The average marks of a group of 20 students on a test is reduced by 4 when the topper who scored 90 marks is replaced by a new student. How many marks did the new student have?

Sure, let's solve this step by step:

Step 1: Calculate the total marks of the original group of students. The average marks of the original group of 20 students is reduced by 4 when the topper is replaced. This means the original average was 4 marks higher. We don't know this original average, but we can express it in terms of the reduced average. Let's call the reduced average 'A'. Therefore, the original average was 'A+4'.

The total marks of the group is the average marks times the number of students. So, the total marks of the original group was 20*(A+4).

Step 2: Calculate the total marks of the new group of students. The total marks of the new group is also the average marks times the number of students. So, the total marks of the new group is 20*A.

Step 3: Calculate the marks of the new student. The change in total marks when the topper is replaced by the new student is the marks of the topper minus the marks of the new student. This is also equal to the total marks of the original group minus the total marks of the new group.

So, 90 (marks of the topper) - X (marks of the new student) = 20*(A+4) - 20*A.

Solving this equation for X gives us the marks of the new student.

Step 4: Substitute the given values into the equation. We know that the average was reduced by 4 marks when the topper was replaced. So, A+4 - A = 4.

Substituting this into the equation from step 3 gives us:

90 - X = 20*4.

Solving this equation for X gives us X = 90 - 80 = 10.

So, the new student had 10 marks.