The average marks for three batches of the students having $70,50$ and $30$ students respectively are $50,55$ and $45$. Find the average marks of the $150$ students taken together.
$A) 50.67$
$B) 40.67$
$C) 50.60$
$D) 40.60$

Hint: Here we use the formula of average marks for each of the students separately which will give us the sum of the marks obtained by students in each batch. Then calculating the sum of marks obtained by students in all three batches we calculate the average marks of all the students.

Formula used: Average mark ${\text{ = }}\dfrac{{{\text{sum of the mark}}}}{{{\text{number of student}}}}$

Complete step by step solution:
Let us assume three batches separately.
Now using the formula, we can write the sum of the marks of the batch $ = $ average marks of the batch $ \times $ number of the students in the batch ……………………$(1)$
First we have to find Batch $1:$
Average marks of the students in the batch $1 = 70$
Number of students in batch $1 = 50$
Using equation $(1)$
We can write
$\therefore $Sum of marks in batch $1 = 70 \times 50$
On multiplying the terms and we get,
Therefore the sum of marks in batch $1 = 3500$
Again we find Batch $2$:
Average marks of students in batch $2 = 50$
Number of students in batch $2 = 55$
Using equation $(1)$
$\therefore $ Sum of the marks in batch $2 = 50 \times 55$
Let us multiply the terms and we get,
Sum of the marks in batch $2 = 2750$
Also, we can find the Batch $3$:
Average marks of the students in the batch $3 = 30$
Number of students in the batch $3 = 45$
Using equation $(1)$
We can write,
Sum of the marks in batch $3 = 30 \times 45$
$\therefore $ Sum of the marks in batch $3 = 1350$
Now, we have a sum of marks of three batches of students as $3500 + 2750 + 1350 = 7600$
Therefore the sum of marks of all the three batches $ = 150$ (given)
Substituting the values of the formula of the average marks
Average marks of all the students $ = \dfrac{{7600}}{{150}}$
After dividing the fraction \[\; = 50.6666 \ldots \ldots .\]
Therefore we can write the value for the fraction after round of the decimal place $ = 50.67$

Hence the correct option is $(A)$

Note: Students might try to find the solution by applying the formula of average marks to the given average marks of the three batches that is $ = \dfrac{{50 + 55 + 45}}{3}$
Average marks $ = \dfrac{{150}}{3}$
on dividing the term and we get,$ = 50$
This would be the wrong method because it gives us the average marks.