Understanding the Impact of Multiplying Each Number by a Constant on the Average
When dealing with statistical data, it's often necessary to manipulate certain values and understand how these manipulations affect the overall average. This article will explore a common mathematical scenario: what happens to the average when each of a set of numbers is multiplied by a constant. Specifically, we will examine an example involving ten numbers with an average of 7 and demonstrate how multiplying each of these numbers by 8 affects the new average.
Introduction to the Problem
Scenario: Consider a set of 10 numbers whose average is 7. What would be the new average if each of these numbers is multiplied by 8?
Step-by-Step Solution
Step 1: Calculate the Original Sum
The first step is to determine the total sum of the 10 numbers. We know that the average of any set of numbers is defined as the sum of the numbers divided by the count of the numbers. Mathematically, this can be expressed as:
Average Sum / Total number of values
Given that the average (A) is 7 and the total number of values (N) is 10, we can calculate the sum (S) as follows:
S A × N
S 7 × 10
S 70
Step 2: Multiply the Sum by the Constant
Next, we multiply the sum by the constant (k) which, in this case, is 8. This multiplication effectively scales each number in the set by the same factor:
New Sum S × k
New Sum 70 × 8
New Sum 560
Step 3: Calculate the New Average
Finally, we need to compute the new average using the new sum and the same number of values (10). The formula for the new average (A') is:
A' New Sum / N
A' 560 / 10
A' 56
Therefore, the new average after multiplying each of the 10 numbers by 8 is 56.
Key Concepts and Practical Implications
Understanding the effect of multiplying each number in a set by a constant on the average is crucial in various real-world applications, such as data analysis, finance, and scientific research. This property, often referred to as the "constant multiplication property of averages," can be summarized as:
If each number in a set is multiplied by a constant k, the new average will also be multiplied by k.
Mathematically, this can be represented as:
New Average (Original Sum × k) / N (Average × k)
Proving the Concept with Different Examples
Example 1: Multiplying by 5
Let's consider another example where each of the 10 numbers is multiplied by 5 and the average of the original set was 8:
Original Average 8
Original Sum 8 × 10 80
New Sum 80 × 5 400
New Average 400 / 10 40
Example 2: Multiplying by 8
In the given problem where each number is multiplied by 8 and the original average was 7:
Original Average 7
Original Sum 7 × 10 70
New Sum 70 × 8 560
New Average 560 / 10 56
Conclusion
In conclusion, when each number in a set is multiplied by a constant, the new average is simply the original average multiplied by the same constant. This property simplifies many statistical computations and provides a powerful tool for analyzing and manipulating data sets.