Calculating Score Differences Analyzing Student Averages And Ratios
In this article, we will delve into a problem concerning the average marks of students in a class. Understanding the concept of averages is crucial in various fields, including academics, statistics, and data analysis. We aim to dissect the problem step by step, providing a clear and concise solution. The problem involves calculating the marks of two remaining students given the average marks of the entire class and subgroups within it. By breaking down the problem into smaller parts, we can systematically arrive at the answer. This exercise will not only help in solving this particular problem but also enhance problem-solving skills in general.
The average marks of 20 students is 75. Among them, the average marks of 2 students are 50, and the average marks of 16 students from the remaining group are 70. The marks of the remaining two students are in the ratio 1:3. The task is to find the difference between the marks of these two students. This problem requires a good grasp of average calculations and ratio concepts. We will explore how to use the given information effectively to determine the unknown marks.
Breaking Down the Problem
To effectively tackle this problem, we will break it down into manageable parts. Here’s how we will approach it:
- Calculate the total marks of all 20 students.
- Calculate the total marks of the first 2 students.
- Calculate the total marks of the next 16 students.
- Determine the combined marks of the remaining 2 students.
- Use the given ratio to find the individual marks of the remaining students.
- Calculate the difference between these marks.
This structured approach will help us to avoid confusion and ensure accuracy in our calculations. Each step builds upon the previous one, leading us towards the final solution.
1. Calculate the Total Marks of All 20 Students
The average marks of 20 students is 75. To find the total marks, we multiply the average by the number of students:
Total marks = Average marks × Number of students Total marks = 75 × 20 = 1500
Therefore, the total marks of all 20 students are 1500. This is our baseline figure, which we will use in subsequent calculations. Understanding the relationship between average and total is key to solving problems of this nature.
2. Calculate the Total Marks of the First 2 Students
The average marks of the first 2 students are 50. Similarly, we calculate their total marks:
Total marks of 2 students = Average marks × Number of students Total marks of 2 students = 50 × 2 = 100
Thus, the first two students scored a total of 100 marks. This subtotal will be subtracted from the overall total to narrow down the marks of the remaining students. Breaking down the problem into smaller, solvable parts makes the overall calculation more manageable.
3. Calculate the Total Marks of the Next 16 Students
The average marks of the next 16 students are 70. Their total marks are:
Total marks of 16 students = Average marks × Number of students Total marks of 16 students = 70 × 16 = 1120
So, these 16 students scored a total of 1120 marks. We now have the total marks for two subgroups, which will help us find the combined marks of the last two students. Accurate multiplication is essential in these calculations to avoid errors.
4. Determine the Combined Marks of the Remaining 2 Students
To find the combined marks of the remaining 2 students, we subtract the total marks of the first 2 students and the next 16 students from the overall total:
Combined marks of 2 students = Total marks of all students - (Total marks of first 2 students + Total marks of 16 students) Combined marks of 2 students = 1500 - (100 + 1120) Combined marks of 2 students = 1500 - 1220 = 280
Hence, the remaining two students scored a combined total of 280 marks. This figure is crucial as it sets the stage for using the given ratio to find individual scores. Subtraction plays a key role in isolating the marks of these two students.
5. Use the Given Ratio to Find the Individual Marks of the Remaining Students
The marks of the remaining two students are in the ratio 1:3. Let the marks of the first student be x and the marks of the second student be 3x. Their combined marks are 280:
x + 3x = 280 4x = 280 x = 280 / 4 x = 70
So, the marks of the first student are 70. The marks of the second student are:
3x = 3 × 70 = 210
Therefore, the marks of the two students are 70 and 210, respectively. Understanding ratios and their application is fundamental in solving this part of the problem.
6. Calculate the Difference Between These Marks
Finally, we find the difference between the marks of the two students:
Difference = Marks of second student - Marks of first student Difference = 210 - 70 = 140
Thus, the difference between the marks of the two students is 140. This completes the solution to the problem. Subtraction is the final step in determining the difference between the two scores.
In conclusion, the difference between the marks of the remaining two students is 140. By systematically breaking down the problem and addressing each part step by step, we were able to arrive at the solution accurately. This problem underscores the importance of understanding averages, ratios, and basic arithmetic operations in mathematical problem-solving. The ability to break down complex problems into simpler steps is a valuable skill in mathematics and beyond. This exercise not only provides a solution to a specific problem but also enhances overall analytical and problem-solving capabilities. The methodical approach demonstrated here can be applied to a wide range of mathematical problems, making it a useful technique for students and professionals alike. Consistent practice and a clear understanding of fundamental concepts are key to mastering such problems. We hope this detailed analysis has provided a clear understanding of how to approach and solve problems involving averages and ratios effectively. Problem-solving skills are not just about finding the right answer; they are about developing a systematic approach to tackle challenges.
