Let the passing marks be denoted by $P$.

Step 1: A's marks calculation

A got $33 \frac{1}{3}\%$ more than the passing marks, so A's marks are: $A = P \times \left(1 + \frac{33 \frac{1}{3}}{100}\right) = P \times \left(1 + \frac{1}{3}\right) = P \times \frac{4}{3}$ Thus, A's marks are $A = \frac{4}{3}P$.

Step 2: B's marks calculation

B's marks are $62.5\%$ of A's marks. Converting $62.5\%$ to a fraction gives $\frac{5}{8}$, so: $B = \frac{5}{8}A = \frac{5}{8} \times \frac{4}{3}P = \frac{20}{24}P = \frac{5}{6}P$ Thus, B's marks are $B = \frac{5}{6}P$.

Step 3: B failed by 50 marks

We are told that B failed by 50 marks, meaning: $B = P - 50$ Substitute $B = \frac{5}{6}P$ into this equation: $\frac{5}{6}P = P - 50$

Step 4: Solving for $P$

Rearrange the equation to solve for $P$: $P - \frac{5}{6}P = 50$ $\frac{1}{6}P = 50$ Multiply both sides by 6: $P = 50 \times 6 = 300$

Thus, the passing marks are $\boxed{300}$.

Let me know if you want further details on any step!

Additional Questions:

1. What would A's marks be if the passing marks were 400?
2. How much did B score in this exam based on the calculated passing marks?
3. If the passing marks increased by 10%, how much would A score?
4. What percentage did B score relative to the passing marks?
5. What would B's score be if he failed by 70 marks instead?

Tip:

To solve percentage-based problems efficiently, always convert percentages to fractions or decimals to simplify the equations.