In an examination, a student who gets $20\%$ of the maximum marks fails by $5$ marks. Another student who scores $30\%$ of the maximum marks, gets $20$ marks more than the passing marks. The percentage required for passing is :

Let the maximum marks of the examination be $M$.
Step 1: Set up the equation for passing marks
From the first student's condition:

$$\text{Passing Marks}=20\%\text{ of }M+5=0.2M+5$$

From the second student's condition:

$$\text{Passing Marks}=30\%\text{ of }M-20=0.3M-20$$

**Step 2: Equate both conditions to find maximum marks ($M$)**

$$0.2M+5=0.3M-20$$

$$5+20=0.3M-0.2M$$

$$25=0.1M$$

$$M=\frac{25}{0.1}=250$$

So, the maximum marks are $250$.

Step 3: Calculate the passing marks
Substitute $M=250$ into the first condition:

$$\text{Passing Marks}=20\%\text{ of }250+5=\left(\frac{20}{100}\times 250\right)+5=50+5=55$$

Step 4: Find the passing percentage

$$\text{Passing Percentage}=\left(\frac{\text{Passing Marks}}{\text{Maximum Marks}}\right)\times 100\%$$

$$\text{Passing Percentage}=\left(\frac{55}{250}\right)\times 100\%=\frac{55\times 2}{5}=11\times 2=22\%$$