Raushan's Exam Scores Analysis

This question requires us to determine the percentage of pass marks in an examination. We are given information about a student, Raushan, and his performance in two different hypothetical scenarios.

• Scenario 1: Raushan scored 25% marks and failed by 64 marks.
• Scenario 2: If Raushan had scored 40% marks, he would have secured 32 marks more than the pass marks.

Our goal is to find the pass marks as a percentage of the total marks.

Total Marks Calculation from Raushan's Scores

Let's define the variables:

• Let M represent the total marks in the examination.
• Let P represent the pass marks required for the examination.

Now, we translate the given information into mathematical equations:

From Scenario 1:

• Raushan's score = 25% of M = $0.25 \times M$.
• Since he failed by 64 marks, his score is 64 marks less than the pass marks: $0.25 \times M = P - 64$.
• Rearranging this equation, we get an expression for P: $P = 0.25 \times M + 64$ (Equation 1).

From Scenario 2:

• Raushan's score = 40% of M = $0.40 \times M$.
• He secured 32 marks more than the pass marks: $0.40 \times M = P + 32$.
• Rearranging this equation, we get another expression for P: $P = 0.40 \times M - 32$ (Equation 2).

We now have two equations representing the pass marks (P). By setting them equal to each other, we can solve for the total marks (M):

$0.25 \times M + 64 = 0.40 \times M - 32$

Let's rearrange the equation to solve for M:

$64 + 32 = 0.40 \times M - 0.25 \times M$

$96 = 0.15 \times M$

To find M, we divide 96 by 0.15:

$M = \frac{96}{0.15}$

$M = \frac{9600}{15}$

$M = 640$

So, the total marks in the examination are 640.

Pass Marks Determination

Now that we know the total marks (M = 640), we can calculate the pass marks (P) using either Equation 1 or Equation 2.

Using Equation 1:

$P = 0.25 \times M + 64$

$P = 0.25 \times 640 + 64$

$P = 160 + 64$

$P = 224$

Using Equation 2 for verification:

$P = 0.40 \times M - 32$

$P = 0.40 \times 640 - 32$

$P = 256 - 32$

$P = 224$

The pass marks required for the examination are 224.

Pass Marks Percentage Calculation

The final step is to calculate the percentage of pass marks. This is done by dividing the pass marks (P) by the total marks (M) and multiplying the result by 100.

Pass Marks Percentage $= \frac{P}{M} \times 100$

Substitute the values P = 224 and M = 640:

Pass Marks Percentage $= \frac{224}{640} \times 100$

Simplify the fraction $\frac{224}{640}$:

$\frac{224}{640} = \frac{7 \times 32}{20 \times 32} = \frac{7}{20}$

Now, convert the fraction to a percentage:

Pass Marks Percentage $= \frac{7}{20} \times 100$

Pass Marks Percentage $= 7 \times 5$

Pass Marks Percentage $= 35\%$

Final Exam Pass Percentage

The percentage of pass marks in the examination is 35%.