Hello, hven191!
This doesn't require any fancy formulas . . .
The probability for students at a certain university
to pass Mathematics and Economics are 3/7 and 5/7 respectively.
One student failed both subjects and four students passed both subjects.
Find the number of students who took the test. Make a Venn diagram of the students who passed. Code:
*---------------------------------------*
| |
| *-----------------------* |
| | Math | |
| | only *---------------+-------* |
| | M | Both | | |
| | | 4 | | |
| | | | | |
| *-------+---------------* Eco | |
| | E only | |
| *-----------------------* |
| Neither 1 |
*---------------------------------------*
Let 
= number of students that passed Math only.
Let 
= number of students that passed Economics only.
We know there are 4 that passed both, and 1 that failed both.
The total number of students is: . 
The number of students that passed Math is: . 
The probability that a student passed Math is 
We have: . ![\frac{M+4}{M+E+5} \:=\:\frac{3}{7} \quad\Rightarrow\quad 4M - 3E \:=\:-13\;\;{\color{blue}[1]}](http://latex.codecogs.com/png.latex?\frac{M+4}{M+E+5} \:=\:\frac{3}{7} \quad\Rightarrow\quad 4M - 3E \:=\:-13\;\;{\color{blue}[1]})
The number of students that passed Economics is 
The probability that a student passes Economics is 
We have: . ![\frac{E+4}{M+E+5} \:=\:\frac{5}{7} \quad\Rightarrow\quad 5M - 2E \:=\:3\;\;{\color{blue}[2]}](http://latex.codecogs.com/png.latex?\frac{E+4}{M+E+5} \:=\:\frac{5}{7} \quad\Rightarrow\quad 5M - 2E \:=\:3\;\;{\color{blue}[2]} )
![\begin{array}{cccccc}\text{Multiply {\color{blue}[1]} by -2:} & \text{-}8M + 6E &=& 26 & {\color{blue}[3]}\\<br />
\text{Multiply {\color{blue}[2]} by 3:} & 15M - 6E &=& 9 & {\color{blue}[4]}\end{array}](http://latex.codecogs.com/png.latex?\begin{array}{cccccc}\text{Multiply {\color{blue}[1]} by -2:} & \text{-}8M + 6E &=& 26 & {\color{blue}[3]}\\<br />
\text{Multiply {\color{blue}[2]} by 3:} & 15M - 6E &=& 9 & {\color{blue}[4]}\end{array})
![\text{Add {\color{blue}[3]} and {\color{blue}[4]}: }\;7M \:=\:35 \quad\Rightarrow\quad\boxed{ M \:=\:5}](http://latex.codecogs.com/png.latex?\text{Add {\color{blue}[3]} and {\color{blue}[4]}: }\;7M \:=\:35 \quad\Rightarrow\quad\boxed{ M \:=\:5})
Substitute into [1]: .  - 3E \:=\:-13 \quad\Rightarrow\quad\boxed{ E \:=\:11})
Therefore, the number of students is: . 
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