The question asks for the total marks in an examination, given the following details:

• A student must score 55% to pass the exam.
• The student scored 420 marks but failed by 78 marks.

Let's solve this step by step:

1. The student scored 420 marks but needed 78 more marks to pass, so the passing marks are: $\text{Passing Marks} = 420 + 78 = 498$

2. The passing percentage is 55%. Therefore, if 55% corresponds to 498 marks, we can find the total marks (let's call it $T$) by using the relation: $55\% \text{ of } T = 498$

3. Converting 55% to a decimal: $0.55 \times T = 498$

4. Solving for $T$: $T = \frac{498}{0.55} = 905.45$

Since the total marks must be a whole number, we round $905.45$ to the nearest whole number, which is 905.

Answer:

The total marks for the examination are 905.

Would you like more details or clarifications on any step? Here are some related questions for practice:

1. If a student scores 40% on the exam, how many marks will they score out of 905?
2. If the passing percentage was 60%, what would be the new passing marks?
3. If the student had scored 10 more marks, how many marks would they have failed by?
4. How much percentage would the student need to score an exact 500 marks?
5. If the total marks were 1000, and the passing percentage remained 55%, what would be the new passing marks?

Tip: Always double-check percentages and units to ensure calculations are aligned with the problem requirements.