Understanding and Solving Multi-Group Average Problems in Mathematics
In mathematics, understanding and solving multi-group average problems can be crucial for various applications, from statistical analysis to real-world scenario modeling. This article will guide you through a detailed example, explaining how to solve a complex average problem using algebraic methods. By walking you through the process, we aim to demystify the steps involved and provide a comprehensive solution.
Problem Presentation: Analyzing Averages of Partial Groups
Let's consider a specific problem where the average of 17 numbers is 45. Additionally, the average of the first 9 numbers is 51, and the average of the last 9 numbers is 36. Given this information, what is the ninth number?
Given Data and Equations
We have the following pieces of information:
The average of 17 numbers is 45. The average of the first 9 numbers is 51. The average of the last 9 numbers is 36.We can express these as equations:
1. ( frac{x_1 x_2 ldots x_{17}}{17} 45 )
2. ( frac{x_1 x_2 ldots x_9}{9} 51 )
3. ( frac{x_9 x_{10} ldots x_{17}}{9} 36 )
Step-by-Step Solution
Let's start by working through the equations step by step:
From the first equation, we have:
( x_1 x_2 ldots x_{17} 45 times 17 765 )
From the second equation, we have:
( x_1 x_2 ldots x_9 51 times 9 459 )
From the third equation, we have:
( x_9 x_{10} ldots x_{17} 36 times 9 324 )
In the given problem, we need to find the value of the ninth number, ( x_9 ).
First, let's find the sum of the first 9 numbers and the sum of the last 9 numbers. From the above equations, we know:
( text{Sum of first 9 numbers} 459 )
( text{Sum of last 9 numbers} 324 )
Now, we can find the sum of the middle part (excluding the ninth number) by subtracting the sum of the last 9 numbers from the total sum:
( text{Sum of first 9 numbers} 765 - 324 441 )
To find the ninth number, we subtract the sum of the first 8 numbers from the sum of the first 9 numbers:
( x_9 441 - 459 -18 )
Therefore, the ninth number is ( -18 ).
Alternative Solution Method
Another way to solve this problem is by using a different set of equations and algebraic manipulation:
1. ( (n1 n2 ... n17)/17 45 )
2. ( (n1 n2 ... n9)/9 51 )
3. ( (n9 n10 ... n17)/9 36 )
By summing equations 2 and 3, we get:
( (n1 n2 ... n9) (n9 n10 ... n17) 459 324 )
( (n1 n2 ... n17) 2n9 783 )
Subtracting the total sum of 17 numbers (from equation 1) from this, we get:
( 765 2n9 783 )
( 2n9 18 )
( n9 18 )
This confirms that the ninth number is ( 18 ).
Conclusion
In this article, we have walked through the process of solving a multi-group average problem. By understanding the given data, setting up the appropriate equations, and performing algebraic manipulations, we were able to find the correct solution. This problem-solving technique can be applied to a wide range of similar problems, making it a valuable skill for students and professionals in fields such as mathematics, statistics, and data analysis.