Understanding the Impact of Incrementing Numbers on Their Average
When working with numerical data, understanding how changes in the values affect the average or arithmetic mean is a fundamental concept. This guide will explore a specific problem: what happens to the average of 8 numbers when each number is increased by a fixed amount. By following step-by-step methods, you will gain insights into how to calculate the new average and why it behaves the way it does.
Problem Statement
The problem states that the average of 8 numbers is 25. Each of these numbers is to be increased by 15. What is the new average after this change?
Step-by-Step Solution
Let's break down the problem into manageable steps to find the new average.
Step 1: Find the Original Sum of the Numbers
The formula for the average is given by:
Average Sum of Numbers / Number of Numbers
Given that the average of 8 numbers is 25, we can calculate the sum of these numbers:
Sum of Numbers Average × Number of Numbers
25 × 8
200
Step 2: Find the New Sum After Increasing Each Number by 15
To find the new sum, we need to add 15 to each of the 8 numbers. Therefore, we need to multiply the increase by the number of numbers:
New Sum Old Sum (Number of Numbers × Increase)
200 (8 × 15)
200 120
320
Step 3: Find the New Average
The new average can be found by dividing the new sum by the number of numbers:
New Average New Sum / Number of Numbers
320 / 8
40
Conclusion
Thus, the new average of the 8 numbers after increasing each by 15 is 40. This result can be generalized using the following principle:
Given a set of numbers with an average of X, if each number in the set is increased by a constant Y, the new average becomes X Y.
Generalized Formula
The formula can be expressed mathematically as:
Y ax b
Then, ean{y}{n} aean{x}{n} b
implies ean{y} aean{x} b
Setting a1 and b15 we see that:
ean{y} 25 15 40
Additional Insights
This process can be useful in scenarios such as adjusting financial data, evaluating test scores, or any other set of numerical values where the average needs to be recalculated after a uniform adjustment. Understanding this concept helps in making more accurate predictions and decisions in data analysis.