Question

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

Answer 1

Hint: In this question, we are given average marks of 3 batches of students. Using the number of students in each batch and their average, first find the sum of the marks of students of each batch. Next, add the sum of all the three batches. Then, add the number of students in each class. Last step is to divide the total sum of marks of three batches by the total number of students. Simplify and you will get your answer.

Formula used: Average = $\dfrac{{{\text{Sum of observations}}}}{{{\text{Number of observations}}}}$

Complete step-by-step solution:
We have to find the average marks of all the students. We will find the sum of the marks of each batch individually first. Let us find it one by one:
$ \Rightarrow $Batch 1:
Number of students = $55$
Average = $50$
Putting in the formula: Average = $\dfrac{{{\text{Sum of observations}}}}{{{\text{Number of observations}}}}$,
$ \Rightarrow $$50 = \dfrac{{{\text{Su}}{{\text{m}}_1}}}{{55}}$
On solving, we get,
$ \Rightarrow {\text{Su}}{{\text{m}}_1} = 50 \times 55 = 2750$
$ \Rightarrow $Batch 2:
Number of students = $60$
Average = $55$
Putting in the formula: Average = $\dfrac{{{\text{Sum of observations}}}}{{{\text{Number of observations}}}}$,
$ \Rightarrow 55 = \dfrac{{{\text{Su}}{{\text{m}}_2}}}{{60}}$
$ \Rightarrow {\text{Su}}{{\text{m}}_2} = 55 \times 60 = 3300$
$ \Rightarrow $Batch 3:
Number of students = $45$
Average = $60$
Putting in the formula: Average = $\dfrac{{{\text{Sum of observations}}}}{{{\text{Number of observations}}}}$,
$ \Rightarrow 60 = \dfrac{{{\text{Su}}{{\text{m}}_3}}}{{45}}$
$ \Rightarrow {\text{Su}}{{\text{m}}_3} = 60 \times 45 = 2700$
Next step is to find the total sum of the marks of students of three batches. For this, we will add the 3 sums that we have found above.
$ \Rightarrow $Total sum = ${\text{Su}}{{\text{m}}_1}{\text{ + Su}}{{\text{m}}_2}{\text{ + Su}}{{\text{m}}_3}$
$ \Rightarrow $Total sum = $2750 + 3300 + 2700$
$ \Rightarrow $Total sum = $8750$
Now, we will find total students.
Total students = $55 + 60 + 45 = 160$
Next, we will put them in the formula.
$ \Rightarrow $Average = $\dfrac{{{\text{Total sum}}}}{{{\text{Total students}}}}$
Putting all the values,
$ \Rightarrow $Average = $\dfrac{{8750}}{{160}} = 54.6875$
We will take decimals upto 2 digits only.

Hence, our final answer is option B) $54.68$

Note: This method that we used is called the weighted average method. Instead of finding out everything, you could have just put the values in the formula and find the final answer. Its formula is as follows: ${A_w} = \dfrac{{\sum\limits_{i = 1}^3 {{w_i}{x_i}} }}{{\sum\limits_{i = 1}^3 {{x_i}} }}$
In this method, every value has a different weight or in layman's language, we can call it different importance. The average is multiplied with its weight and divided by the total of weights. Following this method, we get our average.