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# All the students in an algebra class took a 100-point test. Five stude

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Math Expert
Joined: 02 Sep 2009
Posts: 59728
All the students in an algebra class took a 100-point test. Five stude  [#permalink]

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22 Mar 2019, 00:24
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Difficulty:

65% (hard)

Question Stats:

53% (02:26) correct 47% (02:17) wrong based on 58 sessions

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All the students in an algebra class took a 100-point test. Five students scored 100, each student scored at least 60, and the mean score was 76. What is the smallest possible number of students in the class?

(A) 10
(B) 11
(C) 12
(D) 13
(E) 14

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Posts: 110
Re: All the students in an algebra class took a 100-point test. Five stude  [#permalink]

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22 Mar 2019, 00:45
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There are two ways of approaching this problem... First is by answer choices...

Say if I have got 12(because it's the middle value) students in the class... The sum of their scores would be 12*76 = 912...
Out of 12 students 5 got 100... Now 912-500 = 412..
Each student scored atleast 60... Which means the minimum value of the sum of marks scored by remaining 7 students should be 7*60 = 420.. and 420>412.. which means our answer should be the next greatest value I.e 13...

Alternatively this can also be solved algebraically... Say there are "n" students in the class... The total or the sum of marks scored by n students is 76n..
5 students got 100 , their sum is 500... The remaining (n-5) atleast got 60...

(500+(n-5)60)= 76n
500 +60n -300 = 76n
500-300 = 76n-60n
200 = 16n
N= 12.5... rounding up to 13..

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Re: All the students in an algebra class took a 100-point test. Five stude  [#permalink]

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22 Mar 2019, 02:38
Bunuel wrote:
All the students in an algebra class took a 100-point test. Five students scored 100, each student scored at least 60, and the mean score was 76. What is the smallest possible number of students in the class?

(A) 10
(B) 11
(C) 12
(D) 13
(E) 14

5*100+(x-5)*60 = 76x
solve for x = 12.5 ~ 13
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Re: All the students in an algebra class took a 100-point test. Five stude  [#permalink]

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22 Jun 2019, 08:16
5 Students with 100 P
Mean at 76

We are 120 (5*24) points away from the mean when we start out, this means we have to find low scores to bring the mean back down to 76.

The difference from the mean to the minimum score of the students is 16 (76-60).

Distance from Mean (120) divided by the amount the new students can decrease the mean (16) = 120/16 = 7.5
Thus, we need 8 children with lower than mean scores.

8 new children + 5 existing children = class size of 13
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Re: All the students in an algebra class took a 100-point test. Five stude  [#permalink]

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25 Jun 2019, 04:09
This is a good question on the concepts of mean and minimization. Since the mean of the data set is fixed, the total is fixed as well. To minimize the number of students, we will have to reach this total, while keeping in mind that the smallest score any student obtained is 60.

Let the number of students be x. Since the mean score is 76, the sum of their scores is 76x.
Of the x students, 5 students scored 100, which totals to 500. Therefore, the total of the remaining students should be 76x – 500, with the smallest score being 60.
We can now plug in values for x from the answer options, starting with C.

If x = 12, the number of students who didn’t get a 100 = 7. Since each student got at least 60 points, the minimum total score by these 7 students is 420. The sum total of all the points = 500 + 420 = 920.

But 76 * 12 = 912. This is lesser than the minimum sum total required. This means, 12 is not the required minimum number of students.

If 12 is not the smallest possible number, 10 and 11 cannot be the smallest possible number. Options A, B and C can be eliminated.

If x = 13, total points = 76 * 13 = 988. Also, 500 + 8*60 = 500 + 480 = 980. This satisfies all the constraints defined in the question. So, the minimum possible number of students is 13.

Since, we are trying to minimize the number of students, while keeping the total points fixed, we also need to ensure that every student (other than the ones who scored 100) gets close to the minimum score, 60.

Hope this helps!
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Re: All the students in an algebra class took a 100-point test. Five stude   [#permalink] 25 Jun 2019, 04:09
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