this article is from totalgadha.com
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 22a 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.
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