Analyzing the Examination Results Problem

This problem requires us to determine the number of students who achieved a third division pass based on the overall pass percentage, the number of students who failed, and the breakdown of successful candidates by division.

Understanding the Given Information:

• Pass Percentage: 80% of the total students passed the examination.
• Fail Count: A specific number, 90 students, did not pass the examination.
• Division Breakdown (of successful candidates):
  • 27% of those who passed scored a first division.
  • 43% of those who passed scored a second division.
• Objective: Calculate the number of students who passed in the third division.

Step-by-Step Calculation and Explanation

1. Determining the Total Number of Students

First, we need to find the total number of students in the class. We know 80% passed, which implies 20% failed.

Percentage of students who failed = \(100\% - \text{Pass Percentage}\)

\( \text{Percentage failed} = 100\% - 80\% = 20\% \)

We are given that the number of students who failed is 90.

Let the total number of students in the class be represented by '\(T\)'.

We can set up the equation based on the failed students:

\( 20\% \text{ of } T = 90 \)

Converting the percentage to a decimal or fraction:

\( \frac{20}{100} \times T = 90 \)

Solving for '\(T\)':

\( T = \frac{90 \times 100}{20} \)

\( T = 90 \times 5 \)

\( T = 450 \)

So, there are 450 students in the class.

2. Calculating the Number of Successful Candidates

Next, we find out how many students passed the examination. This is 80% of the total students.

Number of students who passed = \(80\% \text{ of } 450\).

\( \text{Number passed} = \frac{80}{100} \times 450 \)

\( \text{Number passed} = \frac{4}{5} \times 450 \)

\( \text{Number passed} = 4 \times 90 \)

\( \text{Number passed} = 360 \)

Alternatively, we can subtract the number of failed students from the total number of students:

\( \text{Number passed} = \text{Total students} - \text{Failed students} \)

\( \text{Number passed} = 450 - 90 = 360 \)

There were 360 successful candidates.

3. Calculating the Percentage of Students Passing in the Third Division

The problem provides the distribution of marks among the successful candidates (the 360 students who passed).

• Percentage passing in first division = 27%
• Percentage passing in second division = 43%

Let's find the combined percentage of successful candidates who passed in either the first or second division:

\( \text{Percentage in 1st or 2nd division} = 27\% + 43\% = 70\% \)

The remaining percentage of successful candidates must have passed in the third division. Therefore:

\( \text{Percentage in 3rd division} = 100\% - (\text{Percentage in 1st or 2nd division}) \)

\( \text{Percentage in 3rd division} = 100\% - 70\% = 30\% \)

This means 30% of the successful candidates passed in the third division.

4. Calculating the Number of Students Passing in the Third Division

Finally, we calculate the actual number of students who passed in the third division.

Number of students in the third division = \(30\%\) of the successful candidates (360).

\( \text{Number in 3rd division} = 30\% \text{ of } 360 \)

\( \text{Number in 3rd division} = \frac{30}{100} \times 360 \)

\( \text{Number in 3rd division} = \frac{3}{10} \times 360 \)

\( \text{Number in 3rd division} = 3 \times 36 \)

\( \text{Number in 3rd division} = 108 \)

Final Answer

The calculation shows that 108 students from the class passed in the third division.