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# There are 28 students in a class. The first student draws a line of le

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Math Expert
Joined: 02 Sep 2009
Posts: 46194
There are 28 students in a class. The first student draws a line of le [#permalink]

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21 May 2017, 06:07
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Difficulty:

55% (hard)

Question Stats:

82% (01:12) correct 18% (02:31) wrong based on 17 sessions

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There are 28 students in a class. The first student draws a line of length ‘a’ units on the black board of the class. The second student draws a line of length ‘b’ units (a is not equal to b) on the black board of the class. The third student draws a line whose length is the average of the lengths of the two lines that are drawn by the first and second student on the black board. The fourth student draws a line whose length is the average of the lengths of the lines that are drawn by the first, second and third student. This process of drawing lines on the black board continues till the twenty-eighth student of the class. How many students drew the line having the same length?

A. 28
B. 27
C. 26
D. 25
E. 24

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Re: There are 28 students in a class. The first student draws a line of le [#permalink]

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21 May 2017, 07:35
Consider value of a as 3, and b as 1(since a is not equal to b)
The average of a,b is 2(making it the 3rd value)
The 4rth value is an average of a=3,b=1 and average of a,b(2) which is 2, again!
This pattern repeats till the 28th person who will have the same length(2)

Excluding a and b every other student in the class has the same length.
So 26 students have the same length(Option C)
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There are 28 students in a class. The first student draws a line of le [#permalink]

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21 May 2017, 07:41
Bunuel wrote:
There are 28 students in a class. The first student draws a line of length ‘a’ units on the black board of the class. The second student draws a line of length ‘b’ units (a is not equal to b) on the black board of the class. The third student draws a line whose length is the average of the lengths of the two lines that are drawn by the first and second student on the black board. The fourth student draws a line whose length is the average of the lengths of the lines that are drawn by the first, second and third student. This process of drawing lines on the black board continues till the twenty-eighth student of the class. How many students drew the line having the same length?

A. 28
B. 27
C. 26
D. 25
E. 24

1st turn: $$a$$
2nd turn: $$b$$
3rd turn: $$\frac{a+b}{2}$$
4th turn: $$\frac{a+b+\frac{a+b}{2}}{3}=\frac{a+b}{2}$$
5th turn: $$\frac{a+b+\frac{a+b}{2}+\frac{a+b}{2}}{4}=\frac{a+b}{2}$$

Assume that in $$k$$th-turn, the length of the line is $$l_k=\frac{l_1+l_2+...+l_{k-1}}{k-1}=\frac{a+b}{2} \implies l_1+l_2+...+l_{k-1}=\frac{(a+b)(k-1)}{2}$$

In next turn, $$(k+1)$$th-turn, the length of the line will be
$$\frac{l_1+l_2+...+l_{k-1}+l_k}{k}=\frac{(l_1+l_2+...+l_{k-1})+l_k}{k}=\frac{\frac{(a+b)(k-1)}{2}+\frac{a+b}{2}}{k}=\frac{(a+b)(k)}{2k}=\frac{a+b}{2}$$.

Hence, these lines will have the same length: $$l_3, l_4, ..., l_{28}$$. Hence the answer will be C. 26
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There are 28 students in a class. The first student draws a line of le   [#permalink] 21 May 2017, 07:41
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