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The next set of medium/hard DS questions. I'll post OA's with detailed explanations after some discussion. Please, post your solutions along with the answers.

1. What is the product of three consecutive integers?

(1) At least one of the integers is positive (2) The sum of the integers is less than 6

4. Two machines, A and B, each working at a constant rate, can complete a certain task working together in 6 days. In how many days, working alone, can machine A complete the task?

(1) The average time A and B can complete the task working alone is 12.5 days. (2) It would take machine A 5 more days to complete the task alone than it would take for machine B to complete the task

5. Set A={3-2x, 3-x, 3, 3+x, 3+2x}, where x is an integer. Is the standard deviation of set A more than the standard deviation of set B={3-2x, 3-x, 3, 3+x, 3+2x, y}

(1) The standard deviation of set A is positive (2) y=3

6. The ratio of the number of employees of three companies X, Y and Z is 3:4:8, respectively. Is the average age of all employees in these companies less than 40 years?

(1) The total age of all the employees in these companies is 600 (2) The average age of employees in X, Y, and Z, is 40, 20, and 50, respectively.

7. Was the average (arithmetic mean) temperature in city A in March less than the average (arithmetic mean) temperature in city B in March?

(1) The median temperature in City A in March was less than the median temperature in city B (2) The ratio of the average temperatures in A and B in March was 3 to 4, respectively

8. Two marbles are drawn from a jar with 10 marbles. If all marbles are either red of blue, is the probability that both marbles selected will be red greater than 3/5?

(1) The probability that both marbles selected will be blue is less than 1/10 (2) At least 60% of the marbles in the jar are red

1. What is the product of three consecutive integers? (1) At least one of the integers is positive (2) The sum of the integers is less than 6

1) Set of 3 integers can be (-1,0,1,), (0,1,2), (1,2,3), (2,3,4), ……, (200,201,202),……………….. Not sufficient.

2) Set of 3 integers can be (0,1,2), (-1,0,1,), (-2,-1,0), (-3,-2,-1), ……, (-102,-101,-100),……………….. Not sufficient.

Together: Only possible sets are (-1,0,1,) and (0,1,2)for which at least one of the integers is positive and sum of the integers is less than 6. For both of these sets, product of elements is 0. Sufficient.

2. If x and y are both positive integers and x>y, what the remainder when x is divided by y? (1) y is a two-digit prime number (2) x=qy+9, for some positive integer q

1) Information on y is only given. No information on x. Not sufficient. 2) x/y = q + 9/y. For y > 9, remainder will be 9 when x is divided by y. But remainder will be different for other y’s, Not sufficient.

Together: y>9 --> remainder will be 9 when x is divided by y. Sufficient.

3. The length of the median BD in triangle ABC is 12 centimeters, what is the length of side AC? (1) ABC is an isosceles triangle (2) AC^2 = AB^2 + BC^2

1) There is no information on angles and which two sides are equal. Not sufficient.

2) It says that angle at B is right angle and AC is hypotenuse. If any right-angled triangle is inscribed in a circle, the hypotenuse of the triangle must be diameter of circle and the median extending to the hypotenuse of the triangle must be radius of the circle -->Median is equal to half of the hypotenuse --> AC = 24. Sufficient.

4. Two machines, A and B, each working at a constant rate, can complete a certain task working together in 6 days. In how many days, working alone, can machine A complete the task? (1) The average time A and B can complete the task working alone is 12.5 days. (2) It would take machine A 5 more days to complete the task alone than it would take for machine B to complete the task

Assuming A and B takes x and y days respectively to finish the work alone. In 6 days, work done by A = 6/x, work done by B = 6/y, total work done by A and B together = 6/x + 6/y 6/x + 6/y = 1 --> 6x + 6y = xy …. (eq1)

1) (x+y)/2 = 12.5 --> x+y = 25 Substituting this in eq1, 6*25 = x(25-x) --> x^2 -25x + 150 = 0 --> (x-10)(x-15) = 0 --> x = 10 or 15 Not sufficient.

2) y = x-5 Substituting this in eq1, 6x + 6x – 30 = x(x-5) --> x^2 -17x + 30 = 0 --> (x-2)(x-15) = 0 --> x = 2 or 15 If x = 2, y become -3 which is impossible. --> x = 15. Sufficient.

5. Set A={3-2x, 3-x, 3, 3+x, 3+2x}, where x is an integer. Is the standard deviation of set A more than the standard deviation of set B={3-2x, 3-x, 3, 3+x, 3+2x, y} (1) The standard deviation of set A is positive (2) y=3

1) Standard deviation can be positive or zero (in case of x=0). Here the information only says that x is not zero and all elements have different values. But, without knowing y, we cannot conclude. Not sufficient. 2) This says mean of both the sets is 3, no. of elements are 5 and 6 for A and B respectively. If x is not equal to 0, elements in set A will be more spread than those in set B. If x = 0, Standard deviation of both the sets will be zero. Not sufficient.

Together: x is not equal to 0 --> elements in set A will be more spread than those in set B --> standard deviation of set A is more than that of set B. Sufficient.

6. The ratio of the number of employees of three companies X, Y and Z is 3:4:8, respectively. Is the average age of all employees in these companies less than 40 years? (1) The total age of all the employees in these companies is 600 (2) The average age of employees in X, Y, and Z, is 40, 20, and 50, respectively.

Assuming number of employees in X, Y, Z are 3n, 4n, 8n respectively. So, total number of employees = 15x

1) We know the total age, but we do not know total number of employees or number of employees of any single company. Not sufficient. 2) Average age = (3x*40 + 4x*20+8x*50)/15x = (120+80+400)/15 = 600/15 = 40. Sufficient.

7. Was the average (arithmetic mean) temperature in city A in March less than the average (arithmetic mean) temperature in city B in March? (1) The median temperature in City A in March was less than the median temperature in city B (2) The ratio of the average temperatures in A and B in March was 3 to 4, respectively

1) This says nothing about temperatures of other days. Not sufficient. 2) This says that the average temperature in city A in March was less than the average temperature in city B in March. Sufficient.

8. Two marbles are drawn from a jar with 10 marbles. If all marbles are either red of blue, is the probability that both marbles selected will be red greater than 3/5? (1) The probability that both marbles selected will be blue is less than 1/10 (2) At least 60% of the marbles in the jar are red

1) If there are 4 blue marbles, probability of drawing 2 blue marbles = 4/10 * 3/9 = 2/15 > 1/10 If there are 3 blue marbles, probability of drawing 2 blue marbles = 3/10 * 2/9 = 1/15 < 1/10 So, there are maximum 3 blue marbles and at least 7 red marbles in the jar. If there are 7 red marbles, probability of drawing 2 red marbles = 7/10 * 6/9 = 7/15 < 3/5 If there are 8 red marbles, probability of drawing 2 red marbles = 8/10 * 7/9 = 28/45 > 3/5 Not sufficient.

2) As seen from calculations against option 1, with 6 or 7 red marbles, probability of drawing 2 red marbles will be less than 3/5, and 8 or more red marbles, probability of drawing 2 red marbles will be more than 3/5. Not sufficient.

Together: There is no added information found from option 1 and 2 together. Number of red marbles can be 7 or 8 or 9 10 which may result in probability of drawing 2 red marbles to be less than or more than 3/5. Not sufficient.

10. What is the value of the media of set A? (1) No number in set A is less than the average (arithmetic mean) of set A. (2) The average (arithmetic mean) of set A is equal to the range of set A.

1) This means all elements in the set have the same value and that is same to mean and median as well. But, we do not know the number. Not sufficient. 2) There are many sets for which average and range will be the same. Examples are: [10,10,30,30] where median is 20, or [2,2,3,3,5] where median is 3. Not sufficient.

Together: This is possible only when all elements have the same value as 0 --> average = range = 0 --> median = 0

Answer is C.

Note: I believe the question should be read as “What is the value of the median of set A?”

The next set of medium/hard DS questions. I'll post OA's with detailed explanations after some discussion. Please, post your solutions along with the answers.

1. What is the product of three consecutive integers?

(1) At least one of the integers is positive (2) The sum of the integers is less than 6

2. If x and y are both positive integers and x>y, what the remainder when x is divided by y?

(1) y is a two-digit prime number (2) x=qy+9, for some positive integer q

3. The length of the median BD in triangle ABC is 12 centimeters, what is the length of side AC?

(1) ABC is an isosceles triangle (2) AC^2 = AB^2 + BC^2

4. Two machines, A and B, each working at a constant rate, can complete a certain task working together in 6 days. In how many days, working alone, can machine A complete the task?

(1) The average time A and B can complete the task working alone is 12.5 days. (2) It would take machine A 5 more days to complete the task alone than it would take for machine B to complete the task

5. Set A={3-2x, 3-x, 3, 3+x, 3+2x}, where x is an integer. Is the standard deviation of set A more than the standard deviation of set B={3-2x, 3-x, 3, 3+x, 3+2x, y}

(1) The standard deviation of set A is positive (2) y=3

6. The ratio of the number of employees of three companies X, Y and Z is 3:4:8, respectively. Is the average age of all employees in these companies less than 40 years?

(1) The total age of all the employees in these companies is 600 (2) The average age of employees in X, Y, and Z, is 40, 20, and 50, respectively.

7. Was the average (arithmetic mean) temperature in city A in March less than the average (arithmetic mean) temperature in city B in March?

(1) The median temperature in City A in March was less than the median temperature in city B (2) The ratio of the average temperatures in A and B in March was 3 to 4, respectively

8. Two marbles are drawn from a jar with 10 marbles. If all marbles are either red of blue, is the probability that both marbles selected will be red greater than 3/5?

(1) The probability that both marbles selected will be blue is less than 1/10 (2) At least 60% of the marbles in the jar are red

9. If x is an integer, is x^2>2x?

(1) x is a prime number. (2) x^2 is a multiple of 9.

10. What is the value of the media of set A?

(1) No number in set A is less than the average (arithmetic mean) of set A. (2) The average (arithmetic mean) of set A is equal to the range of set A.

The next set of medium/hard DS questions. I'll post OA's with detailed explanations after some discussion. Please, post your solutions along with the answers.

1. What is the product of three consecutive integers?

(1) At least one of the integers is positive (2) The sum of the integers is less than 6

We have to find out the product of n*(n+1)*(n+2)..... stmt. 1 -- At least one of the integers is positive --- Not sufficient stmt. 2 -- The sum of the integers is less than 6, that means n+n+1+n+2<6, so 3n+3<6. therefore, n<1, so with the help of stmt 2 we will get different values... not sufficient.

now 1+2 ----- At least one of the integers is positive & n<1 ...... we will get the desired answer. hence, C.

6. The ratio of the number of employees of three companies X, Y and Z is 3:4:8, respectively. Is the average age of all employees in these companies less than 40 years?

(1) The total age of all the employees in these companies is 600 (2) The average age of employees in X, Y, and Z, is 40, 20, and 50, respectively.

----------------------------------------

As given in the Question, we have the ratio of the number of employees of three companies X, Y and Z is 3:4:8 & the Question asks us that Is 3x+4y+8z/15 <40 or Is 3x+4y+8z < 600 ....

Remember that this is a Yes/No question ....

stmt. 1 ---- it clearly tells us that 3x+4y+8z =600 ... therefore it says that 3x+4y+8z = 600 but not < 600.

stmt. 2 ---- after calculating, it also clearly says that 3x+4y+8z=600 but not < 600.

Hence, D... Both stmts are sufficient.

----------------------------------------------------------- 7. Was the average (arithmetic mean) temperature in city A in March less than the average (arithmetic mean) temperature in city B in March?

(1) The median temperature in City A in March was less than the median temperature in city B

(2) The ratio of the average temperatures in A and B in March was 3 to 4, respectively

1. (1) Insufficient. It could be {0,1,2} and the product is 0, or it could be {1,2,3} and their product is 6. (2) Insufficient. It could be {0,1,2} and the product is 0, or it could be {-3,-2,-1} and the product is -6

(1)+(2) Sufficient. Suppose the least number is n, so the second is (n+1) and the third is (n+2). Their sum must be less than 6: n+(n+1)+(n+2)<6 or 3n+3<6 or n<1. By first statement at least one of the integers must be positive, it means that the largest is positive: n+2>0 or n>-2. So n=-1 or n=0. Therefore, there are two possible sets {-1, 0, 1} or {0, 1, 2}. The product anyway is 0.

The correct answer is C.

2. (1) Insufficient. For x=11 and y=10 the remainder when x is divided by y is 1, for x=12 and y=2 the remainder when x is divied by y is 2. (2) Insufficient. The main point here is the remainder must be less than the divisor. We don't know is y>9 or y<=9. For y=1 the remainder when x is divided by y is 0, for y=2, x=2q+9 the remainder is 1 when x is divided by 2.

(1)+(2) Sufficient. If y is a two-digit number the when qy is divided by y there is remainder 0, and when 9 is divided by y the remainder is 9, since 9<y. So, x when divided by y gives the remainder 9.

The correct answer C _________________

I'm happy, if I make math for you slightly clearer And yes, I like kudos:)

1. What is the product of three consecutive integers?

(1) At least one of the integers is positive.

We can have three cases: (i) All three integers are positive. In this case the product will obviously be positive. (ii) Two of the integers are positive: {0, 1, 2}. In this case the product will be zero. (iii) Only one of the integers is positive: {-1, 0, 1}. In this case the product will be zero.

Not sufficient.

(2) The sum of the integers is less than 6. Clearly insufficient, consider {-1, 0, 1} and {-3, -2, -1}.

(1)+(2) The second statement implies that we cannot have case (i) from (1), since the least sum of three positive consecutive integers is 1+2+3=6. Thus we have either case (ii) or case (iii). Therefore the product of the integers is zero. Sufficient.

2. If x and y are both positive integers and x>y, what the remainder when x is divided by y?

If \(x\) and \(y\) are positive integers, there exist unique integers \(q\) and \(r\), called the quotient and remainder, respectively, such that \(y =divisor*quotient+remainder= xq + r\) and \(0\leq{r}<x\).

(1) y is a two-digit prime number. Clearly insufficient since we know nothinf about x.

(2) x=qy+9, for some positive integer q. It's tempting to say that this statement is sufficient and \(r=9\), since given equation is very similar to \(y =divisor*quotient+remainder= xq + r\) . But we don't know whether \(y>9\): remainder must be less than divisor.

For example: If \(x=10\) and \(y=1\) then \(10=1*1+9\), then the remainder upon division 10 by 1 is zero. If \(x=11\) and \(y=2\) then \(11=1*2+9\), then the remainder upon division 11 by 2 is one. Not sufficient.

(1)+(2) From (2) we have that \(x=qy+9\) and from (1) that y is more than 9 (since it's a two-digit number), so we have direct formula of remainder, as given above. Sufficient.

3. The length of the median BD in triangle ABC is 12 centimeters, what is the length of side AC?

(1) ABC is an isosceles triangle. Clearly insufficient.

(2) AC^2 = AB^2 + BC^2. This statement implies that ABC is a right triangle and AC is its hypotenuse. Important property: median from right angle is half of the hypotenuse, hence BD=12=AC/2, from which we have that AC=24. Sufficient.

4. Two machines, A and B, each working at a constant rate, can complete a certain task working together in 6 days. In how many days, working alone, can machine A complete the task?

Given that 1/A+1/B=1/6, where A is the time needed for machine A to complete the task working alone and B is the time needed for machine B to complete the task working alone.

(1) The average time A and B can complete the task working alone is 12.5 days. This statement implies that A+B=2*12.5=25. Now, since we don't know which machine works faster then even if we substitute B with 25-A (1/A + 1/(25-A) = 1/6) we must get two different answers for A and B: A<B and A>B. Not sufficient.

(2) It would take machine A 5 more days to complete the task alone than it would take for machine B to complete the task. A=B+5, so we have that 1/A+1/(A-5)=1/6. From this we can find that A=2 (not a valid solution since in this case B will be negative) or A=15. Sufficient.

5. Set A={3-2x, 3-x, 3, 3+x, 3+2x}, where x is an integer. Is the standard deviation of set A more than the standard deviation of set B={3-2x, 3-x, 3, 3+x, 3+2x, y}

(1) The standard deviation of set A is positive. We know that the standard deviation of any set is more than or equal to zero. The standard deviation of a set is zero only when the set consists of identical elements. So, this statement implies that set A does NOT consists of identical elements or that x does not equal to zero. Still this statement is not sufficient to answer the question.

(2) y=3. The mean of set A is 3. Now, if \(x\neq{0}\) for example if x=1, then the standard deviation of B would be smaller that the standard deviation A, since the elements of B would be less widespread than the element of set A. But if x=0, then A={3, 3, 3, 3, 3} and B={3, 3, 3, 3, 3, 3}, so both will have the standard deviation of zero. Bot sufficient.

(1)+(2) Since from (1) \(x\neq{0}\), then adding a new element which equals to the mean will shrink the standard deviation, thus SD(A)>SD(B). Sufficient.

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