### Nov 13, 2014

## Data Sufficiency

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This article talks about one of the most dreaded areas in CAT, more specifically in GMAT.

DS questions consist of two parts:

1. Question

2. Data

Introduction: The beauty of data sufficiency lies in the process of getting the answer of the questions without actually solving it. The objective of data sufficiency problems is to check the fundamental clarity of the subject and in this process we have to understand that whether the given data is sufficient to find the answer or not.

Format of Data Sufficiency questions: The structure of data sufficiency is very simple and it clearly says that whether the data is sufficient to answer the question or not. The standard format of data sufficiency is.

Question ————————————-?

I- First statement

II- Second statement

Now we have to verify that the given two statements are sufficient to answer the question or not.

Approach forData sufficiency questions: Most of the students struggle in DS not because they lack the concepts but because of an unstructured approach. The trick here is that we need not necessarily have to find the answer of the question but to select the right options. So even if you get the answer, at times, there are incorrect answer options that are selected.. If we follow the right approach we can increase the chances of getting it right. Below is an indicative framework which must be strictly followed for every DS question.

To maximize the accuracy level for data sufficiency questions, there is a systematic three step approach that has to be followed.

First step: Always read the first statement and check whether the first statement alone is sufficient to answer the question or not.

Second step: Always read the second statement and check whether the second statement alone is sufficient to answer the question or not.

Third step: If the first statement and the second statement independently are not sufficient then combine both the statement and check whether the data is sufficient to answer the question or not. If it is sufficient, then we have an answer otherwise the answer is data insufficient.

How to select the answer from the given options:

To select the answer from the given choices understand the table which is given in the standard form.

1) Statement (I) ALONE is sufficient, but statement (II) alone is not sufficient.

2) Statement (I) ALONE is sufficient, but statement (II) alone is not sufficient.

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

4) EACH statement ALONE is sufficient.

5) Statement (I) and (II) TOGETHER are NOT sufficient to answer the question asked, and additional data is needed.

How to select the answer from the given options:

Options Statement I Statement II Both I and II

1 (Y) (N) ( )

2 (N) (Y) ( )

3 (N) (N) ( Y)

4 (Y) (Y) ( )

5 (N) (N) (Data Insufficient )

(Y) Statement alone sufficient (N) Statement alone is not sufficient

To select the answer from the given choices understand the table which is given above in the standard form.

Understanding 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 alone is not sufficient to give a UNIQUE solution.

Option 2: If the data/information of statement B alone is sufficient to give a UNIQUE solution of the question and data/information of statement A alone is not sufficient to give a UNIQUE solution.

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.

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

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.

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 Aman, Baman and Chaman together have $ 15000. How many rupees Chaman alone has? (A) Aman and Baman together have $. 10000

(B) Chaman has 40% of the total amount

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

From statement B, Rancho has 40% of 15000 = $ 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.

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 assumptions.

Alex and Rocky 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 Rocky is 20m/s

(B) Speed of Alex 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.

4.**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 answer. Hence we will mark option 5.

5. **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.

6.** No calculations:**

It is absolutely not required to do any kind of calculation we don’t have to find the exact answer, we are suppose to check whether the given data is sufficient to answer the question or not.

**Examples:**

1.What is the value of a, given a and b are both real numbers in base 10?

(A) a + b = 12

(B) a – b = 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.

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

(A) x^2 – 3x + 2 = 0

(B) x^3 – 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.

Few problems with application of above rules:

What is the value of x?

(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

Selwyn, Alex, Peter, Rocky and Sherry ran a 1000 metres race. Who won the race? (A) Rocky finished after Peter and Sherry, but before Selwyn and Alex.

(B) There are two person between Sherry and Selwyn.

From statement A: We cannot find unique answer. The winner is either Peter or Sherry. 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, therefore 5x + 3z = 19 putting in equation given in question

2y = 27-19 , therefore 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 or 5x + 2y = 18 putting in equation given in question statement implies 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

Brownie Points

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

• 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 wrong option.

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

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

**Practice Exercise **

Directions for Questions 1 to 25: Each question is followed by two statements, A and B. Answer each question using the following instructions:

Choose 1 if the question can be answered by using one of the statements alone but not by using the other statement alone.

Choose 2 if the question can be answered by using either of the statements alone.

Choose 3 if the question can be answered by using both statements together but not by either statement alone.

Choose 4 if the question cannot be answered on the basis of the two statements.

1. If x is an integer, is (54+27)/x also an integer?

A. 6 ≤ x ≤ 81

B. x is a multiple of 3.

2. Is x^2 greater than x?

A. x^2 is greater than 1.

B. x is greater than –1.

3. Are all of the numbers in a certain list of 15 numbers equal?

A. The sum of all of the numbers in the list is 60.

B. The sum of any 3 numbers in the list is 12.

4. If b is the product of three consecutive positive integers c, c + 1, and c + 2, is ‘b’ a multiple of 24?

A. b is a multiple of 3.

B. c is odd.

5. A raincoat and an umbrella cost a total of $53.50. What is the cost of the raincoat?

A. If the cost of the raincoat were to increase by 10 percent, the raincoat and the umbrella would cost a total of $58.00.

B. The cost of the raincoat is $2.50 more than 5 times the cost of the umbrella.

6. Material A costs Rs 3 per kilogram, and material B costs Rs 5 per kilogram. If 10 kilograms of material K consists of x kilograms of material A and y kilograms of material B, is x > y?

A. y > 4.

B. The cost of the 10 kilograms of material K is less than Rs 40.

7. What is the value of the integer n?

A. n(n + 2) = 15

B. (n + 2)^n = 125

8. In a school election, if each of the 900 voters voted for either Edith or Jose (but not both), what percent of the female voters in this election voted for Jose?

A. Eighty percent of the female voters voted for Edith.

B. Sixty percent of the male voters voted for Jose.

9. In a cricket match, the ‘man of the match’ award is given to the player scoring the highest number of runs. In case of a tie, the player (out of those locked in the tie) who has taken the higher number of catches is chosen. Even thereafter if there is a tie, the player (out of those locked in the tie) who has dropped fewer catches is selected. Aakash, Biplab, and Chirag who were contenders for the award dropped at least one catch each. Biplab dropped 2 catches more than Aakash did, scored 50, and took 2 catches. Chirag got two chances to catch and dropped both. Who was the ‘man of the match’?

A. Chirag made 15 runs less than both Aakash and Biplab.

B. The catches dropped by Biplab are 1 more than the catches taken by Aakash.

10. Four friends. A, B, C, and D got the top four ranks in a competitive examination, but A did not get the first, B did not get the second, C did not get the third, and D did not get the fourth rank. Who secured which rank?

A. Neither A nor D were among the first 2.

B. Neither B nor C was third or fourth.

11. The members of a local club contributed equally to pay Rs.600 towards a donation. How much did each one pay?

A. If there had been five fewer members, each one would have paid an additional Rs10.

B. There were at least 20 members in the club, and each one paid no more than Rs. 30.

12. A family has only one kid. The father says “after ‘n’ years, my age will be 4 times the age of my kid.” The mother says “after ‘n’ years, my age will be 3 times that of my kid.” What will be the combined ages of the parents after ‘n’ years?

A. The age difference between the parents is 10 years.

B. After ‘n’ years the kid is going to be twice as old as she is now.

13. What are the values of m and n?

A. n is an even integer, m is an odd integer, and m is greater than n.

B. Product of m and n is 30.

14. Is Country X’s GDP higher than country Y’s GDP?

A. GDPs of the countries X and Y have grown over the past five years at compounded annual rate of 5% and 6% respectively.

B. Five years ago, GDP of country X was higher than that of country Y.

15. What is the value of X?

A. X and Y are unequal even integers, less than 10, and X/Y is an odd integer.

B. X and Y are even integers, each less than 10, and product of X and Y is 12.

16. On a given day a boat ferried 1500 passengers across the river in twelve hours. How many round trips did it make?

A. The boat can carry two hundred passengers at any time.

B. It takes 40 minutes each way and 20 minutes of waiting time at each terminal.

17. A square is inscribed in a circle. What is the difference between the area of the circle and that of the square?

A. The diameter of the circle is 25√2 cm.

B. The side of the square is 25 cm.

18. Two friends, Ram and Gopal, bought apples from a wholesale dealer. How many apples did they buy?

A. Ram bought one-half the number of apples that Gopal bought.

B. The wholesale dealer had a stock of 500 apples.

19. Four candidates for an award obtain distinct scores in a test. Each of the four casts a vote to choose the winner of the award. The candidate who gets the largest number of votes wins the award. In case of a tie in the voting process, the candidate with the highest score wins the award. Who wins the award?

A. The candidate with top three scores each vote for the top scorer amongst the other three.

B. The candidate with the lowest score votes for the player with the second highest score.

20. Zakib spends 30% of his income on his children’s education, 20% on recreation and 10% on healthcare. The corresponding percentages for Supriyo are 40%. 25% and 13%. Who spends more on children’s education?

A. Zakib spends more on recreation than Supriyo.

B. Supriyo spends more on healthcare than Zakib.

21. Tarak is standing 2 steps to the left of a red mark and 3 steps to the right of a blue mark. He tosses a coin. If it comes up heads, he moves one step to the right; otherwise he moves one step to the left. He keeps doing this until he reaches one of the two marks, and then he stops. At which mark does he stop?

A. He stops after 21 coin tosses.

B. He obtains three more tails than heads.

22. In a class of 30 students, Rashmi secured the third rank among the girls while her brother Kumar studying in the same class secured the sixth rank in the whole class. Between the two who had a better overall rank?

A. Kumar was among the top 25 % of the boys merit list in the class in which 60% were boys.

B. There were three boys among the top five rank holders and three girls among the top ten rank holders.

23. Nandini paid for an article using currency notes of denominations Re. 1, Rs. 2, Rs. 5 and Rs. 10 using at least one note of each denomination. The total number of five and ten rupee notes used was one more than the total number of one and two rupee notes used. What was the price of the article?

A. Nandini used a total of 13 currency notes.

B. The price of the article was a multiple of Rs. 10.

24. Ravi spent less than Rs. 75 to buy one kilogram each of potato, onion, and gourd. Which one of the three vegetables bought was the costliest?

A. 2 kg potato and 1 kg gourd cost less than 1 kg potato and 2 kg gourd.

B. 1 kg potato and 2 kg onion together cost the same as 1 kg onion and 2 kg gourd.