Mary and three other students took a math test. Each of their scores w

Question

Mary and three other students took a math test. Each of their scores was a non-negative integer. The teacher announced that the average score (of the 4 students) was 20. Mary immediately knew that all of the other three students scored below average. What is the minimum score that Mary could have gotten to be certain of the above situation?

A. 40
B. 41
C. 60
D. 61
E. 80

A. 40
B. 41
C. 60
D. 61
E. 80

ArtyomYom · Accepted Answer

1. We know, that average is 20.
2. Mary give information, that no one of other 3 student don't get more than 20

Let's say that we have Mary, student a, student b and student c

THE ONLY VARIANT , when we are sure , that no one of students a,b and c get more than 20 each is when the SUM of
their results is no more than 20

a + b + c < 20

Why is it so?

Let's imagine variant:
a+b+c = 30
So, 2 possibilities exist:
1. variant
Student a get 10
b get 10
c get 10
10+10+10=30
2. variant 
a get 28
b get 1
c get 1
28+1+1=30

As we know Mary dont know results of 3 other students, but she KNOW that
NO ONE of them get result more than 20.
So, we have the only variant when a + b + c < 20

Next we calculate Mary result:
We know that 
a+b+c = 19

(19+M)/4 = 20

M = 61

Answ D

IanStewart · Answer

The four scores sum to 80. Mary is completely sure, seeing her score, that the other three scores were all below average. So Mary is completely sure that no one else got a score of 20 or higher. If Mary only got a 60, then it would be possible that the four scores were 60, 20, 0 and 0, and Mary could not be sure that the three other scores were below average (since 20 would be exactly equal to the average). But if Mary gets a 61 or higher, then no one else could have a score higher than 19, and all three of the other scores must be below average. So 61 is the answer.

nick1816 · Answer

Mary immediately knew that all of the other three students scored below average; Hence, Mary score must be greater than (20*4 -20 = 60).

The minimum score that Mary could have gotten to be certain of the above situation=61( Maximum score that any other student can get is 19 which is smaller than the average)

Karmesh · Answer

Can you please explain this.

TIA

abhiroopm · Answer

Hi, I have a question.

The question says 3 students scored below average. What if all the 3 students scored the same marks? In which case the answer could be : 80 - 13*3 = 41. 41 is also an option.

smartchin77 · Answer

Since the worst possible case can be 19,0,0, the minimum sum of the possible scores is 19
Therefore Marry should score at least 80-19= 61

abhiroopm · Answer

I am still not clear about this point. Our aim is to find the lowest possible score for mary. If i were to test from the given answer options then in the case of b = 41

we would get that the rest of the 3 students have cumulatively scored 39 marks. In the simplest case, 3 three could have scored 13 each which is below the average score OR if I were to maximise then one student could have student could have scored 19 and the other two could be any score that adds up to 20, in which case their individual scores would still be below the average.

So the lowest score mary could've gotten can be 41, right?

IanStewart · Answer

If Mary gets 41, then if you were to maximize the second-highest score, that second highest score could be 39, and the remaining two scores could be zero. It seems you're assuming in advance that the remaining three students were below-average (that's the only reason the maximum second-best score would be 19), but then you're "begging the question", in logical terms -- assuming the answer to the question before you answer it.

Mary needs to be absolutely certain, upon seeing her score, that no other student could have a score of 20 or higher. Since some scores can be 0, the only way Mary can be certain of that is if she knows the sum of the three other scores is 19 or less.

Abir77 · Answer

Mary and three other students took a math test. Each of their scores was a non-negative integer. The teacher announced that the average score (of the 4 students) was 20. Mary immediately knew that all of the other three students scored below average. What is the minimum score that Mary could have gotten to be certain of the above situation?

A. 40
B. 41
C. 60
D. 61
E. 80

Doesn't it mean each of the 3 students got less than 20 ?
i mean why are we considering cumulative mark of 3 students instead of considering them individually???

aviddd · Answer

We consider the cumulative score because (Cumulative score + Mary's score) / 4 = 80 (mean)

ShajjadH · Answer

Let,
The three students are a, b and c respectively.
To find minimum number of Mary, we have to calculate the maximum total number of the other three students.
As we know Mary don’t know results of 3 other students, but she KNOW that
NO ONE of them get result more than 20.
So, we have the only variant when a + b + c < 20
In that case, the total maximum number of the other three students should be,
=> 19 < 20
So, a+ b +c = 19
=> (19+M)/4 = 20
=> M = 61

Crytiocanalyst · Answer

For amry to be completely ceratin we need Mary to score a value that can lead to an average that's 20 and others if even if they score 1,1,1 each 
let us assume a value that's 60 if it's going to be 60 then 20 ,0 ,0 can lead to an average 20 which has non negative integers therefore in order to be completely sure Mary should at least take 61 to stay above the average therefore IMO D

Fdambro294 · Answer

The concept of “over-under” with respect to the arithmetic mean is as follows:

The “surplus” created by any data points that exceed the mean must be canceled by the “deficits” of the elements that fall below the mean.

If Mary were to get a 60 on the test, this would represent a surplus of +40 above the Mean (20).

Since each person received a NON-Negative score, we could have the case in which:

Person A = 0 ——- (deficit of -20)
Person B = 0 ——- (deficit of -20)

A and B’s total deficit of -40 would cancel out Mary’s surplus of + 40.

Person C could then receive the Mean score of 20; i.e., neither a surplus nor a deficit.

Mary would need to score AT LEAST a 61 to ensure that all 3 people must have scores below the mean to counteract her net “surplus” of +41.

With a 61 being +41 above the Mean, the lowest scores that two people can get are 0 and 0, thus having a total deficit of -40 below the mean.

The 3rd person would HAVE TO score below the mean to counteract the additional +1 not canceled out by the other two people.

In such a case in which Mary scores a 61, the 3 people would have to score below the Mean of 20 (in order to maintain the Mean of 20).

*D* 61

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