Understanding the Sum and Average of Sets of Numbers
In contemporary analytics and data analysis, we often encounter situations where we need to manipulate and understand sets of numbers based on given averages. This article will guide you through a practical example where we calculate the average of the first two numbers given the averages of a set of five numbers and the last three of those numbers.
The Problem Statement and Given Data
The given problem involves a set of five numbers with known averages for different subsets. Specifically, the average of all five numbers is known to be 9, and the average of the last three numbers is 5. The steps to solve this problem involve finding the average of the first two numbers. Let's break down the problem and the steps required to solve it.
Step 1: Calculating the Total Sum of All Five Numbers
The first step in solving this problem is to use the given average of the five numbers to find their total sum. Given that the average of five numbers is 9, we can set up the following equation:
[ text{Average of 5 numbers} 9 Rightarrow text{Sum of 5 numbers} 9 times 5 45 ]Step 2: Calculating the Total Sum of the Last Three Numbers
Similarly, we use the given average for the last three numbers to find their total sum. Given that the average of the last three numbers is 5, we set up the following equation:
[ text{Average of last 3 numbers} 5 Rightarrow text{Sum of last 3 numbers} 5 times 3 15 ]Step 3: Determining the Sum of the First Two Numbers
With the sums of the entire set and the last three numbers, we can now find the sum of the first two numbers by subtracting the sum of the last three numbers from the total sum of all five numbers:
[ text{Sum of first 2 numbers} text{Total sum of 5 numbers} - text{Sum of last 3 numbers} ] 45 - 15 30 ]Step 4: Calculating the Average of the First Two Numbers
Finally, to find the average of the first two numbers, we divide the sum of the first two numbers by 2:
[ text{Average of first 2 numbers} frac{30}{2} 15 ]Conclusion
In conclusion, the average of the first two numbers is 15, as derived from the given information. This process demonstrates the importance of understanding set arithmetic and how to manipulate data based on given averages. The steps outlined here can be applied to a variety of similar problems in data analysis and numerical problem solving.
Common Applications
This type of problem-solving approach is not only useful in academic settings but also has practical applications in fields such as finance, engineering, and computer science. By mastering these techniques, you can better analyze and manipulate data sets in real-world situations.