November 20, 2018 November 20, 2018 09:00 AM PST 10:00 AM PST The reward for signing up with the registration form and attending the chat is: 6 free examPAL quizzes to practice your new skills after the chat. November 20, 2018 November 20, 2018 06:00 PM EST 07:00 PM EST What people who reach the high 700's do differently? We're going to share insights, tips and strategies from data we collected on over 50,000 students who used examPAL.
Author 
Message 
TAGS:

Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 50661

Six numbers are randomly selected and placed within a set. If the set
[#permalink]
Show Tags
10 Jun 2018, 04:00
Question Stats:
22% (01:38) correct 78% (02:45) wrong based on 64 sessions
HideShow timer Statistics



Math Expert
Joined: 02 Aug 2009
Posts: 7037

Six numbers are randomly selected and placed within a set. If the set
[#permalink]
Show Tags
10 Jun 2018, 19:40
Bunuel wrote: Six numbers are randomly selected and placed within a set. If the set has a range of 16, a median of 6, a mean of 7 and a mode of 7, what is the greatest of the six numbers?
(1) The sum of the two smallest numbers is onefifth of the sum of the two greatest numbers (2) The middle two numbers are 5 and 7 Great question.. Let the numbers be a,b,c,d,e,f Given info.. A) fa=16 B) (c+d)/2=6 C) a,b,c are surely less than 7 D) mode means that 7 is atleast twice but the largest that is 'f' is >7 as it has to get the average to 7 and a,b,c are<7 E) so d and e are 7 each F) now (c+d)/2=7 and d is 7 , so c is 5.. G) numbers become a,b,5,7,7,f We have to find 'f'Let's see what each statement tells us.. (1) The sum of the two smallest numbers is onefifth of the sum of the two greatest numbers This tells us that \(a+b=\frac{7+ f}{5}\)..(i) Also \(\frac{a+b+5+7+7+f}{6}=7....a+b+f=42(19)=23\)...(ii) From (i) & (ii).. \(\frac{(7+f)}{5}+f=23......7+f+5f=115.....6f=108...f=18\) So largest number is 18 Rather all numbers can be found a=1816=2 & b =3 2,3,5,7,7,18 Sufficient (2) The middle two numbers are 5 and 7 Nothing new.. It has already been found in initial statement Insufficient A
_________________
1) Absolute modulus : http://gmatclub.com/forum/absolutemodulusabetterunderstanding210849.html#p1622372 2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html 3) effects of arithmetic operations : https://gmatclub.com/forum/effectsofarithmeticoperationsonfractions269413.html
GMAT online Tutor



Intern
Joined: 05 Feb 2017
Posts: 39

Six numbers are randomly selected and placed within a set. If the set
[#permalink]
Show Tags
10 Jun 2018, 19:50
Hi Chetan, I think C should be 7. How did you deduce that c<7 because a scenario can be possible where c and d both can be 7 with the mean still 7. I agree that a,b will be less than 7 and f>7. I assume all c d e will have the same value I.e 7. Correct me if I am wrong? Sent from my iPhone using GMAT Club Forum mobile app



Math Expert
Joined: 02 Aug 2009
Posts: 7037

Six numbers are randomly selected and placed within a set. If the set
[#permalink]
Show Tags
10 Jun 2018, 20:09
prerakgoel03 wrote: Hi Chetan, I think C should be 7. How did you deduce that c<7 because a scenario can be possible where c and d both can be 7 with the mean still 7. I agree that a,b will be less than 7 and f>7. I assume all c d e will have the same value I.e 7. Correct me if I am wrong? Sent from my iPhone using GMAT Club Forum mobile appC and d would depend on median which is 6.. So c can be at the most 6 where even d is 6, otherwise d6=6c.. a,b,c,6,d,e,f Let me further amplify the terms 1) median .. it is the middle value If odd numbers the middle value in the set.. a,b,c..... B is the median If the set contains an even number of elements, the average of two middle values..a,b,c,d....(b+c)/2 2) mode... The number which shows up the maximum time in the set.. A,b,c,d,d,e...d is mode 3) mean.. It is the average of numbers.. a,b,c.....(a+b+c)/3 I am sure you would know this but are getting confused in mode and median
_________________
1) Absolute modulus : http://gmatclub.com/forum/absolutemodulusabetterunderstanding210849.html#p1622372 2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html 3) effects of arithmetic operations : https://gmatclub.com/forum/effectsofarithmeticoperationsonfractions269413.html
GMAT online Tutor



Director
Status: Learning stage
Joined: 01 Oct 2017
Posts: 930
WE: Supply Chain Management (Energy and Utilities)

Six numbers are randomly selected and placed within a set. If the set
[#permalink]
Show Tags
10 Jun 2018, 20:15
prerakgoel03 wrote: Hi Chetan, I think C should be 7. How did you deduce that c<7 because a scenario can be possible where c and d both can be 7 with the mean still 7. I agree that a,b will be less than 7 and f>7. I assume all c d e will have the same value I.e 7. Correct me if I am wrong? Sent from my iPhone using GMAT Club Forum mobile appHi prerakgoel03, If c=d=e=7 (as per your observation), Is median of 6 numbers 7? we need to keep the median value as 6 as per question. Here , we have even numbers of elements, n=6, For even numbers of elements, median=Average of \((\frac{n}{2})\)th term and \((\frac{n}{2} +1)\)th term Hence , Median=Average of 3rd & 4th term we have deduced that 4th term=d=7 So, 6=\((\frac{c+7)}{2}\) Or, c=127=5 which is less than 7. Hope it helps.
_________________
Regards,
PKN
Rise above the storm, you will find the sunshine



Intern
Joined: 05 Feb 2017
Posts: 39

Six numbers are randomly selected and placed within a set. If the set
[#permalink]
Show Tags
10 Jun 2018, 20:20
I think I misjudged the value as 7. Thanks for pointing out. I am clear now . Sent from my iPhone using GMAT Club Forum mobile app



Director
Status: Learning stage
Joined: 01 Oct 2017
Posts: 930
WE: Supply Chain Management (Energy and Utilities)

Re: Six numbers are randomly selected and placed within a set. If the set
[#permalink]
Show Tags
10 Jun 2018, 20:22
chetan2u wrote: Bunuel wrote: Six numbers are randomly selected and placed within a set. If the set has a range of 16, a median of 6, a mean of 7 and a mode of 7, what is the greatest of the six numbers?
(1) The sum of the two smallest numbers is onefifth of the sum of the two greatest numbers (2) The middle two numbers are 5 and 7 Great question.. Let the numbers be a,b,c,d,e,f Given info.. A) fe=16B) (c+d)/2=6 C) a,b,c are surely less than 7 D) mode means that 7 is atleast twice but the largest that is 'f' is >7 as it has to get the average to 7 and a,b,c are<7 E) so d and e are 7 each F) now (c+d)/2=7 and d is 7 , so c is 5.. G) numbers become a,b,5,7,7,f We have to find 'f'Let's see what each statement tells us.. (1) The sum of the two smallest numbers is onefifth of the sum of the two greatest numbers This tells us that \(a+b=\frac{7+ f}{5}\)..(i) Also \(\frac{a+b+5+7+7+f}{6}=7....a+b+f=42(19)=23\)...(ii) From (i) & (ii).. \(\frac{(7+f)}{5}+f=23......7+f+5f=115.....6f=108...f=18\) So largest number is 18 Rather all numbers can be found a=1816=2 & b =3 2,3,5,7,7,18 Sufficient (2) The middle two numbers are 5 and 7 Nothing new.. It has already been found in initial statement Insufficient A Hi chetan2u, Sir, I have obtained the six numbers as: 2, 3, 5, 7, 7, 18 I think the expression (A)highlighted above should be fa=16. Kindly correct me.
_________________
Regards,
PKN
Rise above the storm, you will find the sunshine



Math Expert
Joined: 02 Aug 2009
Posts: 7037

Re: Six numbers are randomly selected and placed within a set. If the set
[#permalink]
Show Tags
10 Jun 2018, 20:33
PKN wrote: chetan2u wrote: Bunuel wrote: Six numbers are randomly selected and placed within a set. If the set has a range of 16, a median of 6, a mean of 7 and a mode of 7, what is the greatest of the six numbers?
(1) The sum of the two smallest numbers is onefifth of the sum of the two greatest numbers (2) The middle two numbers are 5 and 7 Great question.. Let the numbers be a,b,c,d,e,f Given info.. A) fe=16B) (c+d)/2=6 C) a,b,c are surely less than 7 D) mode means that 7 is atleast twice but the largest that is 'f' is >7 as it has to get the average to 7 and a,b,c are<7 E) so d and e are 7 each F) now (c+d)/2=7 and d is 7 , so c is 5.. G) numbers become a,b,5,7,7,f We have to find 'f'Let's see what each statement tells us.. (1) The sum of the two smallest numbers is onefifth of the sum of the two greatest numbers This tells us that \(a+b=\frac{7+ f}{5}\)..(i) Also \(\frac{a+b+5+7+7+f}{6}=7....a+b+f=42(19)=23\)...(ii) From (i) & (ii).. \(\frac{(7+f)}{5}+f=23......7+f+5f=115.....6f=108...f=18\) So largest number is 18 Rather all numbers can be found a=1816=2 & b =3 2,3,5,7,7,18 Sufficient (2) The middle two numbers are 5 and 7 Nothing new.. It has already been found in initial statement Insufficient A Hi chetan2u, Sir, I have obtained the six numbers as: 2, 3, 5, 7, 7, 18 I think the expression (A)highlighted above should be fa=16. Kindly correct me. Thanks a lot.. It is a typo.. Range is fa
_________________
1) Absolute modulus : http://gmatclub.com/forum/absolutemodulusabetterunderstanding210849.html#p1622372 2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html 3) effects of arithmetic operations : https://gmatclub.com/forum/effectsofarithmeticoperationsonfractions269413.html
GMAT online Tutor



Manager
Joined: 16 Jan 2018
Posts: 57
Concentration: Finance, Technology

Six numbers are randomly selected and placed within a set. If the set
[#permalink]
Show Tags
11 Jun 2018, 17:05
Six numbers are randomly selected and placed within a set. If the set has a range of 16, a median of 6, a mean of 7 and a mode of 7, what is the greatest of the six numbers?
(1) The sum of the two smallest numbers is onefifth of the sum of the two greatest numbers (2) The middle two numbers are 5 and 7
let the number in asc. order be  a,b,c,d,e,f
1) Sufficient
5(a+b)=e+f c+d=12 (from median) fa=16 (from range) a+b+c+d+e+f=42 (from mean)
6(a+b)=30 so, a+b=5
e+f=25
now, e and f both cant be 7. so definitely e is 7 which also means d is 7.
now we can just fill in the series 2,3,5,7,7,18
2) No new info.
Answer A



Director
Joined: 14 Dec 2017
Posts: 511

Six numbers are randomly selected and placed within a set. If the set
[#permalink]
Show Tags
12 Jun 2018, 00:14
Bunuel wrote: Six numbers are randomly selected and placed within a set. If the set has a range of 16, a median of 6, a mean of 7 and a mode of 7, what is the greatest of the six numbers?
(1) The sum of the two smallest numbers is onefifth of the sum of the two greatest numbers (2) The middle two numbers are 5 and 7 Let the six numbers be a, b, c, d, e, f, in ascending order. a as smallest & f as greatest. Given, Range f  a = 16, Median (c + d)/2 = 6, c + d = 12, giving (c,d) = (5,7) or (4,8) or (3,9)...etc. for (c < d) Mode 7, hence 7 is repeated at least twice. Mean (a + b + c + d + e + f)/6 = 7 Hence a + b + c + d + e + f = 42.....(i) Required is f = ? Statement 1: e + f = 5 (a + b) we get, eq.(i) as, 1/5*(e + f) + 12 + e + f = 42 e + f = 25 Now, lets take (c,d) to satisfy (e,f), such that e + f = 25 If (c,d) = (3,9), we get the numbers, as a,b,3,9,e,f, for this we cant get Mode as 7. hence discard this (c,d) =(3,9) From this we can see that, d has to be 7, or else the numbers don't fit the constraints. Hence (c,d) = (5,7) & therefore e = 7, to satisfy that Mode = 7, Substituting e = 7 in e + f = 25, we get f = 18. Statement 1 is Sufficient. Statement 2: The middle two numbers are 5 and 7, hence we have c = 5 & d = 7, since c < d So we have the numbers as, a, b, 5, 7, e, f a + b +5 + 7 + e + f = 42 a + b + e + f = 30, Now since mode is 7, the only possibility we have is e =7, we cannot take f = 7 as it will inflate a, b & the constraints will not be met. So now we have, a + b + f = 23, Hence we can try various combination for a, b & f to satisfy the constraints, a = 3, b = 4, f = 16, we get the numbers as 3, 4, 5, 7, 7, 16, doesn't satisfy range constraint. or a = 2, b = 3, f = 18, we get the numbers as 2, 3, 5, 7, 7, 18, satisfying all the constraints. or a = 1, b = 2, f = 20, we get the numbers as 1, 2, 5, 7, 7, 20, again doesn't satisfy range constraint. Hence we get, only f = 18 satisfying the constraints. Statement 2 is Sufficient. Answer D. Thanks, GyM Ps  chetan2u, kindly review, i must have messed up somewhere, however cannot figure out where.
_________________
New to GMAT Club  https://gmatclub.com/forum/newtogmatclubneedhelp271131.html#p2098335



Math Expert
Joined: 02 Aug 2009
Posts: 7037

Re: Six numbers are randomly selected and placed within a set. If the set
[#permalink]
Show Tags
12 Jun 2018, 03:27
GyMrAT wrote: Bunuel wrote: Six numbers are randomly selected and placed within a set. If the set has a range of 16, a median of 6, a mean of 7 and a mode of 7, what is the greatest of the six numbers?
(1) The sum of the two smallest numbers is onefifth of the sum of the two greatest numbers (2) The middle two numbers are 5 and 7 Let the six numbers be a, b, c, d, e, f, in ascending order. a as smallest & f as greatest. Given, Range f  a = 16, Median (c + d)/2 = 6, c + d = 12, giving (c,d) = (5,7) or (4,8) or (3,9)...etc. for (c < d) Mode 7, hence 7 is repeated at least twice. Mean (a + b + c + d + e + f)/6 = 7 Hence a + b + c + d + e + f = 42.....(i) Required is f = ? Statement 1: e + f = 5 (a + b) we get, eq.(i) as, 1/5*(e + f) + 12 + e + f = 42 e + f = 25 Now, lets take (c,d) to satisfy (e,f), such that e + f = 25 If (c,d) = (3,9), we get the numbers, as a,b,3,9,e,f, for this we cant get Mode as 7. hence discard this (c,d) =(3,9) From this we can see that, d has to be 7, or else the numbers don't fit the constraints. Hence (c,d) = (5,7) & therefore e = 7, to satisfy that Mode = 7, Substituting e = 7 in e + f = 25, we get f = 18. Statement 1 is Sufficient. Statement 2: The middle two numbers are 5 and 7, hence we have c = 5 & d = 7, since c < d So we have the numbers as, a, b, 5, 7, e, f a + b +5 + 7 + e + f = 42 a + b + e + f = 30, Now since mode is 7, the only possibility we have is e =7, we cannot take f = 7 as it will inflate a, b & the constraints will not be met. So now we have, a + b + f = 23, Hence we can try various combination for a, b & f to satisfy the constraints, a = 3, b = 4, f = 16, we get the numbers as 3, 4, 5, 7, 7, 16, doesn't satisfy range constraint. or a = 2, b = 3, f = 18, we get the numbers as 2, 3, 5, 7, 7, 18, satisfying all the constraints. or a = 1, b = 2, f = 20, we get the numbers as 1, 2, 5, 7, 7, 20, again doesn't satisfy range constraint. Hence we get, only f = 18 satisfying the constraints. Statement 2 is Sufficient. Answer D. Thanks, GyM Ps  chetan2u, kindly review, i must have messed up somewhere, however cannot figure out where. Hi Yes, you will have two modes to satisfy then.. 1,5,5,7,7,17 and few more but here mode will be 5 and 7.. But you don't require statement II at all because in the given conditions in the main statement, the middle two have to be 5 & 7.. So B really doesn't give you anything new
_________________
1) Absolute modulus : http://gmatclub.com/forum/absolutemodulusabetterunderstanding210849.html#p1622372 2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html 3) effects of arithmetic operations : https://gmatclub.com/forum/effectsofarithmeticoperationsonfractions269413.html
GMAT online Tutor



Director
Joined: 14 Dec 2017
Posts: 511

Re: Six numbers are randomly selected and placed within a set. If the set
[#permalink]
Show Tags
12 Jun 2018, 04:26
chetan2u wrote: GyMrAT wrote: Bunuel wrote: Six numbers are randomly selected and placed within a set. If the set has a range of 16, a median of 6, a mean of 7 and a mode of 7, what is the greatest of the six numbers?
(1) The sum of the two smallest numbers is onefifth of the sum of the two greatest numbers (2) The middle two numbers are 5 and 7 Let the six numbers be a, b, c, d, e, f, in ascending order. a as smallest & f as greatest. Given, Range f  a = 16, Median (c + d)/2 = 6, c + d = 12, giving (c,d) = (5,7) or (4,8) or (3,9)...etc. for (c < d) Mode 7, hence 7 is repeated at least twice. Mean (a + b + c + d + e + f)/6 = 7 Hence a + b + c + d + e + f = 42.....(i) Required is f = ? Statement 1: e + f = 5 (a + b) we get, eq.(i) as, 1/5*(e + f) + 12 + e + f = 42 e + f = 25 Now, lets take (c,d) to satisfy (e,f), such that e + f = 25 If (c,d) = (3,9), we get the numbers, as a,b,3,9,e,f, for this we cant get Mode as 7. hence discard this (c,d) =(3,9) From this we can see that, d has to be 7, or else the numbers don't fit the constraints. Hence (c,d) = (5,7) & therefore e = 7, to satisfy that Mode = 7, Substituting e = 7 in e + f = 25, we get f = 18. Statement 1 is Sufficient. Statement 2: The middle two numbers are 5 and 7, hence we have c = 5 & d = 7, since c < d So we have the numbers as, a, b, 5, 7, e, f a + b +5 + 7 + e + f = 42 a + b + e + f = 30, Now since mode is 7, the only possibility we have is e =7, we cannot take f = 7 as it will inflate a, b & the constraints will not be met. So now we have, a + b + f = 23, Hence we can try various combination for a, b & f to satisfy the constraints, a = 3, b = 4, f = 16, we get the numbers as 3, 4, 5, 7, 7, 16, doesn't satisfy range constraint. or a = 2, b = 3, f = 18, we get the numbers as 2, 3, 5, 7, 7, 18, satisfying all the constraints. or a = 1, b = 2, f = 20, we get the numbers as 1, 2, 5, 7, 7, 20, again doesn't satisfy range constraint. Hence we get, only f = 18 satisfying the constraints. Statement 2 is Sufficient. Answer D. Thanks, GyM Ps  chetan2u, kindly review, i must have messed up somewhere, however cannot figure out where. Hi Yes, you will have two modes to satisfy then.. 1,5,5,7,7,17 and few more but here mode will be 5 and 7.. But you don't require statement II at all because in the given conditions in the main statement, the middle two have to be 5 & 7.. So B really doesn't give you anything new Thanks for the reply!, however i beg to differ. In my opinion, the given information, in the prompt, provides ambiguity for the middle 2 numbers. We cannot clearly deduce that the middle two numbers will be 5 & 7. Hence once the middle two terms are determined, the other numbers are to be found, such that they satisfy the constraints. Statement 1 helps us to zone in on the middle two numbers & hence help in finding the greatest number, through the relationship between last two terms & first two terms. Statement 2, on the other hand, provides us the middle two terms & hence help in finding the greatest number, by determining that there cannot be two modes & the mean as well as range constraints can be satisfied with only one set of numbers. using statement 2, we cannot say the numbers are, 2.1, 3, 5, 7, 7, 18.1, which satisfies the range constraint but doesn't satisfy the constraint for mean. There will be only one unique set of numbers, containing 5 & 7 as middle terms & satisfying the other constraints. Thanks, GyM
_________________
New to GMAT Club  https://gmatclub.com/forum/newtogmatclubneedhelp271131.html#p2098335



Math Expert
Joined: 02 Aug 2009
Posts: 7037

Re: Six numbers are randomly selected and placed within a set. If the set
[#permalink]
Show Tags
12 Jun 2018, 04:36
GyMrAT wrote: determined, the other numbers are to be found, such that they satisfy the constraints.
Statement 1 helps us to zone in on the middle two numbers & hence help in finding the greatest number, through the relationship between last two terms & first two terms.
Statement 2, on the other hand, provides us the middle two terms & hence help in finding the greatest number, by determining that there cannot be two modes & the mean as well as range constraints can be satisfied with only one set of numbers.
using statement 2, we cannot say the numbers are, 2.1, 3, 5, 7, 7, 18.1, which satisfies the range constraint but doesn't satisfy the constraint for mean. There will be only one unique set of numbers, containing 5 & 7 as middle terms & satisfying the other constraints.
Thanks, GyM Hi.. It is good to differ... But logic for why middle two numbers can be only 5 and 7 is.. Let the numbers be a,b,c,d,e,f Given info.. A) fa=16 B) (c+d)/2=6 C) a,b,c are surely less than 7 D) mode means that 7 is atleast twice but the largest that is 'f' is >7 as it has to get the average to 7 and a,b,c are<7 E) so d and e are 7 each F) now (c+d)/2=7 and d is 7 , so c is 5.. G) numbers become a,b,5,7,7,f You can see if there can be any other middle numbers...
_________________
1) Absolute modulus : http://gmatclub.com/forum/absolutemodulusabetterunderstanding210849.html#p1622372 2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html 3) effects of arithmetic operations : https://gmatclub.com/forum/effectsofarithmeticoperationsonfractions269413.html
GMAT online Tutor



GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 480

Re: Six numbers are randomly selected and placed within a set. If the set
[#permalink]
Show Tags
04 Nov 2018, 11:29
Bunuel wrote: Six numbers are randomly selected and placed within a set. If the set has a range of 16, a median of 6, a mean of 7 and a mode of 7, what is the greatest of the six numbers?
(1) The sum of the two smallest numbers is onefifth of the sum of the two greatest numbers (2) The middle two numbers are 5 and 7
Very nice problem (in which a "structured candidate" has considerable advantage)! \({a_1} \le {a_2} \le \ldots \le {a_6}\,\,\,\,\,\left\{ \matrix{ \,{a_6} = {a_1} + 16\, \hfill \cr \,{a_3} + {a_4} = 12 \hfill \cr \,\sum\nolimits_6 {\, = } \,\,\,42 \hfill \cr \,7{\rm{s}}\,\,{\rm{appear}}\,\,{\rm{most}}\, \hfill \cr} \right.\) \(? = {a_6}\) We "MUST" start with statement (2): it is easier and it will help us "gain knowledge and sensitivity" to what is really going on! \(\left( 2 \right)\,\,\,\,{a_1}\,\,\,{a_2}\,\,\,5\,\,\,7\,\,\,7\,\,\,{a_6}\mathop \ne \limits^{\left( {**} \right)} 7\,\,\,\,\,\left\{ \matrix{ \,{a_6} = {a_1} + 16\, \hfill \cr \,{a_3} + {a_4} = 12\,\,\,\,\left( {{\rm{sure}}} \right) \hfill \cr \,\sum\nolimits_6 {\, = } \,\,\,42\,\,\,\left( * \right) \hfill \cr \,7{\rm{s}}\,\,{\rm{appear}}\,\,{\rm{most}}\,\,\,\,\left( {\,\left( {**} \right)\,\,{a_6} = 7\,\,\, \Rightarrow \,\,\,{a_1} = 7  16 =  9\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,{a_2} = 25 > {a_3}\,\,{\rm{impossible}}\,} \right)\, \hfill \cr} \right.\) \(\left\{ \matrix{ \,{\rm{Take}}\,\,\left( {{a_1},{a_6},{a_2}} \right) = \left( {1,17,5} \right)\,\,\,{\rm{viable}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{a_6} = \,\,17 \hfill \cr \,{\rm{Take}}\,\,\left( {{a_1},{a_6},{a_2}} \right) = \left( {2,18,3} \right)\,\,\,{\rm{viable}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{a_6} = \,\,18\,\, \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{INSUFF}}.\) \(\left( 1 \right)\,\,\,5\left( {{a_1} + {a_2}} \right) = {a_5} + {a_6}\,\,\,\,\left\{ \matrix{ \,{a_6} = {a_1} + 16\, \hfill \cr \,{a_3} + {a_4} = 12 \hfill \cr \,\sum\nolimits_6 {\, = } \,\,\,42 \hfill \cr \,7{\rm{s}}\,\,{\rm{appear}}\,\,{\rm{most}}\, \hfill \cr} \right.\) \({a_3} = 7\,\,\,\, \Rightarrow \,\,{a_4} = 12  7 = 5 < {a_3}\,\,\,\,\, \Rightarrow \,\,\,\,{\rm{impossible}}\) \({a_{`5}} = {a_6} = 7\,\,\,\, \Rightarrow \,\,\,\,\left\{ \matrix{ \,{a_1} = 7  16 =  9 \hfill \cr \,{a_2} = \sum\nolimits_6 {\,  \left[ {{a_1} + \left( {{a_3} + {a_4}} \right) + {a_5} + {a_6}} \right] = } \,\,42  \left( {  9 + 12 + 2 \cdot 7} \right) = 25\,\,\, > \,\,\,{a_5} \hfill \cr} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,{\rm{impossible}}\) \({\rm{Hence}}:\,\,\,\,{a_1}\,\,\,{a_2}\,\,\,{a_3}\,\,\,7\,\,\,7\,\,\,{a_6} \ne 7\,\,\,\,\,\left\{ \matrix{ \,{a_6} = {a_1} + 16\, \hfill \cr \,{a_3} + {a_4} = 12\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{a_3} = 12  7 = 5 \hfill \cr \,\sum\nolimits_6 {\, = } \,\,\,42\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{a_1} + {a_2} + {a_6} = 42  {a_3}  \left( {{a_4} + {a_5}} \right) = 42  5  14 = 23 \hfill \cr \,5\left( {{a_1} + {a_2}} \right) = {a_5} + {a_6}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,5\left( {23  {a_6}} \right) = 7 + {a_6}\,\,\,\,\, \Rightarrow \,\,\,\,? = {a_6}\,\,\,{\rm{unique}} \hfill \cr} \right.\) This solution follows the notations and rationale taught in the GMATH method. Regards, Fabio.
_________________
Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version) Course release PROMO : finish our test drive till 30/Nov with (at least) 50 correct answers out of 92 (12questions Mock included) to gain a 50% discount!




Re: Six numbers are randomly selected and placed within a set. If the set &nbs
[#permalink]
04 Nov 2018, 11:29






