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Let $100x$ be the total number of student in a class.

The girls are in a class $ = 60\% = 100x \times \frac{60}{100} = 60x $

And, the boys are in a class $ = 100\% – 60\% = 40\% = 100x \times \frac{40}{100} = 40x$

According to the question :

There are $30$ more girls than boys.

$ 60x = 40x + 30 $

$\Rightarrow 60x – 40x = 30$

$ \Rightarrow 20x = 30 $

$ \Rightarrow \boxed{x = \frac{3}{2}} $

The number of student in a class $ = 100x = 100 \times \frac{3}{2} = 150 $

Therefore,

- The total number of girls in a class $ = 60x = 60 \times \frac{3}{2} = 90 $
- The total number of boys in a class $ = 40x = 40 \times \frac{3}{2} = 60$

If $68\%$ of the students, including $30$ boys pass an examination.

Then, total number of students who passes in the exam (including $30$ boys) $ = 150 \times \frac{68}{100} = 102 $

Let $k$ be the number of girls who passed in the exam.

$ 30 \; \text{(Boys)} + k \; \text{(Girls)} = 102 $

$ \Rightarrow k = 102 – 30 $

$ \Rightarrow k = 72 $

The total number of girls in a class $ = 90 $

Therefore,

- Number of girls who passed in the exam $ = 72 $
- Number of girls who failed in the exam $ = 90 – 72 = 18 $

$ \therefore$ The percentage of the girls who failed in the exam$ = \frac{18}{90} \times 100 = 20\%.$

Correct Answer $: 20$