Question

In an examination, $80\%$ of the students passed in English, $85\%$ in Mathematics and$75\%$ in both English and Mathematics. If $40$students failed in both the subjects, find the total number of students.

• A. $350$
• B. $400$
• C. $450$
• D. $500$

Answer 1

The correct option is B

$400$

Find the total number of students

Given : Students passed in English is $80\%$, students passed in Mathematics is $85\%$ and students passed in both English and Mathematics$75\%$

Students failed in both the subjects $=40$

Let $x$ be the total number of students.

Let$n\left(A\right)$ denotes the number of students passed in English and$n\left(B\right)$ denotes the number of students passed in Mathematics .

So,

$n\left(A\right)=\left(\frac{80}{100}\right)x,n\left(B\right)=\left(\frac{85}{100}\right)x,n(A\cap B)=\left(\frac{75}{100}\right)x$

$\therefore$Total number of students passed $n(A\cup B)=n\left(A\right)+n\left(B\right)–n(A\cap B)$

$$\begin{gathered} =\left(\frac{80}{100}\right)x+\left(\frac{85}{100}\right)x-\left(\frac{75}{100}\right)x \\ =\left(\frac{90}{100}\right)x \\ =\frac{9}{10}x \end{gathered}$$

$\therefore$Number of students failed$=x-\left(\frac{9}{10}x\right)$

$=\frac{\mathrm{x}}{10}$

Given, $40$ students failed in both the subjects.

Therefore,

$$\begin{gathered} \frac{x}{10}=40 \\ \Rightarrow x=400 \end{gathered}$$

Hence, the correct answer is is option (B) .