CAT Data Sufficiency Questions

No. of athletes which are top academic performers: 10
According to statement a) Sixty per cent of the top academic performers were not athletes  so 40% of top academic performers are athletes which we know is 10. So total no. of top academic performers are 25. 
Hence statement a is alone to answer the question.
while we cant deduce anything from statement b .

From the information given in the question, D has a better rank than E and B has a better rank than C. Also, C = 4 or 5.
From statement A, A = 5 => C = 4. But we do not know if D or B is number 1.
From statement B, B = 3 or 4. Again, we cannot determine the number 1 rank precisely.
By using both the statements together, we know C = 4, A = 5, B = 3 and D has a better rank than E.
So, D = 1 and E = 2. Therefore, the question can be answered by using both the statements together.
Hence, option D.

Let the total number of employees be 100k.
Number of males = 30k. Number of females = 70k.
Number of female engineers = 7k.
From statement A, number of male engineers = 25k - 7k = 18k.
From statement B, number of male engineers = (6/5)*7k.
Therefore, percentage of male engineers can be calculated.
So, the question can be answered by either statement alone.

From statement A, we know that the club scored four goals in the second half but we do not know the number of goals scored by the opposition. So, we cannot answer the question.
From statement B, we know that the opposition scored 4 goals in the match but we do not know the number of goals scored by Mahindra and Mahindra club. So, we cannot answer the question.
Using both the statements, we know that the opposition scored 4 goals in the match and Mahindra and Mahindra scored 4 goals in the second half. But, we do not know if the score was 3-0 or 4-1 at the end of first half. If it was 4-1, then Mahindra and Mahindra would win the match after the second half. But if the score was 3-0 at the end of first half, the match would end in a draw after the second half.
Therefore, the question cannot be answered even by using both the statements together.

Let w1 and w2 be average of both the groups respectively.

Since average of whole class is 45. $$\frac{50*w1+50*w2}{100} = 45 $$ => w1 + w2 = 90

And if we consider case A we have w2-w1 = 1. From the two equations average weight can be found out.

Further using the equations deduced from given condition which are : $$\frac{50*w1-l+h}{50} = w2$$ and $$\frac{50*w2+l-h}{50} = w1$$ where l and h is weight of poonam and deepak respectively. However, both of the above equations are effectively the same. Hence we have two equations and 3 unknowns, thus we cannot find the weights by using statement I alone.

If we consider statement B, then we can get another equation. So we will have 3 equations and then we can solve them for the 3 variables.
So both the equations are required to answer the question.

Using statement A alone, we can determine the inner volume of the tank to be at least 4/3 $$\pi$$ (125-64) = 256 $$m^3$$.
But we do not know if it is more than 400 $$m^3$$. So, using statement A alone, we cannot determine the answer.
Using statement B alone, volume of tank = 30000 kg / 3gm/cc = $$10^7 * 10^{-6}$$  = 10 $$m^3$$
So, inner volume = 4/3 $$\pi$$ 125 - 10 > 400 $$m^3$$
So, using statement B alone, we can answer the question

We know that the $$x^2 + y^2 + z^2$$ will be minimum when x = y = z = 89/3
Since that is not an integer, we can consider integer values that are closest to 89/3
Let x = y = 30 and z = 29 => $$x^2 + y^2 + z^2$$ = 2641
If x = y = 29 and z = 31,$$x^2 + y^2 + z^2$$ = 2643
So, the minimum is 2641
So, the question can be answered using statment 1 alone
Using statement 2 alone, we cannot answer the question.
Option a)

If the side of the square is x cm, then the maximum length of OM is $$\sqrt2x$$
The minimum length of OL is x.
So, OM can never be 2 times OL
So, using statement A alone, we can conclude that Rahim is unable to draw the square.
Using statement B alone, we cannot answer the question.
So, option a) is the correct answer.

We have to find out if 3Z > 4S => Z > 1.33S

According to statement 1, Z/5 > S/4 => Z > 1.25S

According to statement 2, 13S > 10Z => Z < 1.3S

So, using statement 2 alone, we can answer the question

Let the persons be A,B,C and D in order of the scores tha is, A scores the highest and D the least.

Statement A: A votes for B , B and C votes for A
Even if D votes for B,then also A wins as A has higher score as compared to B.
A alone is sufficient.

StatementB: D votes for B 
Nothing can be inferred from this statement.

We know that 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.

Considering statement A we have 18 B and 12 G and out of 18 boys kumar is among top 4 . But in this case we can't get the relative posiiton of rashmi and kumar. 

Consider statement B. We know that among top 5 , 3 are boys and 2 are girls .So rashmi is not among top5 and we know that Kumar's rank is 6 . So Rashmi is definately below Kumar. Hence, the question can be answered by using one of the statements alone but not by using the other statement alone.

Consider Statement A:

B_ _x_R

If the person has to reach red, then he has to take 2 steps in the most simplest manner. Now suppose he first takes left step and then tkes a right step.  He has taken 2 steps and has to take further 2 steps in order to reach red. In this way, a person has to take even number of steps to reach red. In the similar manner, a person has to take three steps to reach blue in the simplest manner. It means he would have to take odd number of steps to reach blue and since 21 is an odd number, therefore the person would reach blue step.

Consider Statement B,

It is given that the person obtains 3 more tails as compared to heads. Take the case of 1 head and four tails. In this case, he would reach blue step.Now take the case of 2 heads and 5 tails. In this case 2 heads are neutralised by 2 tails. The ramaining portion is 3 tails which would help to reach blue steps. In this manner, no matter how many heads are, three more tails would make sure that the person reaches blue corner.  

Using statement 1 alone, 2P + G < P + 2G

=> P < G

Using statement 2 alone, P + 2O = O + 2G

=> P + O = 2G

So, using either statement alone, we cannot find the answer to the question.

However, by using both the statements together, we can determine that P < G < O

So, option c) is the answer.

Let the no. of 1,2,5,10 Rs notes be x,y,z,w.

According to given condition we have z+w=x+y+1 .

Now considering statement A

The possible values are 4+3=3+3+1 .

But in this case the data is not sufficient to find price of article as the both 5 and 10 rs can have any of the 4 or 3 notes.

Hence statement A is not sufficient.

Now considering statement B ,

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

For this, more than 1 possibility of no. of notes of a particular value is there. Also 2 statements together can't give the total price of the article. 

Statement B is redundant. We already know that M has 5 siblings from the question. From statement A alone, we know that F has 2 brothers and M has 2 brothers and 3 sisters.

Considering statement A. The game ended normally.  Here we know that last 2 results were heads and the person receives 100 Rs. and since there was overall loss of 50 Rs we can calculate no. of matches which ended in tails or non consecutive heads . hence , statement a is sufficient to answer.
Considering statement B. The total number of tails obtained in the game was 138. So we know person lost here rs.138 and we can add any no. of single heads in between those tails such that the overall loss will be 50Rs. Hence this statement is also sufficient to answer.

Considering statements a we can deduce that she got letter S in her 21st purchase in previous 20 puchase she had either A, O or P . Hence statement a is not enough to answer. If we consider statement b we will know that out of 20 purchases 18 are vowel so 2 purchases contained letter P . So both statements are necessary to answer the question.

Let the distance be x. Time taken by A is T, so time taken by B is T+60 and time taken by C is T+90.
From statement A, we know that the ratio of speeds of A and C is x:(x-375) = x/T : x/(T+90)
We have only 1 equation and 2 unknows. We need the second statement to find the values. So, the question can be answered by using both the statements together.

If b = 4, then the answer to the question is no. If b = 32, then the answer is yes. So, using statement A alone, we cannot answer the question.
Using statement B alone, we can conclude that $$a^{44} < b^{11}$$

Using statement A, sum of roots = 0 and product of roots = 0, so b = c = 0.
Using statement B, sum of roots = x - 1/2 = -b/4 and product of roots = c/4 = b/4 = -x/2
So, we can calculate the values of b and c using either statement alone

Let the radius be r. Using statement B alone, r, r-5 and 2.5 form a right-angled triangle. So, we can answer the question using statement B alone.

Consider the first statement:
When the common ratio is less than 1 we can apply the formula of sum of infinite terms.
So, LHS = $$1/(a^2 - 1)$$
RHS = $$a/(a^2 - 1)$$
If a<1 then LHS<RHS

If a = 1,then LHS = RHS

So, we cannot answer the question using statement 1 alone

Using statement 2 alone, we know that a = 1/2. So, RHS > LHS
Hence, option a)

From statement 1 alone, we can infer that the triangle ABC is an equilateral triangle with side = 2 cm
Similarly, from statement 2 alone, we can infer that the triangle ABC is an equilateral of side 2 cm
So, the question can be answered using either statement alone

Let amount received by Jagdish, Punit and Girish be j,p,g rs. such that j+p+g = 38500. According to statement A , j= 2*(p+g)/9 . Using this alone we can't answer the question.
Now consider statement B , we have p = 3*(j+g)/11. Using this alone we can't answer the question. But considering both statements together we can answer the question. Hence, option C is the correct answer.

Considering statement B there are 2 possible score values before last 5 mins  possible. Korea 3:1 India or Korea 2:0 India. Now considering statement A , in one case India wins (as the score becomes India 4:3 Korea) and in other there is a draw(as the score becomes India 3:3 Korea). Hence, The question cannot be answered by either of the statements.

Considering statement A 12 students can be written as 4 students + 8 students. Hence Even if only 4 students are added the teacher can divide students in teams of 8. Now considering statement B , we cant perfectly tell if after adding teacher can divide in teams of 8 . Hence only A is enough to answer the question. Hence , The question can be answered by one of the statement alone but not by the other.

Consider statement 1:$$(x+y)(1/x + 1/y) = 4$$

$$(x+y)^2/xy = 4$$

$$x^2+y^2+2xy = 4xy$$

$$x^2+y^2+2xy-4xy = 0$$

$$(x-y)^2 = 0$$

x = y.

Consider statement 2:$$(x-50)^2 = (y-50)^2$$

(x-y)(x+y-100) = 0

Either x = y or x+y = 100.

Statement 1 is sufficient whereas statement 2 is not sufficient to answer the question.

Let the initial selling price be s1 and changed value be s2, also wholesale cost price be c. According to the given condition we have s1=1.28*c. By statement A we have s2=0.9*s1 and s2-c = 10 . Hence, wholesale cost price can be found out.
Considering statement B we have just s2=50. Using this we can't find c. Hence, the question can be answered by one of the statements alone but not by the other.

Consider statement A alone and assume that half of the people scored 600 and half of the people scored 499, so average is not 500

Also there is a possibility that average can be 500 when half of the people score about 500 and the other half score below 500.

Hence we won't get unique answer using A.

Consider statement B alone: We can't find average just by knowing the highest and the lowest score.

Even by using both the statements, we cannot find the answer to the question. Option d) is the correct answer.

Statement A: -1<x<1
-3<x-2<-1
So 1<mod(x-2)<3
It is definitely greater than 1

Statement B:  -2<x-1<2
-3<x-2<1
which implies mod(x-2) might be greater than or less than 1

So only one statement is sufficient.

Let x,y,z be the no. of people speaking only French, only Russian and both respectively. By statement A we have x+y+z = 300 and z = 196. Using this we can't find number of people in a club who speak only French.
Now considering statement B we have x = 58. So using both the statements together we can find number of people in a club who speak only French. Hence, option C is the correct answer.

Using either statement alone, we cannot answer the question.
The factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30. From the information given in both statements, the value of m is 15, and n is 2. So, option c).

This is a previous year's CAT question. We can also consider the negative values, which gives us other possibilities. So, a single answer can't be obtained. 

Even by using both the statements, we do not know whether country X has a higher GDP than country Y because we do not know the absolute values of the GDPs 5 years back. So, the question cannot be answered.

If we take the first statement alone,
X and Y can be among 2, 4, 6, and 8.
It is given that X/Y is an odd integer. So, X must be greater than Y.
If we take different pairs of (X,Y) among (8, 2), (8, 4), (8, 6), (6, 2), (6, 4), (4, 2), only (6, 2) satisfies the above condition of X/Y being an odd integer.
So, we can uniquely determine the values of X and Y which are 6 and 2 respectively.

If we take the second statement alone,
12 = 2 * 2 * 3
We want 12 to be the product of two even numbers.
The only possibility is when the numbers are 2 and 6.
However, we don't know any relation between X and Y.
So, both of them can be either 2 or 6.
Therefore, statement II alone is not sufficient to determine the value of X.

Hence, option A is the correct answer.

The question asked is about the number of round trips made in 12 hours. This can be answered if we know the time of travel and the time of waiting. So, the question can be answered by using statement 2 alone but not by using statement 1 alone.

We cannot answer the question using either of the statements alone. If we consider both the statements i.e. transfer rate is 6 kilobytes per second and the size of the software is 4.5*1024 kilobytes = 4608 kilobytes then download time = 4608 / 6 seconds. Hence, the question can be answered by using both the statements together, but cannot be answered by using either statement alone.

In these type of questions, we don't need to find out the exact answer but just check if the answer can be calculated.

When a square is inscribed inside a circle, the diagonal of the square is a diameter of the circle. Hence, if either the circle diameter or side of the square is given, then we can determine the areas of both the circle and square as well as the difference between their areas.

Hence, the question can be answered by using either statement alone.

Consider the first statement. We just know the relation between number of apples of bought by the two guys. Hence, the question cannot be answered using this statement.

Based on the total number of available apples, we cannot determine how many of them were bought by Ram and Gopal. 

Using both statements together, we don't know how many apples the two of them bought together. Hence, we cannot answer the question using even both the statements together.

If we consider statemnet A. X is greater than at least one of Y and Z. Here X>Z or X>Y, both are possible.

By statement b. Y is greater than at least one of X and Z. Here  Y>X or Y>Z.

If we consider bith the statements together then  X>Y>Z or X,Y>Z.

In both cases Z is the smallest.

Considering statement A, X(X+3)<0 . Here values of x which satisfy are X=-2 , -1 .
Considering statement B, X(X-3)>0 . Here values of x which satisfy are X = ..,-2,-1,4,5,6,... So using only statement 1 we know that uniquely modulus of x has to be less than 3. Hence,  the question can be answered by one of the statements alone, but cannot be answered by using the other statement alone.

Let a,b and c be length of PR, QR and PQ.

By 2nd statement we know that c=18 and also area = (a*b*c)/9 and by using statement A we have area = inradius*(a+b+c)/c.

Using both the equations we can find out value of PR + RQ . Hence option C.

Let the purchase price be P and sale price be S. So total brokerage = 0.01P + 0.01S. 

If we consider statement A we have S = 1.05*P.

Hence we can find brokerage value in terms of P.

then total profit = S - brokerage = Constant * P.

Then profit per rupee would be dividing last equation by P

Hence statement A is sufficient.

Using statement B we cant make out the profit percentage.

Hence , option A. 

We have (2 + 0) + (-5 + -6)

Now using statement B it can be written as (0 + 2) + (-5 + -6).

Using trhe statement A we can get the answer to the required question.

Hence,the question can be answered by using both the statements together, but cannot be answered by using either statement alone.

Using option B we know that since the constant terms are non-zero and unique , the lines do not intersect at the origin. Also since other coefficients are not equal , the lines can be parallel or intersecting . Hence, the question cannot be answered even by using both statements together.

Using statement A we know that C is midpoint of both BD and AE . Also using statement B we know that C is midpoint of BD and AE. Hence , the question can be answered by using either statement alone.

Considering 1st statement we know that number of people watching Q alone is 900 and both P and Q is 100. But we cant find the number of people watching P. Also, using statement B alone, we cannot find the number of people watching P. But if we consider B along with A , we can definately get total no. of people watching P . Hence, the question can be answered by using both the statements together, but cannot be answered by using either statement alone.

Mumbai and No-man’s-land would be in different time zones. Using both the statements together we would just know the time mumbai local time when ghosh babu reaches at island. Hence, the question cannot be answered even by using both statements together.

According to statement A, |X - Y| = 6

According to statement B, XY = 6k

we have 2 equations and 3 variables, which cannot be solved => Cannot be answered using both the statements.

Using statement A, we cannot infer the number of students in the class.

Using statement B, we can say that the sum of weights of heaviest and the lighest persons is equal to twice the average weight of the class. But we cannot infer the number of students in the class.

Using both the statements as well, we do not have any information through which we can find the answer.

Hence, option D is the answer.

Using statement 1, we come to know that if the inner radius is r, then the outer radius is r+1

Using statement 2, Volume = 20 l = 20000$$cm^3$$

$$\frac{4}{3} \pi (r+1)^3-r^3$$ = 20000

We can calculate r using this equation.

Using only statement A, we know that the shape of th field is rectangle. We cannot infer anything else with only statement A.

Using only statement B, we know that C is one of the vertices, but we do not know how many vertices exist.

Using both A and B, we know that the field is rectangular in shape and the length and breadth are 300 metres and 100 metres.

Hence, we can find the value of the diagonal which is the maximum distance between any two points inside the field.

Hence, option C is the answer.

Growth = Change in value on Y-axis / Change in value on X-axis = Tangent of the angle made by the line with the X-axis

In statement 1, since we know both the angles, we can calculate the growth for both the periods and hence the ratio of growths.

Using statement 2 alone, we cannot determine the answer. Option a) is the correct answer.

From statement I, We can draw the following figure

From the figure we can say that in triangle GCD we know 2 sides GC and CD, so we can find the third side
Also we can find the angle GDC(Using trigonometric ratios)

Similarly in triangle CDE, we know CE and DE, so we can find angle CDE.
From this we can get the angle FDE. Now in right angled triangle FDE, we can find the angle DFE.
So we can answer from statement I
Now let us consider statement II

By using trigonometric relations angle CFG can be calculated.
From this we can get angle DFE
Iin triangle CGD we can find angles GDC(Using trigonometric ratios)
Now GDC = CDJ
Thus we can find EDH and from that we can find EHD.
Hence the question can be answered from statement II alone.

ax + by = c

dx + ey = f

Using statement A, we can modify the equation ax + by + c to k(dx+ey=f) 

=> We cannot find the point (x,y).

Using statement B, we can change the equations to x + y = c and 2x + 2y = f (f not equal to 2c)

As the slopes are equal and the intercepts are not equal, we can say that the equations do not have a solution.

Hence, we can answer the question using statement B but not A.

The value of b is unknown in this question.

Using Statement a, we can say that B is not a mathematician. A may be or may not be a mathematician as we do not know whether he can add three numbers correctly or not. Similarly C may be or may not be a mathematician as we do not know the value of b.

Using statement b, We cannot find out for sure if A and C are mathematicians or not.

So the question cannot be answered even by using both statements together.

Using only the first statement, we know that A and B passed the exam. We can't infer anything about C and D.

Using only the second statement, we know that C and D failed the exam. We can't infer anything about A and B.

Using both the statements, we can say that A and B passed the exam and C and D failed the exam.

Both x = 4 and x = 16 satisfy the condition in statement A.

Using only statement B, we cannot find the unique value of x.

Using both A and B, we can infer that x = 4

Hence, option C is the answer.

Using only statement A, we can say that at least 10 white seeds were used. But we cannot find the exact number.

Using only statement B, we cannot make any inference.

Using both the statements, we can only make the same conclusion as we made using statement A.

Hence, the question cannot be answered by using both the statements together.

Only statement A: Angles OAC, ACB, CBO are right angles => OACB is a rectangle. OC = AB and OC is radius => AB is equal to radius. Hence, we can find the answer using statement A only.

Only statement B: The point B can change even though the point A s fixed at a distance of 5 units on x-axis from origin. Hence, length of AB changes => We cannot find the answer using statement B only.

Let's consider the statements one by one.

Statement 1: n is a multiple of HCF(3,5,7,9) = 315. Hence, n can be 315, 630, 945 .... Hence we cannot say if n is odd or even.

Statement 2: 0<n<400, we cannot say if n is odd or even.

Taking both statements together, n=315. Hence, it is odd.

Thus the question can be answered using both statements together.

The definition of the given function is expressed in statement 2, where  $$a \circledast b=\frac{a+b}{a}$$

Hence $$2\circledast3=\frac{2+3}{2}$$.

Hence the answer can be determined by b alone. A does not give any releant information.

Let the total questions be x and the questions solved by Radha and Rani be y and z respectively.

According to the first statement, y+z = 20% of x

Accoding to the second statement, y = 5/3rd of z

There are 2 equations and three variables. Hence the solution cannot be determined.

Let price of tea be t, coffee be c and cold drink be d.

Statement 1: c=t+5, we cannot infer value of t from this information

Statement 2: c=d-5 and d=3t, we cannot infer the value of t from this information

Using the statements together, c=t+5 and c=3t-5.

=>t+5=3t-5

=>2t=10

=> t=5

Hence, we can answer the questions using both statements together.

We cannot uniquely determine a value for a using either statement alone.

Using both statements together, we get an inconsistent set of equations. Hence, there is no solution using both statements together.

Thus, the question cannot be answered using both statements together.

Let the number of girls be g and boys be b. Hence, b+g=150

Statement 1: 50g+25b=4900 => 2g+b=196. Solving the two equations, g=46. Hence, answer can be obtained using statement 1 alone.

Statement 2: 30b+30g=4500 => b+g=150. As both the equations are equal, we cannot find a unique solution.

Thus, the question can be answered using one of the statements alone and not the other.

From the statements we need to infer which envelope contains L2.

From statement 1, we know that C2 should open E1.

From statement 2, we cannot tell which envelope contains L2.

Hence, the question can be answered using one of the statements alone and not the other.

Let's take the two statements individually:

Statement 1: If the second book is in Rack 3, then the fourth book must be in the only remaining odd numbered rack i.e. rack 1. Hence, we can infer the position of book 4 using the first statement.

Statement 2: If the third book is in Rack 2, then the first book must be in Rack 4. However, we don't know anything about the position of Book 4. Hence, we cannot infer the position of book 4 using the second statement.

Hence, the question can be answered using one of the statements but not the other.

Statement 1: a = (X+Y)/2 and $$a^2 = XY$$. Hence, $$a=(X+a^2/X)/2$$. Hence, we can re-arrange the equation to represent X in terms of a

Statement 2: There is no relation given between X and a. Hence, we cannot find value of X in terms of a.

Hence, the question can be answered by one statement alone and not the other.

We know that r1>r2.

Statement 1: $$2\pi(r1-r2) $$ = k. We cannot determine r1 from this information

Statement 2: $$\pi(r1^2-r2^2)$$ = m. We cannot determine r1 from this information.

Using both statements together, (r1+r2)/2 = m/k. We now have two linear equations with two variables. Hence, we can determine the value of r1 in terms of m and k.

Hence, the answer can be found using both statements together.

Considering first statement only, nothing absolute can be said about the value of $$a^3 + b^3$$ similarly considering 2nd statement only nothing absolute can be said about the value of $$a^3 + b^3$$.
When we consider both statements together two values of (a+b) can be possible as $$\sqrt{28}$$ and $$-\sqrt{28}$$
By which two possible values of $$a^3 + b^3$$. Nothing absolute can be said about it.
Hence answer will be D).

99 = 9 * 11 => If the number is divisible by 9 and 11, then the number is divisible by 99.

=> We can answer the question using only statement I.

The divisibility rule of 9 is that the sum of the digits in the number must be divisible by 9.

The divisibility rule of 11 is that the difference of sums of digits in odd places and digits in even places must be a multiple of 11.

According to statement II, if the number is reversed then the number is divisible by 9 and 11 => Divisible by 99.

Now, on reversing the number again, the number is still divible by 99 because the sum of the digits remains the same and also the difference of sums of digits in odd places and digits in even places remains the same.

Hence, the question can be answered using either of the statements alone.

Considering first statement alone, we can calculate 1/3rd of the complete distance as $$\sqrt{1^2 + 3^2}$$ Hence, we can evaluate complete distance too.

Considering second statement alone, we can calculate 2/3rd of the complete distance as $$\sqrt{2^2 + 6^2}$$ Hence, we can evaluate complete distance too.

So answer will be B as complete distance can be calculated by using any of the statement alone.

Considering first statement alone, 
As there are 2 variables x and y, and one equation we can't say anything absolute about values of x and y.
Considering second statement alone,
With 2 variables and one equation, we can't say anything absolute about variables x and y.
Even considering both statements together, we can't find values of x and y as equation in 2nd statement is same as in 1st statement after multiplying it 3.5

Considering first statement alone:
Nothing absolute can be said about the hats of Q and R, as Q might have black hat or white hat similarly R might have white or black hat as seen by P.
Considering second statement alone:
Nothing absolute can be said about the hats of P and R as they can have either Black or White caps
Even considering both statements together, we were not be able to reduce the possibilities of having black hats and white hats 
Hence answer will be D.

Let the speed of the car be x.

Using both the statements, we get a quadratic equation in x.

This equation will have 2 roots, one positive root and one negative root. So, we have to consider only the positive root.

The question can, therefore, be answered by using both the statements together.

Considering 1st statement alone:
As height is reduced to 1/3rd of it, radius will also get reduced 1/3rd of complete cone
Now we know volume is directly proportional to product of square of radius and height , we can calculate ratio of volumes.
Similarly considering 2nd statement alone, we can calculate ratio of volumes as we will have sufficient information regarding radius and height.
So answer will be B

 

Considering 1st statement alone:
We can find only one coordinate regarding both lines, which is insufficient information to calculate area.
Considering 2nd statement alone:
We have only one coordinate , regarding one line with insufficient information to calculate area.
Now if we consider both statements together, we can find points where lines are meeting co-ordinate axis and then area.
So our answer will be C.
 

Let the cost price of the Chair be x and that of the table be y.
Using statement 1)
Hence we get the total selling price = 1.15x + 1.2y and profit = 0.15x + 0.2y 
Using this alone we can't find the cost price of the chair.
Using Statement 2)
Total cost price = 1.10 x + 1.2y
Using Statement 1 and 2 together,
We get profit in second case = 0.05x.
Reduction in profit = (0.15x + 0.2y) - 0.05x = 0.1x + 0.2y = 20
Hence we can't deduce the value of x even after using both statements together.

From Statement 1, we know the speed of each runner.

From Statement 2, we can calculate the length of the track as we know the speed of runner every second.

But the answer cannot be determined as we do not know the direction of each runner.

Considering 1st statement alone:
When total number of widgets is 20000
let's say share of type-1 = x
and share of type-2 = y
So x+y = 20000
and 1.1x+0.94y = 20000
Hence we can find value of x and y by statement 1 alone.
Now considering second statement alone;
$$x+y = 20000$$
$$\frac{x}{y} = \frac{2}{1}$$
Hence with second statement also, we can find value of x and y
So answer will be B.
 

In either of the statements, a numerical value is not given.

With the given information, we can only find the radius in terms of N and T.

Hence, the radius cannot be determined using both the statements as well.

The proportions of liquid A in vessels 1, 2 and 3 are $$\frac{3}{8}$$, $$\frac{2}{5}$$ and $$\frac{4}{7}$$ respectively.

LCM of 8, 5 and 7 is 280 => Assume each container ha 280 litres.

=> Quantity of liquid A in vessels 1, 2 and 3 are 105, 112 and 160 respectively.

=> Total volume of A = 377 litres

Total volume of B = 840 - 377 = 463 litres

Hence, ratio can be found using both the statements and neither statement alone can be used to find the ratio.

For (a-b+c) > (a+b-c)
Considering first statement alone
if b<0, we can't say anything affirmative about inequality as information about c is unavailable.
Considering second statement alone, 
if c>0, we can't say anything about b.
Now if we consider both statement together 
i.e. b<0 and c>0, then term (-b+c) will be positive and additive to a whether term (b-c) will be negative and subtracting from a.
Hence (a-b+c) > (a+b-c).
And answer will be D.

$$(α^2 +β^2)$$ can be reduced to $$(α+β)^2 - 2(αβ)$$
Now as we can see, we need both values of (α+β) and (αβ) to solve the equation.
Hence answer will be D.

Considering first statement alone:
When loss is 25% on cost price, selling price will be $$\frac{3}{4}$$th of cost price but we can't get the absolute value of cost price.
Considering second statement alone:
As this statement can also be derived from 1st statement, but we need some other informations to get value of cost price
So answer will be A.

Cosidering first statement alone:
sp = $$\frac{6cp}{5}$$ (Where cp is cost price and sp is selling price)
But nothing absolute value can be derived from this information only
Now considering second statement alone:
sp = $$\frac{5cp}{4}$$ (Where sp is selling price and cp is cost price)
cp is equals to 250 rs.
Now we can derive value of sp from above equation.
Hence answer will be C.

Considering given statements, we can't find an absolute number of triangles as lines in a plane can be concurrent also with several other possibilities.
So answer will be A.

Considering statement 1 alone, together with all the information we also need information about the time period of compounding interest which is given in second statement.
Hence we can find value of assets using both the statements together.
So answer will be D.

Anil's age is a prime number in both 1996 and 1998. Hence, Anil's age in 1996 and 1998 are twin primes ( Prime numbers that differ by 2). Anil's age in 1996 and 1998 can be (11,13),(17,19) ,(29,31),.......and so on. Since a unique set cannot be determined, option A is the right answer.

Considering first and second statement alone, we can't say about time difference as information will not be sufficient.
Now considering both statements together, we can say that if journey duration is of 5 hours and flight departure time is 9:00 a.m., arrival time according to new york will be 2:00 p.m. but it is given that in London, arrival time is 10:00 a.m. Hence, the time difference will be of 4 hours between NewYork and London.

Considering first statement and second statement alone, information will not suffice to find the sequence of stations,
But if we consider both statements together we can tell about the sequence of stations as between A and B, Mr. Thomas travels with Mr. Murthy but not with Mr. Rahman. That means Mr. Thomas must have departed from station C and arrived at station D.
And it is also given that between C and D there are two segments. Hence, E lies between C and D, and Mr. Rahman departed from station D.

Taking only 1st statement into consideration, nothing certain can be said about product Q
Similarly with the only 2nd statement, we have only 1 information about product Q and R.
But considering both statements together we can say with sufficient information that product R will be more beneficial than product Q

Using first statement alone, we won't be able to find distance as insufficient data is there.
Now using second statement alone, we can find distance d as follows:
velocity before engine failure = $$\frac{40}{2}$$ = 20 km/h and after engine failure = 5 km/hr
And we know total time exceeds 45 min.
Hence 2 + $$\frac{d}{5}$$ = $$\frac{40}{60}$$ + $$\frac{40+d}{20}$$
or $$d = \frac{40}{9}$$  km.

Considering 1st statement alone, $$x^2 + x$$ is greater than 50, a two digit number and x is a prime number, then it has to be 7.

On the other hand, we cannot uniquely determine the value of x using statement B alone. Option a) is the correct answer.

As Avg. of 3 quotations is 110, so the sum of them will be 330
Now considering 1st statement alone, if the lowest quotation is 100, then highest can not exceed 129 because then the sum of all three will exceed from 330.
So we can get the answer by considering the first statement only.

Considering first statement only, given information will not suffice to answer that how many people read both Indian express and Times of India as data regarding intersection of all and data regarding number of people reading other 2 newspapers is not given
Now if we consider only second statement then we can answer about people reading both Indian express and Times of India as follows:
Total readers = 200 = 100 (indian express) + 120 (Times of india)  + 50 (Reading none) - Reading both.
So Reading both newspapers = 270-200 = 70

Given information X=3(Y-3)
Now considering first statement alone, we will have following equation:
Y+17 = X
On solving two eq. we can find the value of X.
Hence answer will be A.

Considering the first statement only, we will get radius > 14. Hence, we can't find the value of radius with this information only.
Now considering the second statement only, we will get radius <= 16.​Hence, we can't find the value of radius with this information only.
Now considering both statements simultaneously, we can't find an absolute value of radius.
Hence, answer will be D
 

Given: P - R = 2370
or P = 2370 + R ;
So if we consider only 1st statement, R can be 2 or 6 as 2372 and 2376 both will be divisible by 4.
And if we consider only 2nd statement, R will be having an absolute value of 6 as 2376 will be divisible by 6.
Hence by taking only 2nd statement into consideration, we can find unique value of R.
So answer will be B).

Concept: Try to find a way with which we can find an absolute/unique answer. 

Concept: Try to see whether we can have a "UNIQUE/ABSOLUTE" answer to given question by applying statements with 3 posibilities.
 If only 1st statement is taken into consideration.
 If only 2nd statement is taken into consideration.
 If both statements are taken into consideration.
In given question with first possibility, we can't have a unique answer as there can be 3 cases. Childs greater than 5 years, childs equal to 5 years and childs less than 5 years and with first possibility we know only that childs greater than 5 years get 5 choclates nothing about other two cases.
With second possibility too, we can't find a unique answer as we can't say anything about childs greater than 5 years
Now with third possibility, we are having a unique answer to given question because we have information about all childs which are either greater than 5 year , lesser or equal to 5 years. Childs older than 5 years will be 5, childs equal to 5 years are 2 and childs younger than 5 years will be 1.

So answer will be C)

1. If only 1st statment taken into consideration:
$$y=0.30x$$ (Where y= distance from Madras to Trivandrum and x= distance from Calcutta to Madras)
Nothing absolute can be said about average speed from Madras to Trivendrum.
2. If only 2nd statement taken into consideration:
$$\frac{y}{t_y} = 2\frac{x}{t_x}$$ ( Where t_x and t_y are time taken for calcutta to madras and madras to trivendrum respectively.)
And nothing absolute can be said about average speed from Madras to Trivendrum.
3. Now when both statements taken into consideration:
We will have $$y=0.30x$$;  $$\frac{y}{t_y} = 2\frac{x}{t_x}$$ and $$\frac{x+y}{t_x+t_y}$$ = 40 (Given)
After solving above three equations we can find a unique value for average speed from Madras to Trivendrum i.e. 
 $$\frac{y}{t_y} = \frac{920}{13}$$
Hence answer will be C.

If only first statement taken into consideration:

y-x = 2 where x, y and z are three odd integers.

Nothing absolute can be said about divisibility of x+z with 4.

Now if only second statement taken into consideration :

z-y=2 and nothing absolute can be said about divisibility of x+z with 4.

If we consider both statement simoultaneously then

we get x+z = 2y

Now as we know y is an odd integer , so 2y will never get a factor of 4.

Hence x+z is not divisible by 4.

As we find an absolute answer by using both statements Hence answer will be C)

Considering first statement only,current unit price of P1 is twice that of P2 and, Rates of P1 and P2 are either decreasing or P1 has constant rate. Now we can't say that which product will be costlier aftre 5 years as nothing certain information is there about rates of decreasing. Similarly if we consider only 2nd statement or consider both statements simultaneously, nothing certain can be said about rate of decreasing hence D will be answer

Taking only first statement into consideration:
Nothing absolute can be said about Z and X
Now taking only second statement into consideration:
$$W \leq V$$
As it is given that X is older than Y; Z is younger than W and V is as old as Y.
and through second statement we know that W is younger or as old as V.
So using above four information we can conclude that Z is not older than X.
Z is either younger or as old as X.
Hence answer will be B).

As it is clear that total time of journey will be = Time taken at stop-overs + Time taken while travelling with avg. speed 66 km/h
Hence both informations are needed to solve the question. So our answer will be D.

Given equation can be resolved to $$\frac{1}{\frac{1}{x} + \frac{1}{y}}$$ ($$xy \neq 0$$ and $$x \neq y$$ )
Now for $$\frac{1}{\frac{1}{x} + \frac{1}{y}}$$ to be greater than 1,
1/x + 1/y has to be less than 1.
For this to be true both x and y should be greater than 2.
Statement I doesn’t give any information about this,
but statement II clearly specifies this. Hence, only
statement II is required to answer the given question.
So it can be answered using statement 2 alone.