressively pursuing GMAT’s traditional stronghold.

IIMA to set up additional campus in Hyderabad

The Indian Institute of Management (IIM), Ahmedabad is looking at setting up a campus at Hyderabad to offer executive education. In a press notification dated June 29, the institute spokesperson said, “At the instance of the Government of Andhra Pradesh, the Indian Institute of Management, Ahmedabad is engaged in discussions with the Government of Andhra Pradesh for setting up an IIMA campus in Hyderabad.”

The school has recommended the AP government to allocate 150 acres of land and Rs 100 crores in funds for establishing the campus. The recommended area is even larger than the IIM’s original campus at Ahmedabad, which is spread over a total of 116 acres.



“The Institute plans to begin operations by offering Executive Education Programs and subsequently it may explore the possibility of starting academic programs on the campus. The Institute would attempt to link the campus at Ahmedabad with the Hyderabad campus through use of technology, for seamless operations between the two campuses,” added the press statement.

IIMA will be the second IIM to set up an additional campus, the first being IIM Lucknow which set up a satellite campus in Noida in 2005 to offer part-time and executive programs.

The Hyderabad campus materializes as part of Andhra Pradesh’s Late Chief Minister YS Rajasekhara Reddy’s agenda to make Hyderabad an education hub. During his regime, he had aggressively wooed institutions such as IIM Bangalore, Institute of Management Technology (IMT) Ghaziabad, NMIMS Mumbai and Indian Institute of Science (IISc) Bangalore to set up campuses in the state. Of these, IMT and NMIMS had accepted the invitations and received land allocations, while IIM Bangalore had declined the offer citing faculty constraints. While IIMA had received the offer more than a year ago, the institute had held back on a commitment due to the political turmoil in Andhra after YS Rajasekhara Reddy’s death followed by the Telengana protests.