Question

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

Answer 1

Hint: Assume three batches A, B, C and \[{{x}_{1}},{{x}_{2}},................,{{x}_{55}}\] , \[{{y}_{1}},{{y}_{2}},.............,{{y}_{60}}\] , and \[{{z}_{1}},{{z}_{2}},.............,{{z}_{45}}\] are the marks obtained by the students of batch A, B, and C respectively. Use the formula, Average marks obtained by students = \[\dfrac{\text{Summation of marks obtained by students}}{\text{Total number of students}}\] and calculate the average marks of students of batch A, batch B, and batch C. Now, compare it with the average marks of students of batch A, batch B, and batch C to obtain three equations. When all the batches are taken together then the number of students is 160 and the marks obtained by the students when batches A, B, and C are taken together is \[{{x}_{1}},{{x}_{2}},................,{{x}_{55}},{{y}_{1}},{{y}_{2}},....................,{{y}_{60}},{{z}_{1}},{{z}_{2}},............,{{z}_{45}}\] . Now, solve it further and calculate the average marks of the students when batch A, batch B, and batch C are taken together.

Complete step-by-step solution
According to the question, we are given that there are three batches of 55, 60, and 45 students. The average marks of all the students of three batches are 50, 55, and 60 respectively.
First of all, let us assume three batches A, B, C and \[{{x}_{1}},{{x}_{2}},................,{{x}_{55}}\] , \[{{y}_{1}},{{y}_{2}},.............,{{y}_{60}}\] , and \[{{z}_{1}},{{z}_{2}},.............,{{z}_{45}}\] are the marks obtained by the students of batch A, B, and C respectively.
We are given that,
The number of students of batch A = 55 …………………………….. (1)
The number of students of batch B = 60 ……………….…………….. (2)
The number of students of batch C = 45 ……………….………….… (3)
The average marks of students of batch A = 50 …………………… (4)
The average marks of students of batch B = 55 ………….………… (5)
The average marks of students of batch C = 60 ……………………. (6)
We know the formula, Average marks obtained by students = \[\dfrac{Summation\,of\,marks\,obtained\,by\,students\,}{Total\,number\,of\,students}\] ……………………………………..(7)
Now, using the formula shown in equation (7), we get
The average marks of students of batch A = \[\dfrac{{{x}_{1}}+{{x}_{2}}+........+{{x}_{55}}}{55}\] ………………………………………….(8)
From equation (4), we also have the average marks of students of batch A.
On comparing equation (4) and equation (8), we get
\[\begin{align}
  & \Rightarrow \dfrac{{{x}_{1}}+{{x}_{2}}+........+{{x}_{55}}}{55}=50 \\
 & \Rightarrow \left( {{x}_{1}}+{{x}_{2}}+........+{{x}_{55}} \right)=50\times 55 \\
\end{align}\]
\[\Rightarrow \left( {{x}_{1}}+{{x}_{2}}+........+{{x}_{55}} \right)=2750\] …………………………………………..(9)
Similarly, using the formula shown in equation (7), we get
The average marks of students of batch B = \[\dfrac{{{y}_{1}}+{{y}_{2}}+........+{{y}_{60}}}{60}\] ………………………………………….(10)
From equation (5), we also have the average marks of students of batch B.
On comparing equation (5) and equation (10), we get
\[\begin{align}
  & \Rightarrow \dfrac{{{y}_{1}}+{{y}_{2}}+........+{{y}_{60}}}{60}=55 \\
 & \Rightarrow \left( {{y}_{1}}+{{y}_{2}}+........+{{y}_{60}} \right)=55\times 60 \\
\end{align}\]
\[\Rightarrow \left( {{y}_{1}}+{{y}_{2}}+........+{{y}_{60}} \right)=3300\] …………………………………………..(11)
Similarly, using the formula shown in equation (7), we get
The average marks of students of batch C = \[\dfrac{{{z}_{1}}+{{z}_{2}}+........+{{z}_{45}}}{45}\] ………………………………………….(12)
From equation (6), we also have the average marks of students of batch C.
On comparing equation (6) and equation (12), we get
\[\begin{align}
  & \Rightarrow \dfrac{{{z}_{1}}+{{z}_{2}}+........+{{z}_{45}}}{45}=55 \\
 & \Rightarrow \left( {{z}_{1}}+{{z}_{2}}+........+{{z}_{45}} \right)=60\times 45 \\
\end{align}\]
\[\Rightarrow \left( {{z}_{1}}+{{z}_{2}}+........+{{z}_{45}} \right)=2700\] …………………………………………..(13)
We are asked to find the average of marks of all students i.e., when students of batches A, B, and C are taken together.
The marks obtained by the students when batches A, B, and C are taken together is \[{{x}_{1}},{{x}_{2}},................,{{x}_{55}},{{y}_{1}},{{y}_{2}},....................,{{y}_{60}},{{z}_{1}},{{z}_{2}},............,{{z}_{45}}\] ………………………………(14)
The total number of students = The total number of students of batch A + The total number of students of batch B + The total number of students of batch C ……………………………………………(15)
From equation (4), equation (5), equation (6), and equation (15), we get
The total number of students = \[55+60+45=160\] ………………………………………(16)
Now, using equation (7), equation (14), and equation (16), we get
The average of marks obtained by students when A, B, and C taken together =
\[\dfrac{{{x}_{1}}+{{x}_{2}}+.........+{{x}_{55}}+{{y}_{1}}+{{y}_{2}}+.........+{{y}_{60}}+{{z}_{1}}+{{z}_{2}}+.........+{{z}_{45}}}{160}\] ……………………………………….(14)
From equation (9), equation (11), equation (13), and equation (14), we get
\[=\dfrac{2750+3300+2700}{160}=\dfrac{8750}{160}=54.6875\]
Therefore, the average marks of all the students is 54.68.
Hence, the correct option is (A).

Note: In this question, the hidden meaning of average marks of all students is the average marks of the students when all three batches are combined. Therefore, keep this point into consideration to avoid mistakes.