In a class, there are 70 students, such that:
I) 14 boys passed in Mathematics;

II) 12 boys passed in Physics;

III) 10 boys and 16 girls did not pass in Mathematics or Physics;

IV) 32 are boys;
V) 10 passed in both subjects;

VI) 22 passed only in Mathematics.

How many girls passed only in Physics?
Solution

To solve the problem, we will use the diagram below, where the upper rectangle represents the quantities of boys in each case and the lower one the quantities of girls; in the left circle, the number of students who passed in mathematics, while in the right one, the number who passed in physics, and in the intersection, the number who passed in both.

![](https://cdn.mathpix.com/cropped/2024_05_01_96f63abf6bdef97495eag-04.jpg?height=647&width=809&top_left_y=1564&top_left_x=505)

Now, let's fill in the diagram using the information, in a convenient sequence. By $I I I$, we have:

![](https://cdn.mathpix.com/cropped/2024_05_01_96f63abf6bdef97495eag-05.jpg?height=653&width=825&top_left_y=276&top_left_x=695)

By $I V$ and $I$, since 32 are boys and 14 of them passed in mathematics, then $32-14=18$ did not pass in mathematics, which means that $18-10=8$ of them passed only in physics.

![](https://cdn.mathpix.com/cropped/2024_05_01_96f63abf6bdef97495eag-05.jpg?height=645&width=812&top_left_y=1520&top_left_x=694)

By $I I$, since 12 boys passed in physics, then $12-8=4$ of them also passed in mathematics.

![](https://cdn.mathpix.com/cropped/2024_05_01_96f63abf6bdef97495eag-06.jpg?height=640&width=804&top_left_y=283&top_left_x=505)

By $I$, since 14 boys passed in mathematics, then $14-4=10$ of them passed only in mathematics. With this, we have determined all the quantities related to the boys.

|  | boys |  |  |
| :---: | :---: | :---: | :---: |

By $V$, since 10 students passed in both subjects, then $10-4=6$ girls passed in both.

![](https://cdn.mathpix.com/cropped/2024_05_01_96f63abf6bdef97495eag-07.jpg?height=653&width=825&top_left_y=276&top_left_x=695)

By $V I$, since 22 passed only in mathematics, then $22-10=12$ girls passed only in mathematics.

![](https://cdn.mathpix.com/cropped/2024_05_01_96f63abf6bdef97495eag-07.jpg?height=651&width=825&top_left_y=1385&top_left_x=695)

Since we already have $10+10+4+8+12+6+16=66$ students in the diagram, the number of girls who passed only in physics is $70-66=4$.

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