Solving for the Highest Marks in a Series of Exams: A Step-by-Step Guide
In this article, we will go through a detailed step-by-step approach to solve a problem involving the average marks of a student, specifically in the context of determining the highest marks in a series of exams. This problem not only involves basic arithmetic but also requires the ability to set up and solve algebraic equations. Let's break it down into clear, manageable steps.
Understanding the Problem
The average marks scored by a student in six exams is 96. This is a fundamental piece of information that allows us to determine the total marks scored in all six exams. We can use the formula for the average:
Average Total Marks / Number of Exams
Given that the average is 96 and the number of exams is 6, we can calculate the total marks as follows:
Total Marks Average × Number of Exams
Total Marks 96 × 6 576
Excluding the Highest and Lowest Marks
When we consider the remaining four exams, the average is 94. We can apply the same formula to determine the total marks for these four exams:
Total Marks (Remaining 4 Exams) Average × Number of Exams (Excluding Highest and Lowest)
Total Marks (Remaining 4 Exams) 94 × 4 376
The total marks for the two remaining exams (highest and lowest) can be determined by subtracting the total marks of the remaining four exams from the total marks of all six exams:
Total Marks (Highest and Lowest) Total Marks (All Exams) - Total Marks (Remaining 4 Exams)
Total Marks (Highest and Lowest) 576 - 376 200
Setting Up the Equations
Let's denote the highest marks as x and the lowest marks as y. According to the problem, the difference between the highest and lowest marks is 12:
x - y 12
The sum of the highest and lowest marks is 200, as derived from the total marks of the two exams:
x y 200
We now have a system of two equations:
1. x - y 12
2. x y 200
Solving the System of Equations
We can solve these equations using the method of substitution or elimination. Here, we'll use the elimination method:
First, add the two equations:
(x - y) (x y) 12 200
2x 212
x 106
Now substitute x 106 into one of the equations to find y:", true, "content": "
y x - 12 106 - 12 94
Therefore, the highest mark scored by the student is 106, and the lowest mark is 94.
Summary of the Solution
1. Calculate the total marks for all six exams: 576 (96 x 6).
2. Calculate the total marks for the remaining four exams: 376 (94 x 4).
3. Determine the total marks for the two highest and lowest exams: 200 (576 - 376).
4. Set up and solve the system of equations:
5. x - y 12
6. x y 200
7. Add the equations to find x 106.
8. Substitute x 106 into one of the equations to find y 94.
Therefore, the highest score is 106, and the lowest score is 94.
Conclusion
By systematically breaking down the problem and applying basic algebra, we successfully determined the highest and lowest marks. This method is applicable to similar problems and can be used to solve for unknown variables in a series of related equations. Understanding these steps and the underlying math is crucial for mastering the skills required for more complex problem-solving in mathematics and beyond.
Frequently Asked Questions (FAQs)
Q: How do I verify the solution?
A: You can verify the solution by checking if the highest and lowest marks satisfy both equations (x - y 12 and x y 200). In this case, 106 - 94 12 and 106 94 200, confirming the solution is correct.
Q: What are some real-life applications of this problem-solving technique?
A: This technique is applicable in various scenarios, such as financial planning, project management, and data analysis. It helps in understanding complex relationships and making informed decisions based on numerical data.
Q: How can I practice similar problems to improve my skills?
A: You can practice by working through similar problems or creating your own. The key is to understand the underlying concepts and apply them consistently. Online resources, problem sets, and math textbooks are excellent sources for practice.