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# A class has 4 sections P, Q, R and S and the average weights of the st

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A class has 4 sections P, Q, R and S and the average weights of the st  [#permalink]

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29 Jun 2017, 03:25
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Difficulty:

95% (hard)

Question Stats:

49% (02:39) correct 51% (02:57) wrong based on 199 sessions

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A class has 4 sections P, Q, R and S and the average weights of the students in the sections are 45lb, 50lb, 55lb and 65lb respectively. What is the maximum possible number of students in section R if there are 40 students in all sections combined and the average weight of the all students across all the sections is 55lb? It is known that each section has at least one student.

(A) 18
(B) 20
(C) 25
(D) 35
(E) 37

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Re: A class has 4 sections P, Q, R and S and the average weights of the st  [#permalink]

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29 Jun 2017, 03:33
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Bunuel wrote:
A class has 4 sections P, Q, R and S and the average weights of the students in the sections are 45lb, 50lb, 55lb and 65lb respectively. What is the maximum possible number of students in section R if there are 40 students in all sections combined and the average weight of the all students across all the sections is 55lb? It is known that each section has at least one student.

(A) 18
(B) 20
(C) 25
(D) 35
(E) 37

Average is 55
P : Q :R :S
45 : 50 : 55 : 65
We need to minimize the number of students in each section (Each will have one atleast)

R is mean

P has 1 : 10 away from mean
Q has 1 : 5 away from mean

Then S will have 3 students : 3*5 to maintain the mean of overall population

Leftover students in R (Maximum) : 40 - 1- 1-3 = 35
D
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Re: A class has 4 sections P, Q, R and S and the average weights of the st  [#permalink]

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01 Jul 2017, 16:57
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In order to maximize the mean we need to equalize the difference of numbers between the two poles of the data set.

45 has a difference of 10 from 55 and
50 has difference of 5 from 55 therefore, the left side of the mean has a total difference of 15 at the least.

65 has a difference of 10 from the mean, the least difference possible. Therefore, the least average of combined weight to the left should be equal to the least combined weight of the right.

$$(1)10 + 2 (5) = 2 (10) ==> 10 + 10= 20$$

Therefore, a total of 5 students in other sections which bring the number of students in Section R to 25.
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Re: A class has 4 sections P, Q, R and S and the average weights of the st  [#permalink]

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01 Aug 2017, 00:48
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Bunuel wrote:
A class has 4 sections P, Q, R and S and the average weights of the students in the sections are 45lb, 50lb, 55lb and 65lb respectively. What is the maximum possible number of students in section R if there are 40 students in all sections combined and the average weight of the all students across all the sections is 55lb? It is known that each section has at least one student.

(A) 18
(B) 20
(C) 25
(D) 35
(E) 37

We've 45 50 55 65 and the average is 55. Now, around the mean the distribution is -10 (for 45) and -5(for 50) = -15 on the left AND 0(55) and 10(65) = +10 on the right.

We must now try and balance both sides in such a manner so as to maximize number of students for R.
On the right the increment is in steps of 10, next biggest = 20. This means we will give 2 students to section S (10 X 2)
AND on the left we will increment one student in Q. This implies that 2 students in Q [(-10 X 1) + (-5 X 2)]. Therefore, both sides are balanced.

We will get, P has 1; Q has 2 and S has 2. Therefore, Number of students in R = 40 - 1 - 2 - 2 = 35.
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Joined: 11 Feb 2017
Posts: 188
Re: A class has 4 sections P, Q, R and S and the average weights of the st  [#permalink]

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01 Aug 2017, 07:47
Bunuel wrote:
A class has 4 sections P, Q, R and S and the average weights of the students in the sections are 45lb, 50lb, 55lb and 65lb respectively. What is the maximum possible number of students in section R if there are 40 students in all sections combined and the average weight of the all students across all the sections is 55lb? It is known that each section has at least one student.

(A) 18
(B) 20
(C) 25
(D) 35
(E) 37

Can someone explain me this question in the most easiest way to understand?
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Joined: 19 Jun 2017
Posts: 6
Re: A class has 4 sections P, Q, R and S and the average weights of the st  [#permalink]

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08 Aug 2017, 06:30
1
Call p,q,r,s are the number of students in each section respectively. We have p+q+r+s= 40
Total W=45p+50q+55r+65s=55*40
Rewrite the W with respect to average weight of R (55) W=(55-10)p+(55-5)q+55r+(55+10)s=55(p+q+r+s)+10s-5q-10p=W+10s-5q-10p
---> 10s-5q-10p=0 hence s:q:p=2:1:2
Each section must have at least 1 student and max r --> min q =1 --> min s = min p=2 --> max r = 40 -5=35 -->D
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Joined: 16 Sep 2011
Posts: 96
Re: A class has 4 sections P, Q, R and S and the average weights of the st  [#permalink]

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07 Oct 2018, 01:09
Bunuel wrote:
A class has 4 sections P, Q, R and S and the average weights of the students in the sections are 45lb, 50lb, 55lb and 65lb respectively. What is the maximum possible number of students in section R if there are 40 students in all sections combined and the average weight of the all students across all the sections is 55lb? It is known that each section has at least one student.

(A) 18
(B) 20
(C) 25
(D) 35
(E) 37

What to look in an average problem is distance of a object from its average... so here average is 55lb which is section R

distance of P from section R= -10lb (negative as I consider R as centre since average lies at R)
distance of Q from R = - 5lb (negative as I consider R as centre since average lies at R)
distance of S from R = 10lb

Given: Each section has at least one student ..
So 37 is ruled out since if we take 1 from P, 1 from Q, 1 from S and 37 from R....average will not be 55lb....Lets see how?
1 from P = -10 distance from R.....1 from S = +10 distance from R...so both negate each other but we have additional -5 from P...so average will be slightly below 55lb now

Lets take another case 1 from P = -10lb
2 from Q = 2*-5lb = -10lb
and 2 from S= 2*+10lb= 20lb

So overall 1 from P + 2from Q = 2 from S...and average still remains at 55lb in section R

So total students taken= 1from P + 2 from Q + 2 from S = 5

Remaning students in R = 40-5= 35, which is the answer

Please give kudos if it is useful for you to understand..
Re: A class has 4 sections P, Q, R and S and the average weights of the st   [#permalink] 07 Oct 2018, 01:09
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