Answer

Verified

393.6k+ views

**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.

Recently Updated Pages

The base of a right prism is a pentagon whose sides class 10 maths CBSE

A die is thrown Find the probability that the number class 10 maths CBSE

A mans age is six times the age of his son In six years class 10 maths CBSE

A started a business with Rs 21000 and is joined afterwards class 10 maths CBSE

Aasifbhai bought a refrigerator at Rs 10000 After some class 10 maths CBSE

Give a brief history of the mathematician Pythagoras class 10 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Name 10 Living and Non living things class 9 biology CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

Write the 6 fundamental rights of India and explain in detail