Let's denote the maximum marks of the examination by $x$.

Step 1: Understanding the Problem

• The student secures 50% of the maximum marks and fails by 20 marks.
• If the student had scored 60%, they would have passed by 25 marks.

Step 2: Set Up Equations

• The marks obtained when the student scores 50% are $0.5x$.
• The passing marks are $0.5x + 20$ because the student fails by 20 marks.

When the student scores 60%, the marks obtained are $0.6x$.

• The passing marks are also $0.6x - 25$ because the student passes by 25 marks.

Since the passing marks are the same in both cases, we can set up the equation:

$0.5x + 20 = 0.6x - 25$

Step 3: Solve the Equation

Let's solve this equation for $x$.

$0.6x - 0.5x = 20 + 25$

$0.1x = 45$

$x = \frac{45}{0.1} = 450$

So, the maximum marks of the examination are 450.

Would you like more details or have any questions? Here are some related questions:

1. How many marks did the student obtain with 50%?
2. What is the passing percentage for this exam?
3. If the student scored 55%, how many marks would they need to pass?
4. What would be the new passing marks if the student had to pass by 10 marks at 60%?
5. How much would the student need to score to just pass the exam?

Tip: When dealing with percentage problems, it’s helpful to translate word problems into equations to systematically solve for the unknown variable.