Solving for the Number of Passed Students Using Average Marks: A Comprehensive SEO-Optimized Guide

Successfully analyzing and understanding the average marks of students in an exam can provide valuable insights into academic performance. This article delves into a specific problem related to the average marks of students and guides you through the process of solving for the number of students who passed the exam. This information is particularly useful for SEO optimization and improving website content for educational purposes.

Understanding the Problem

Let's consider a scenario where the average marks of 140 students in an exam are 36. We are also provided with the average marks of the students who passed (42) and those who failed (21). The objective is to determine the number of students who passed the exam.

Setting Up the Equations

We define the following variables:

Number of passed students: ( p ) Number of failed students: ( f )

From the problem, we know the following:

Total number of students: ( p f 140 ) Average marks of all students: 36 Total marks of all students: ( 36 times 140 5040 ) Average marks of students who passed: 42 Total marks of students who passed: ( 42p ) Average marks of students who failed: 21 Total marks of students who failed: ( 21f )

Using this information, we can set up the following equation to represent the total marks:

42p 21f 5040

Now we have a system of two equations:

p f 140 42p 21f 5040

Let's solve these equations step by step:

Step 1: Express ( f ) in terms of ( p )

From the first equation:

f 140 - p

Step 2: Substitute ( f ) into the second equation

Substituting ( f ) into the second equation:

42p 21(140 - p) 5040

Step 3: Simplify the equation

Distributing the 21:

42p 2940 - 21p 5040

Combining like terms:

21p 2940 5040

Step 4: Solve for ( p )

Subtracting 2940 from both sides:

21p 5040 - 2940 21p 2100

Dividing by 21:

p frac{2100}{21} 100

Step 5: Find ( f )

Now substitute ( p 100 ) back into the first equation to find ( f ):

f 140 - 100 40

Conclusion

The number of students who passed the exam is 100.

Further Explorations

Let's explore a couple of additional scenarios:

Scenario 2: Solving for the Number of Passed Students Using Different Average Marks

Suppose the average marks of 140 students is 40. The number of students who passed is ( x ) and the number of students who failed is ( 140 - x ). The average marks of passed and failed students are 35 and 10 respectively.

From the problem:

Total marks of all students: ( 40 times 140 5600 ) Total marks of passed students: ( 35x ) Total marks of failed students: ( 10(140 - x) )

The total marks equation is:

35x 10(140 - x) 5600

Step 1: Simplify the equation

35x 1400 - 1 5600 25x 1400 5600

Step 2: Solve for ( x )

25x 5600 - 1400 25x 4200 x frac{4200}{25} 168

Note: The solution ( x 168 ) indicates an impossible scenario because the number of students cannot exceed the total number of students, which is 140. This example highlights the importance of checking for logical consistency in mathematical problems.

Scenario 3: Additional Complex Distribution

Let ( x ) be the number of students who passed and ( y ) be the number of students who failed. The total marks for 100 students is ( 40 times 100 4000 ).

The average marks for passed students is 45, and for failed students is 25. We can set up the following equation:

45x 25y 4000

And the total student equation is:

x y 100

Step 1: Express ( y ) in terms of ( x )

y 100 - x

Step 2: Substitute ( y ) into the first equation

45x 25(100 - x) 4000

Step 3: Simplify and solve for ( x )

45x 2500 - 25x 4000 2 2500 4000 2 4000 - 2500 2 1500 x frac{1500}{20} 75

The number of students who passed is 75.

Conclusion and Key Takeaways

By understanding and applying the principles of average marks, we can effectively solve for the number of passed students in various scenarios. This article provides a step-by-step guide to the problem-solving process and offers insights into the importance of careful calculation and logical consistency in mathematical problem-solving. The final answers are summarized as follows:

Scenario 1: 100 students passed Scenario 2: Impossible scenario (168 is not a valid number of students) Scenario 3: 75 students passed

These guidelines and scenarios are valuable for students, educators, and professionals seeking to enhance their understanding of average marks and their real-world applications.