Understanding and Calculating Averages: A Step-by-Step Guide
In this article, we will delve into the concept of averages and explore a practical example to illustrate how to calculate the average of a set of numbers when a specific number is excluded. Understanding this concept is crucial for various fields, including data analysis, statistics, and even basic problem-solving in mathematics.
Example Problem
The problem at hand involves the following:
The average of 10 numbers is 72.8. The average of the first six numbers is 88.5. The average of the last five numbers is 64.4.We are tasked with finding the average of the remaining numbers when the 6th number is excluded. Let's break this down step-by-step.
Step-by-Step Solution
Let's begin with the given information and follow a structured approach to solve the problem.
Step 1: Find the Sum of All 10 Numbers
We know that the average of 10 numbers is 72.8.
Total Sum of 10 Numbers Average × Number of Numbers
Total Sum of 10 Numbers 72.8 × 10 728
Step 2: Find the Sum of the First 6 Numbers
The average of the first six numbers is 88.5.
Total Sum of the First 6 Numbers Average × Number of Numbers
Total Sum of the First 6 Numbers 88.5 × 6 531
Step 3: Find the Sum of the Last 5 Numbers
The average of the last five numbers is 64.4.
Total Sum of the Last 5 Numbers Average × Number of Numbers
Total Sum of the Last 5 Numbers 64.4 × 5 322
Step 4: Find the Sum of the Remaining Numbers (Excluding the 6th Number)
We now need to find the sum of the remaining numbers after excluding the 6th number. This can be done by subtracting the sum of the first six numbers and the sum of the last five numbers from the total sum of all ten numbers.
Total Sum of the Remaining 9 Numbers Total Sum of All 10 Numbers - Sum of the 6th Number
Total Sum of the Remaining 9 Numbers 728 - 531 - 322 64.4
Total Sum of the Remaining 9 Numbers 728 - 531 - 442.5 64.4
Total Sum of the Remaining 9 Numbers 639.5
Step 5: Calculate the Average of the Remaining 9 Numbers
Finally, we calculate the average of the remaining nine numbers by dividing the total sum of these numbers by the number of remaining numbers.
Average of the Remaining 9 Numbers Total Sum of the Remaining 9 Numbers / Number of Remaining Numbers
Average of the Remaining 9 Numbers 639.5 / 9 71
Therefore, the average of the remaining numbers when the 6th number is excluded is 71.
Conclusion
Understanding how to calculate averages and work with sets of numbers is a fundamental skill in mathematics and data analysis. This example demonstrates the step-by-step process, from finding the total sum of a set of numbers to excluding a specific number and calculating the new average.
Additional Information
This problem-solving technique can be applied to various scenarios, such as:
Data Analysis: Identifying trends in data sets. Statistics: Calculating mean values for large data sets. Problem-Solving: Solving real-world problems that involve manipulating sets of numbers.By following a structured approach like the one demonstrated above, you can tackle similar problems and gain a deeper understanding of the underlying concepts.