Please give an intuitive explanation

A class average mark in an exam is 70. The average of students who scored below 60 is 50. The average of students who scored 60 or more is 75. If the total number of students in this class is 20, how many students scored below 60?

20 in class

mark average is 70

average below 60 is 50

the average of 60 or more is 75

I am going to split the class into 2 groups.  Those who got below 60 (Lower)  and those who got 60 or more.(Upper)

I do no know how many are in each group but I do know that there are 20 students altogether so the lower+upper  =20

The average of the lower marks is  50      so   sum of lower marks  is 50L

The average of the upper marks is  is  75      so   sum of upper marks  is 75U

Now there are 20 students and the average mark is 70

so     (50L+75U)/20   =70

50L+75U   =70*20

2L+3U=(70*20)/25

2L+2U+U=56

But 2L +2U=2* all the students = 40

40+U=56

U=16

There are 16 studens in the top group so there must be 4 students in the top group.

Answer: 4 students scored below 60. 

A class average mark in an exam is 70. The average of students who scored below 60 is 50. The average of students who scored 60 or more is 75. If the total number of students in this class is 20, how many students scored below 60?

I didn't feel that I should give you an answer "intuitively", when simple symbolic Algebra could give me the answer definitively!.

Let the number of students who scored below 60=x

Let the number of students who scored above 60=y

x + y=20

50x + 75y=20 X 70

Solve the following system:
{50 x+75 y = 1400 |     (equation 1)
x+y = 20 |     (equation 2)
Subtract 1/50 × (equation 1) from equation 2:
{50 x+75 y = 1400 |     (equation 1)
0 x-y/2 = -8 |     (equation 2)
Divide equation 1 by 25:
{2 x+3 y = 56 |     (equation 1)
0 x-y/2 = -8 |     (equation 2)
Multiply equation 2 by -2:
{2 x+3 y = 56 |     (equation 1)
0 x+y = 16 |     (equation 2)
Subtract 3 × (equation 2) from equation 1:
{2 x+0 y = 8 |     (equation 1)
0 x+y = 16 |     (equation 2)
Divide equation 1 by 2:
{x+0 y = 4 |     (equation 1)
0 x+y = 16 |     (equation 2)
Collect results:
Answer: | x=4         and          y=16
 

I suppose what you mean by "answer intuitively", is that you need an explanation as to why "Guest #1" got the answers that he/she did? This is a simple explanation for the reslts:

If the number of students who scored below 60 was less than 4, let us say 3, then that would mean: 3 X 50=150 marks out of a total of 20 X 70, or 1,400 marks.Then the difference of 1400 -150=1,250 would have to be divided among the remaining 17 students, whose average score would come down to 1250/17=73.5, which would be below the stated average of 75.

Similarly, if the number of students who scored below 60 was greater than 4, let us say 5, then their average score would be 5 X 50 or 250 marks out of 1400. The difference 1400 - 250=1150, which when divided by the remaining 15 students would come to: 1150/15=76 2/3, which would above the stated average of 75. And that is the reason the answer has to be as calculated by "Guest #1".

Let the number of students who average  50 = x

Let the number of students who averaged  75 = 20-x    

And we know that :

[ 50x  + 75(20-x)] / 20  = 70     simplify

50x + 1500 - 75x   =   1400

100 = 25x     divide both sides by 25

4 = x    this is the number that averaged 50  [ below 60]

I like your answer Chris.  Guest 1 and 2 gave good answers too :)

It is a bit unclear what intuitive means to this student or their teacher.

We have all used some algebra.

I took intuitive to mean that we needed to explain the algebra that we were using.  

I think CPhill and guest 2 interpreted it the same way.  :)

Thanks, Melody......maybe I should have provided a little more detail........my answer wasn't particularly "intuitive".........just standard Algebra

I think yours was faily intuitive to someone with a lot of maths training. Perhaps you could have said a little more for the extended audience, maybe I could have too, but I think your answer was good.   If anyone askes you or me to explain our answers I know we will be more that happy to oblige. I expect our guests would be happy to oblige too :D

I like the method that you used the best. :)  It was really simple.  :)

Hello,

I have a question to those, who give an answer and tried to explain.

If you take x and y as a avarege mark, so how x+y can be 20?