Step 1: Understanding the Concept:
This problem deals with the concept of percentages in the context of examination scoring.
The core idea is that the "Passing Marks" for a specific exam is a constant value.
We have two different students, P and Q, whose performances are measured relative to this constant.
P is below the passing mark, and Q is above it.
By expressing the passing marks in two different ways based on their percentages and absolute mark differences, we can establish an algebraic equality.
Step 2: Key Formula or Approach:
Let the total maximum marks of the exam be represented by \( x \).
Passing Marks = \( (\text{Score of P}) + \text{Shortfall} = (\text{Score of Q}) - \text{Excess} \).
Using percentages:
Passing Marks = \( 30% \text{ of } x + 30 \)
Passing Marks = \( 40% \text{ of } x - 40 \)
Step 3: Detailed Explanation:
Equating the two expressions for passing marks:
\[ \frac{30}{100}x + 30 = \frac{40}{100}x - 40 \]
Convert the fractions to decimals for easier algebraic manipulation:
\[ 0.3x + 30 = 0.4x - 40 \]
To solve for \( x \), move all terms containing \( x \) to one side and constant numbers to the other:
\[ 30 + 40 = 0.4x - 0.3x \]
\[ 70 = 0.1x \]
Divide both sides by 0.1:
\[ x = \frac{70}{0.1} = 700 \]
The maximum marks for the exam are 700.
Now, calculate the actual passing marks using P's data:
Passing Marks = \( 30% \text{ of } 700 + 30 \)
\[ \text{Passing Marks} = 0.3 \times 700 + 30 = 210 + 30 = 240 \]
Finally, determine the pass percentage by finding what percent 240 is of 700:
\[ \text{Pass Percentage} = \left( \frac{\text{Passing Marks}}{\text{Maximum Marks}} \right) \times 100 \]
\[ \text{Pass Percentage} = \left( \frac{240}{700} \right) \times 100 \]
\[ \text{Pass Percentage} = \frac{240}{7} \]
Dividing 240 by 7 gives:
\( 240 \div 7 = 34.2857... % \)
Rounding to the nearest decimal as per the options, we get approximately 34.2%.
Step 4: Final Answer:
The pass percentage for the examination is approximately 34.2%.