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 the students is:
A) 53.33
B) 54.68
C) 55
D) None of these

Answer 1

Hint: In this question we will use the formula for average, Average \[ = \dfrac{{Sum\,of\,all\,terms}}{{No.\,of\,terms}}\]. In the question we have been given the averages for 3 batches of students and also the corresponding number of students in each batch. We will consider the three batches as batch 1,batch 2 and batch 3 respectively. We will then calculate the sum of marks of students in all the three batches by substituting the given values of average and number of students in the formula. We have been asked to find the average marks for all students. Again we will use the same formula, we need total no. of students and the total sum of marks. Total number of students can be calculated by adding the number of students in all 3 batches, Total number of students= No. of students in batch 1 + No. of students in batch 2 + No. of students in batch3. Total number of students= 55+60+45. Now we will need total marks which can again be calculated by adding the sum of marks of all 3 batches. Substituting these values will get us the average for all the students.

Complete step by step answer:
We can see in the question we have been given the values of the number of students and the averages for all the three batches.
By using the formula of average we can calculate the sum of marks for each batch.
Now, we know that average is given by
Average \[ = \dfrac{{Sum\,of\,all\,terms}}{{No.\,of\,terms}}\]
\[ = \dfrac{{Sum\,of\,marks}}{{No.\,of\,students}}\]
Let us apply this formula to all 3 batches.
Let the three batches be batch 1, batch 2, batch 3 respectively.
For Batch 1 we have,
No. of students=55
Average=50
We will apply the formula,
\[\dfrac{{Sum\,of\,marks}}{{No.\,of\,students}} = Average\]
We will substitute the values in the formula.
\[
  \dfrac{{{S_1}}}{{55}} = 50 \\
  {S_1} = 50 \times 55 \\
\]
Now for Batch 2 we have,
No. of students=60
Average=55
Applying the formula to above values we will get,
\[
  \dfrac{{{S_2}}}{{60}} = 55 \\
  {S_2} = 55 \times 60 \\
\]
Lastly we will calculate the sum of marks for batch 3.
We have,
No. of students=45
Average=60
Applying the formula to above values we will get,
\[
  \dfrac{{{S_3}}}{{45}} = 60 \\
  {S_3} = 60 \times 45 \\
\]
Now we need to calculate the average for all the students.
We will first calculate the total number of students.
Total no. of students= Students in batch 1+ Students in batch 2+ Students in batch 3
\[ = 55 + 60 + 45\]
Total no. of students=160.
Now we will calculate the total sum of marks for all students.
total sum of marks for all students= Sum of marks for batch 1+ Sum of marks for batch 2+ Sum of marks for batch 3
\[
   = {S_1} + {S_2} + {S_3} \\
   = 50 \times 55 + 55 \times 60 + 60 \times 45 \\
\]
Now for the total average we will substitute above values in the formula.
Average = \[\dfrac{{50 \times 55 + 55 \times 60 + 60 \times 45}}{{160}}\]
=54.68

So, the correct answer is “Option B”.

Note: When solving the question, start with the formula and substitute all the given and known terms. This will give us clarity of what exactly we need to find. Do not complicate the questions by adding variables or terms when not required, here we did not need a variable for no. of students in each batch as we had the values given to us.