Question

In a History examination, the average for the entire class was 80 marks. If \[10\% \] of the students scored 95 marks and \[20\% \] scored 90 marks, what was the average marks of the remaining students of the class?
A. \[65.5\]
B. \[72.5\]
C. 75
D. 85

Answer 1

Hint:
Here, we have to find the average marks of the remaining students of the class. We will use the concept of percentage to find the marks of the students who have scored 95 marks and 90 marks. Then we will find the average marks of the remaining students of the class. The average is defined as the total sum of observations divided by the number of observations.

Formula Used:
We will use the following formula:
1. The average is given by the formula Average \[ = \] Total sum of observations \[ \div \] Number of observations.
2. The percentage is given by the formula Percentage \[ = \] (Total Number \[ \div \] Number of observations) \[ \times 100\].

Complete step by step solution:
Let \[n\] be the total number of students in a class.
Since \[10\% \] of the students scored 95 marks, then the total marks scored by students will be:
Total marks of the students who have scored 95 marks \[{M_1} = \dfrac{{10}}{{100}} \times n \times 95 = \dfrac{{95n}}{{10}}\]
Since \[20\% \] of the students scored 90 marks, then the total marks scored by the students is given by
Total marks of the students who have scored 90 marks\[{M_2} = \dfrac{{20}}{{100}} \times n \times 90 = \dfrac{{180n}}{{10}}\].
Percentage of the rest of the Students \[ = 100 - 10 - 20 = 70\]
Now we will find the total marks of the students who have scored \[x\] marks.
Total marks of the students who have scored \[x\] marks\[{M_3} = \dfrac{{70}}{{100}} \times n \times x = \dfrac{{7nx}}{{10}}\]
Now, we will add all the marks to find the total Marks of the students in a class.
\[{M_1} + {M_2} + {M_3} = \dfrac{{95}}{{10}}n + \dfrac{{180}}{{10}}n + \dfrac{7}{{10}}n\]
Adding the terms, we get
\[{M_1} + {M_2} + {M_3} = \dfrac{{95 + 180 + 7x}}{{10}}n\]
Again adding the terms, we get
\[ \Rightarrow \] Total Marks of the students in a class\[ = \dfrac{{275 + 7x}}{{10}}n\]
We are given that in a History examination, the average for the entire class was 80 marks
The average is given by the formula Total sum of observations \[ \div \] Number of observations.
  Now, by using the average formula, we get
\[ \Rightarrow {\rm{Average}} = \dfrac{{{M_1} + {M_2} + {M_3}}}{n}\]
Substituting the known values, we get
\[ \Rightarrow {\rm{Average}} = \dfrac{{\dfrac{{275 + 7x}}{{10}}n}}{n}\]
Cancelling out the common terms, we get
\[ \Rightarrow 80 = \dfrac{{275 + 7x}}{{10}}\]
By cross-multiplying, we get
\[ \Rightarrow 800 = 275 + 7x\]
Rewriting the equation, we get
\[ \Rightarrow 7x = 800 - 275\]
\[ \Rightarrow 7x = 525\]
Dividing by 7, on both the sides, we get
\[ \Rightarrow x = \dfrac{{525}}{7}\]
\[ \Rightarrow x = 75\]
Therefore, the average marks of the remaining students of the class is 75.

Hence, option C is the correct option.

Note:
We might make mistakes in using the percentage and average formula simultaneously. First, we should be clear about the use of the Percentage formula and the Average formula. The percentage formula is used to find the total marks of the students in a class where the percentage of students is known whereas the average formula is used to find the average marks of the students in a class.