Question

If the average marks of three batches of 55, 60 and 45 students respectively is 50, 55 and 60 then the average marks of all students is
A. 54.68
B. 53.33
C. 55
D. None of these

Answer 1

Hint: Here we use the formula of average marks for each batch of students separately which will give us the sum of marks obtained by students in each batch. Then calculating the sum of marks obtained by students in all the three batches and the total number of students from three batches we calculate the average marks of all students.
* Average is given by the sum of total observations divided by total number of observations.

Complete step-by-step answer:
Let us assume three batches separately.
We know that average marks are given by the sum of marks of students divided by the number of students.
So, we can write sum of marks of the batch \[ = \]average marks of batch \[ \times \]number of students in the batch. … (1)
Batch 1:
Average marks of students in batch 1 \[ = 50\]
Number of students in batch 1 \[ = 55\]
Using equation (1) we can write
Sum of marks in batch 1 \[ = 50 \times 55\]
\[\therefore \]Sum of marks in batch 1 \[ = 2750\]
Batch 2:
Average marks of students in batch 2 \[ = 55\]
Number of students in batch 2 \[ = 60\]
Using equation (1) we can write
Sum of marks in batch 2 \[ = 55 \times 60\]
\[\therefore \]Sum of marks in batch 2 \[ = 3300\]
Batch 3:
Average marks of students in batch 3 \[ = 60\]
Number of students in batch 3 \[ = 45\]
Using equation (1) we can write
Sum of marks in batch 3 \[ = 60 \times 45\]
\[\therefore \]Sum of marks in batch 3 \[ = 2700\]
So we have a sum of marks from three batches of students as 2750, 3300 and 2700.
Now to calculate average marks of all the students we divide the sum of marks from all the three batches by the total number of students from the three batches.
Sum of marks from all three batches \[ = 2750 + 3300 + 2700\]
\[\therefore \]Sum of marks from three batches\[ = 8750\]
Total number of students from three batches \[ = 55 + 60 + 45\]
\[\therefore \]Total number of students from three batches \[ = 160\]
Substitute the values in formula of average marks
Average marks of all students \[ = \dfrac{{8750}}{{160}}\]
Calculate the fraction.
\[\therefore \] Average marks of all students \[ = 54.68\]

So, option A is correct.

Note: Students might try to find the solution by applying the formula of average marks to the given average marks of three batches, i.e.
they might calculate average \[ = \dfrac{{50 + 55 + 60}}{3}\]
Average marks \[ = \dfrac{{165}}{3} = 55\]
This is the wrong method because it gives us the average of the average marks. Try to avoid these kinds of mistakes.