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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

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.

From F.S 1, if the no of employees is in X,Y,Z is 3k,4k,8k respectively;k is a positive integer; the average age = 600/(3+4+8)k = 40/k, hence a NO for k=1,a YES for k=2. Insufficient.

From F.S 2, the average age of all the employees for no of employees in the given ratio= (3k*40+4k*20+8k*50)/15k ; where k is a positive integer --> 600/15 = 40,hence a NO irrepective of the value of k. Sufficient.

A={3 -2x, 3-x , 3, 3+x, 3+2x} -> mean = 3, deviations from the mean: 2x, x, 0, x, 2x B={3- 2x, 3-x , 3, 3+x, 3+2x, y} -> mean = (15/6 +y/6),

(1) SD of A is positive, then x>0. (no other restrictions on x, so let's take for example x = 1). The deviations of set A are 2, 1,1,2. If y is a gigantic number, and x = 1, it is obvious that the deviations from the mean in set B will be much higher than 2, 1,0,1,2. However, if y = 3, the deviations will also be 2,1,1,2. INSUFFICIENT

(2) y = 3, means that the mean for set B = 3, and deviations are: 2x, x,0,0,x,2x. Therefore SD for B and A are the same. 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

Answer should be C The work is 100% Machine A and B, working together, can finish a work in 6 days. So in a day they are completing 100/6 = 16.66% of work. A + B = 16.66

S1 :- (100/A) + (100/B) = 25 Not Sufficient.

S2 :- (100/A) – (100/B) = 5 Not Sufficient.

S1 + S2 :- Solving two equations we would get A= 10 Sufficient. _________________

Q5) Is the standard deviation of set A {3-2x, 3-x, 3, 3+x, 3+2x} > the standard deviation of set B={3-2x, 3-x, 3, 3+x, 3+2x, y} From Stmt 1, y is not known and only SD > 0 - Not Sufficient If y <= 3 SD of set A > SD of set B and for other values SD of set A < SD of set B

From Stmt 2, y = 3 - Sufficient When y = 3 and for any integer value for x SD is always >= 0 and SD of set A > SD of set B Answer is B _________________

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.

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.

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?

Given that the ratio of the number of employees is 3x:4x:8x, for some positive multiple x.

The questions asks whether (average age)=(total age)/(# of employees)<40, or whether (total age)/(3x+4x+8x)<40, which is the same as: is (total age)<600x?

(1) The total age of all the employees in these companies is 600. The question becomes: is 600<600x? Or is 1<x. We don't know that: f x=1, then the answer is NO but if x>1, then the answer is YES. Not sufficient.

(2) The average age employees in X, Y, and Z, is 40, 20, and 50, respectively. (total age)=40*3x+20*4x+50*8x=600x, so the answer to the question is NO. Sufficient.

Is x(x-2)>0? --> is x<0 or x>2. Basically if x is not 0, 1, or 2 we have an YES answer to the question.

(1) x is a prime number. If x=2 then the answer is NO but if x is some other prime, then the answer is YES. Not sufficient.

(2) x^2 is a multiple of 9. If x=0 then the answer is NO but if x=3, then the answer is YES. Not sufficient.

(1)+(2) Since from (1) x is a prime and from (2) x^2 is a multiple of 9, then x can only be 3. Therefore the answer to the question is YES. Sufficient.

Note that I cannot award more than 5 Kudos to the same person per day, so those of you who have more than 5 correct solutions please PM me tomorrow the links for which I owe you kudos points.

1) Is there a fixed time frame to post the answers ? (Background: I tried posting yesterday, 11 AM IST, but the post was locked)

2) How can I quickly know about similar posts which have open questions accompanied with kudos for answers ? Subscribing to a topic didn't work for me. I am a corporate slave.

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?

Given that the ratio of the number of employees is 3x:4x:8x, for some positive multiple x.

The questions asks whether (average age)=(total age)/(# of employees)<40, or whether (total age)/(3x+4x+8x)<40, which is the same as: is (total age)<600x?

(1) The total age of all the employees in these companies is 600. The question becomes: is 600<600x? Or is 1<x. We don't know that: f x=1, then the answer is NO but if x>1, then the answer is YES. Not sufficient.

(2) The average age employees in X, Y, and Z, is 40, 20, and 50, respectively. (total age)=40*3x+20*4x+50*8x=600x, so the answer to the question is NO. Sufficient.

Answer: B.

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

Hi Bunuel,

Doubt in this Q. Pls help & suggest where am I going Wrong in this Question ......

Q .. 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

Now, it gives the ratio of number of employees i.e.; 3:4:8 & then we can take this as real number of employees like 3 employees in Company X, 4 in Y & 8 in company Z ...therefore, we have 3+4+8=15 , 15 employees in total....& average age of employees in company X must be something, lets say its x for Company X, so the total age for company X will be 3x, similarly, for Company Y the average age is y therefore, total age for company Y will be 4y & for Company Z, the total age will be 8z. & the average of the total ages will be 3x+4y+8z/15 ...... & as the question asks, Is the average age of all employees in these companies less than 40 years ?? i.e.; 3x+4y+8z/15 <40 or 3x+4y+8z<600...... so we have to find out if 3x+4y+8z<600 ..... rephrasing complete.

now statement (1) The total age of all the employees in these companies is 600. that means 3x+4y+8z=600 but we have to check if 3x+4y+8z<600. so it is clearly NO.

statement (2) The average age of employees in X, Y, and Z, is 40, 20, and 50, respectively therefore, total age = 3*40+4*20+8*50 = 600. as we have to check if 3x+4y+8z<600. so it is also a clear NO. & hence , answer is D.

Now please let me know where am I going wrong. Your help will be appreciated. Thanks !!!! _________________

If you don’t make mistakes, you’re not working hard. And Now that’s a Huge mistake.

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?

Given that the ratio of the number of employees is 3x:4x:8x, for some positive multiple x.

The questions asks whether (average age)=(total age)/(# of employees)<40, or whether (total age)/(3x+4x+8x)<40, which is the same as: is (total age)<600x?

(1) The total age of all the employees in these companies is 600. The question becomes: is 600<600x? Or is 1<x. We don't know that: f x=1, then the answer is NO but if x>1, then the answer is YES. Not sufficient.

(2) The average age employees in X, Y, and Z, is 40, 20, and 50, respectively. (total age)=40*3x+20*4x+50*8x=600x, so the answer to the question is NO. Sufficient.

Answer: B.

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

Hi Bunuel,

Doubt in this Q. Pls help & suggest where am I going Wrong in this Question ......

Q .. 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

Now, it gives the ratio of number of employees i.e.; 3:4:8 & then we can take this as real number of employees like 3 employees in Company X, 4 in Y & 8 in company Z ...therefore, we have 3+4+8=15 , 15 employees in total....& average age of employees in company X must be something, lets say its x for Company X, so the total age for company X will be 3x, similarly, for Company Y the average age is y therefore, total age for company Y will be 4y & for Company Z, the total age will be 8z. & the average of the total ages will be 3x+4y+8z/15 ...... & as the question asks, Is the average age of all employees in these companies less than 40 years ?? i.e.; 3x+4y+8z/15 <40 or 3x+4y+8z<600...... so we have to find out if 3x+4y+8z<600 ..... rephrasing complete.

now statement (1) The total age of all the employees in these companies is 600. that means 3x+4y+8z=600 but we have to check if 3x+4y+8z<600. so it is clearly NO.

statement (2) The average age of employees in X, Y, and Z, is 40, 20, and 50, respectively therefore, total age = 3*40+4*20+8*50 = 600. as we have to check if 3x+4y+8z<600. so it is also a clear NO. & hence , answer is D.

Now please let me know where am I going wrong. Your help will be appreciated. Thanks !!!!

Given that the ratio of the number of employees of three companies X, Y and Z is 3:4:8, respectively, so the number of employees could be: 3, 4, 8; 3*2=6, 4*2=8, 8*2=16; 3*3=9, 4*3=12, 8*3=24; 3*4=12, 4*4=16, 8*4=32; ...

Notice that the multiple is the same in each case. Thus the ratio of the number of employees is 3x:4x:8x, for some positive multiple x, not 3x:4y:8z.

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?

Given that the ratio of the number of employees is 3x:4x:8x, for some positive multiple x.

The questions asks whether (average age)=(total age)/(# of employees)<40, or whether (total age)/(3x+4x+8x)<40, which is the same as: is (total age)<600x?

(1) The total age of all the employees in these companies is 600. The question becomes: is 600<600x? Or is 1<x. We don't know that: f x=1, then the answer is NO but if x>1, then the answer is YES. Not sufficient.

(2) The average age employees in X, Y, and Z, is 40, 20, and 50, respectively. (total age)=40*3x+20*4x+50*8x=600x, so the answer to the question is NO. Sufficient.

Answer: B.

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

Hi Bunuel,

Doubt in this Q. Pls help & suggest where am I going Wrong in this Question ......

Q .. 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

Now, it gives the ratio of number of employees i.e.; 3:4:8 & then we can take this as real number of employees like 3 employees in Company X, 4 in Y & 8 in company Z ...therefore, we have 3+4+8=15 , 15 employees in total....& average age of employees in company X must be something, lets say its x for Company X, so the total age for company X will be 3x, similarly, for Company Y the average age is y therefore, total age for company Y will be 4y & for Company Z, the total age will be 8z. & the average of the total ages will be 3x+4y+8z/15 ...... & as the question asks, Is the average age of all employees in these companies less than 40 years ?? i.e.; 3x+4y+8z/15 <40 or 3x+4y+8z<600...... so we have to find out if 3x+4y+8z<600 ..... rephrasing complete.

now statement (1) The total age of all the employees in these companies is 600. that means 3x+4y+8z=600 but we have to check if 3x+4y+8z<600. so it is clearly NO.

statement (2) The average age of employees in X, Y, and Z, is 40, 20, and 50, respectively therefore, total age = 3*40+4*20+8*50 = 600. as we have to check if 3x+4y+8z<600. so it is also a clear NO. & hence , answer is D.

Now please let me know where am I going wrong. Your help will be appreciated. Thanks !!!!

Given that the ratio of the number of employees of three companies X, Y and Z is 3:4:8, respectively, so the number of employees could be: 3, 4, 8; 3*2=6, 4*2=8, 8*2=16; 3*3=9, 4*3=12, 8*3=24; 3*4=12, 4*4=16, 8*4=32; ...

Notice that the multiple is the same in each case. Thus the ratio of the number of employees is 3x:4x:8x, for some positive multiple x, not 3x:4y:8z.

Hope it's clear.

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

HI, Thanks for ur quick reply ......

yes, you are right, the ratio of number of employees is 3x:4x:8x but I didn't mean that the ratio of number of employees is 3x:4y:8z instead I said that 3x+4y+8z can be the total weight of the employees of three companies X,Y, Z if we assume that the average age in Company X is x & average age in Company Y is y & average age in Company Z is z then the total age of employees of company X will be 3x & the total age of employees of company Y will be 4y & the total age of employees of company Z will be 8z & therefore, the total age of all the employees of the three companies will be 3x+4y+8z & then we can calculate the average. Please review & give me some advise on that. Thanks !! in advance !!!! _________________

If you don’t make mistakes, you’re not working hard. And Now that’s a Huge mistake.

Doubt in this Q. Pls help & suggest where am I going Wrong in this Question ......

Q .. 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

Now, it gives the ratio of number of employees i.e.; 3:4:8 & then we can take this as real number of employees like 3 employees in Company X, 4 in Y & 8 in company Z ...therefore, we have 3+4+8=15 , 15 employees in total....& average age of employees in company X must be something, lets say its x for Company X, so the total age for company X will be 3x, similarly, for Company Y the average age is y therefore, total age for company Y will be 4y & for Company Z, the total age will be 8z. & the average of the total ages will be 3x+4y+8z/15 ...... & as the question asks, Is the average age of all employees in these companies less than 40 years ?? i.e.; 3x+4y+8z/15 <40 or 3x+4y+8z<600...... so we have to find out if 3x+4y+8z<600 ..... rephrasing complete.

now statement (1) The total age of all the employees in these companies is 600. that means 3x+4y+8z=600 but we have to check if 3x+4y+8z<600. so it is clearly NO.

statement (2) The average age of employees in X, Y, and Z, is 40, 20, and 50, respectively therefore, total age = 3*40+4*20+8*50 = 600. as we have to check if 3x+4y+8z<600. so it is also a clear NO. & hence , answer is D.

Now please let me know where am I going wrong. Your help will be appreciated. Thanks !!!!

Given that the ratio of the number of employees of three companies X, Y and Z is 3:4:8, respectively, so the number of employees could be: 3, 4, 8; 3*2=6, 4*2=8, 8*2=16; 3*3=9, 4*3=12, 8*3=24; 3*4=12, 4*4=16, 8*4=32; ...

Notice that the multiple is the same in each case. Thus the ratio of the number of employees is 3x:4x:8x, for some positive multiple x, not 3x:4y:8z.

Hope it's clear.

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

HI, Thanks for ur quick reply ......

yes, you are right, the ratio of number of employees is 3x:4x:8x but I didn't mean that the ratio of number of employees is 3x:4y:8z instead I said that 3x+4y+8z can be the total weight of the employees of three companies X,Y, Z if we assume that the average age in Company X is x & average age in Company Y is y & average age in Company Z is z then the total age of employees of company X will be 3x & the total age of employees of company Y will be 4y & the total age of employees of company Z will be 8z & therefore, the total age of all the employees of the three companies will be 3x+4y+8z & then we can calculate the average. Please review & give me some advise on that. Thanks !! in advance !!!!

Can you please tell me which step in the solution (new-ds-set-150653-80.html#p1211908) you don't understand. This way I think it would be easier to explain. _________________

okay ..sir .. as you said in the first line of ur solution to this Q ...

Given that the ratio of the number of employees is 3x:4x:8x, for some positive multiple x.,, I think you have taken x as an integer so that it is flexible to say tat this ratio can be 3:4:8 or 3*2 : 4*2 : 8*2 or 6:8:16 or 9:12:24... I would like to know why can't we change this ratio of number employees to the ratio of weights of employees of the three companies...Can we change the Ratio of number of Employees to the Ratio of weights of Employees of three companies .... Pls Advise , Thanks in Advance !! _________________

If you don’t make mistakes, you’re not working hard. And Now that’s a Huge mistake.

okay ..sir .. as you said in the first line of ur solution to this Q ...

Given that the ratio of the number of employees is 3x:4x:8x, for some positive multiple x.,, I think you have taken x as an integer so that it is flexible to say tat this ratio can be 3:4:8 or 3*2 : 4*2 : 8*2 or 6:8:16 or 9:12:24... I would like to know why can't we change this ratio of number employees to the ratio of weights of employees of the three companies...Can we change the Ratio of number of Employees to the Ratio of weights of Employees of three companies .... Pls Advise , Thanks in Advance !!

What is the ratio of weights?

Anyway, we need to find whether (average age)=(total age)/(# of employees)<40, or whether (total age)/(3x+4x+8x)<40, which is the same as: is (total age)<600x?

okay ..sir .. as you said in the first line of ur solution to this Q ...

Given that the ratio of the number of employees is 3x:4x:8x, for some positive multiple x.,, I think you have taken x as an integer so that it is flexible to say tat this ratio can be 3:4:8 or 3*2 : 4*2 : 8*2 or 6:8:16 or 9:12:24... I would like to know why can't we change this ratio of number employees to the ratio of weights of employees of the three companies...Can we change the Ratio of number of Employees to the Ratio of weights of Employees of three companies .... Pls Advise , Thanks in Advance !!

What is the ratio of weights?

Anyway, we need to find whether (average age)=(total age)/(# of employees)<40, or whether (total age)/(3x+4x+8x)<40, which is the same as: is (total age)<600x?

Does this makes sense?

-------------------------------------------------- I think there is some communication gap b/w us but still sir will try to make u understand what am I trying to say otherwise I will concentrate on ur solution & will try to understand....

First Ratio of # of Employees given as :::::: 3:4:8 ... so for me the # of employees in the three companies can be 3,4,8 or 6,8,16 or 9,12,24 any three numbers that satisfy the ratio 3:4:8. okay.

Second, in company X there must some average weight of employees, right ?? lets say that average weight is a1 okay. so the total weight of employees in company X will be 3*a1 , if we take that there 3 employees in company X. or if u want you can take this as 6 but for that u must satisfy the ratio of # of employees accordingly okay.

now similarly, in company Y there must be some average weight of employees, right ?? lets say that average weight is a2 okay. & therefore the total weight of employees in company Y will be 4*a2.

now similarly, in company Z there must some average weight of employees, right ?? lets say that average weight is a3 okay. & therefore the total weight of employees in company Z will be 8*a3.

Now the total weight of employees in three companies will be 3*a1 + 4*a2 + 8*a3......... okay. & if we are taking number of employees as 6,8,16, this will become 3*2*a1 + 4*2*a2 + 8*2*a3.... & in both the cases the # of employees will change but not the equation as if we take in first case

total weight of employees in three companies is 3*a1 + 4*a2 + 8*a3 so in this case the average weight will be 3*a1 + 4*a2 + 8*a3/3+4+8 ... or 3a1+4a2+8a3/15

& as the Q asks (average age)=(total age)/(# of employees)<40, or whether 3a1+4a2+8a3/15<40, or 3a1+4a2+8a3<15*40 or 3a1+4a2+8a3<600.

similarly for second scenario, if we take #of employees as 6,8,16 maintaining the same ratio of employees as 3:4:8. in this case the total weight will be 3*2*a1 + 4*2*a2 + 8*2*a3 or 6a1+8a2+16a3 & the average will be 6a1+8a2+16a3/6+8+16 or 6a1+8a2+16a3/30

again as the Q asks is (average age)=(total age)/(# of employees)<40, or whether 6a1+8a2+16a3/30, or 6a1+8a2+16a3<30*40 or 6a1+8a2+16a3<1200 or 2 (3a1+4a2+8a3) < 2*600 or 3a1+4a2+8a3<600 ..

but both the statements are disallowing it. so that's how I have this solution. if you think this can't be the way, so I'm really sorry to waste your time. As you are the Master of GMAT Math & you can't be wrong, I will review my logic again & concentrate on your solution to understand it better. Thanks for your precious time . Thanks !!!! _________________

If you don’t make mistakes, you’re not working hard. And Now that’s a Huge mistake.

Harvard asks you to write a post interview reflection (PIR) within 24 hours of your interview. Many have said that there is little you can do in this...