Question

In an examination,the average marks of students in sections A and B are 32 and 60,respectively. The number of students in section A is 10 less than that in section B. If the average marks of all the students across both the sections combined is an integer,then the difference between the maximum and minimum possible number of students in section A is

Answer 1

Correct Answer: 63

Given that the average marks of students in section A and section B are 32 and 60, respectively. Let's denote the number of students in section A as 'x' and in section B as 'y'.
We are also given that the number of students in section A is 10 less than that in section B, which can be expressed as: 
x = y - 10 
The average marks of all students across both sections combined is an integer 'a'. Thus, we have: 
32 < a < 60 
Considering the ratio of the number of students in section A to the number of students in section B, we have: 
\(\frac{x + y}{2} = a\) 
Substituting the value of x from the first equation, we get: 
\(\frac{y - 10 + y}{2} = a \)
\(\frac{2y - 10}{2} = a \)
y - 5 = a 

Now, let's analyze the possible values of 'a': 
1. When both sections have equal numbers of students: 
If x = y, then the average, a, is simply \(\frac{32 + 60}{2} = 46\). 

2. When there are more students in section B: 
In this case, a > 46. 

3. When there are more students in section A: 
In this case, a < 46. 
Since there are 10 more students in section B than in section A, we have 46 < a < 60. 
For extreme values of 'a': 
1. When a = 47: 
The ratio of students in section A to section B is (60 - a) : (a - 32) = 13 : 15. 
This implies 15x - 13x = 10, where x is the common multiplier. 
Solving for x, we get x = 5. 
Total students in section A = 13x = 65. 

2. When a = 56: 
The ratio of students in section A to section B is 1 : 6. 
This implies 6x - x = 10, where x is the common multiplier. 
Solving for x, we get x = 2. 
Total students in section A = x = 2. 

The difference between the maximum and minimum possible number of students in section A is 65 - 2 = 63.