Question

38 students took a test that had 100 as the maximum mark and 35 as passing marks. If the average marks of all the 38 students was 78, find the maximum number of students that could have failed?

Answer: 12 students

Points to ponder

1. It can be thought as distributing a total of 38 × 78 = 2964 amongst 38 students such that the set of students who received less than 35 marks is as high as possible.

2. One simple trivial start will be giving all the students 0 marks. But this way, nothing of the 2964 marks will distributed. And we have to distribute these marks amongst these 38 students … entirely … without anything left with us.

Since we have a lot of marks on hand, 2964, let us distribute more marks to each student …. but we also need maximum of them to fail …. so how many marks should be distributed to each student?

3. Have all the 2964 marks been distributed? Are you forced that some student have to be given 35 or more than 35 marks? And if you are forced, then how much will you want to give to such a student who has passed?

As given in the hint …. we have to distribute 38 × 78 = 2964 marks amongst 38 students, without any marks being left over.

Since maximum have to fail, we will distribute 35– to each student. 35– means just less than 35 i.e. something like 34.999999. For all practical calculation purposes, we can use 35, but in our minds, we will note that the actual value will be infinitesimally small than 35.

Marks distributed = 35 × 38 = 1330.

Marks yet to be distributed = 2964 – 1330 = 1634

Now, that I am forced to give a student more marks i.e. the student will pass, it is best to give him as many more marks as possible, so that the residual marks get exhausted amongst least number of pass student, leaving the number of failed student to be as high as possible.

How many more marks can be given to a student? Simple 100 – 35 = 65.

Thus, if I start giving 65 marks to one student (in addition to the already 35 marks given to each of the 38 students), how many students will it take to distribute the residual 1634 marks?

1634/65 = 25.13 students. This is the minimum number needed to distribute the residual marks. Since students cannot be in fractions, I will need 26 students to distribute the residual marks.

And thus the maximum number of students who will still remain as failed will be 38 – 26 = 12.

Watch the video in this link for 2 more different and shorter approaches to solve the question.