How Multiplying Each Number by a Factor Affects the Average: A Step-by-Step Guide

When working with averages and mathematical operations, understanding how changes in the dataset affect the average is critical. This article explores how multiplying each number in a set by a specific factor impacts the average. We will provide a detailed explanation with examples to ensure a clear view of the process.

Understanding Averages

The average, also known as the arithmetic mean, is calculated by summing a set of numbers and then dividing by the count of numbers in the set. Mathematically, it is represented as:

[[ text{Average} frac{sum x}{N} ] ]

Example 1: Calculating the New Average After Doubling Each Number

Initial Conditions

Suppose the average of five numbers is 55. Here's how we find the new average when each number is multiplied by 2:

Determine the total sum of the original numbers using the formula:

[[ sum x 55 times 5 275 ] ]

Multiply the total sum by the factor (2 in this case) to get the new total:

[[ text{New Total} 2 times 275 550 ] ]

Divide the new total by the count of the numbers to get the new average:

[[ text{New Average} frac{550}{5} 110 ] ]

Therefore, the new average, after each number is multiplied by 2, is 110.

Example 2: Doubling Each Number to Find the New Average

Let's consider the numbers a, b, c, d, and e with an average of 20:

Calculate the total sum of these five numbers:

[[ a b c d e 20 times 5 100 ] ]

Multiply the total sum by the factor (2 in this case) to get the new total:

[[ 2a 2b 2c 2d 2e 2 times 100 200 ] ]

Divide the new total by the count of the numbers to get the new average:

[[ text{New Average} frac{200}{5} 40 ] ]

Thus, the new average is 40 when each number is multiplied by 2.

Example 3: Detailed Calculation

Consider the numbers x, x1, x2, x3, and x4 with an average of 20:

Calculate the total sum of these five numbers:

[[ frac{x x1 x2 x3 x4}{5} 20 ] ] [[ x x1 x2 x3 x4 100 ] ]

Multiply each number by 2 to get the new sum:

[[ 2x 2x1 2x2 2x3 2x4 2 times 100 200 ] ]

Divide the new sum by the count of the numbers to get the new average:

[[ text{New Average} frac{200}{5} 40 ] ]

Hence, the new average is 40 when each number is multiplied by 2.

Conclusion

In conclusion, multiplying each number in a set by a factor (in this case, 2) results in a corresponding change in the average. Specifically, multiplying each number by 2 doubles the average of the set. This principle is useful in various mathematical and real-world scenarios.