In an examination `70%` of the students passed in maths , `60%` of the student passed in English , `40%` of the students passed in both subjects. If `200` students failed in both subjects. Then find the total students.

Options

A. a) `2300`
B. b) `2000`
C. c) `3000`
D. d) `2500`

Answer

To solve the problem step by step, we can use the information provided about the percentages of students passing in each subject and the number of students who failed both subjects.

### Step 1: Define the Variables
Let the total number of students be \( x \).

### Step 2: Calculate the Percentage of Students Passing
- Students who passed in Maths = \( 70\% \) of \( x \) = \( 0.7x \)
- Students who passed in English = \( 60\% \) of \( x \) = \( 0.6x \)
- Students who passed in both subjects = \( 40\% \) of \( x \) = \( 0.4x \)

### Step 3: Use the Principle of Inclusion-Exclusion
To find the number of students who passed in at least one subject, we can use the formula:
\[ \text{Passed in at least one subject} = (\text{Passed in Maths}) + (\text{Passed in English}) - (\text{Passed in both}) \]
Substituting the values:
\[ \text{Passed in at least one subject} = 0.7x + 0.6x - 0.4x = 0.9x \]

### Step 4: Calculate the Number of Students Who Failed in Both Subjects
The number of students who failed in both subjects is given as \( 200 \). Therefore, the number of students who passed in at least one subject can also be expressed as:
\[ \text{Total students} - \text{Failed in both} = x - 200 \]
Setting the two expressions for students who passed in at least one subject equal gives:
\[ 0.9x = x - 200 \]

### Step 5: Solve for \( x \)
Rearranging the equation:
\[ x - 0.9x = 200 \]
\[ 0.1x = 200 \]
Now, divide both sides by \( 0.1 \):
\[ x = \frac{200}{0.1} = 2000 \]

### Step 6: Conclusion
The total number of students is \( 2000 \).