The Impact of Replacing a Number on the Average

Understanding how the average of a set of numbers changes when a number is replaced is a fundamental concept in arithmetic and statistics. In this guide, we will walk through the process of adjusting the average when one of the numbers in a set of 8 is replaced. Specifically, we will explore the situation where the number 14 is replaced with 28, and determine the new average in terms of the original average, denoted as A.

Step-by-Step Calculation

Original Set of Numbers

Given that the average of 8 numbers is A, we start by calculating the total sum of these numbers:

Total Sum of Original Numbers

Since the average of the 8 numbers is A, the total sum of these numbers is:

Total 8A

Adjusting the Total for Replacement

When we replace the number 14 with 28, we need to adjust the total sum accordingly. The adjustment involves subtracting the number being replaced (14) and adding the new number (28). This gives us the following:

New Total 8A - 14 28 8A 14

Calculating the New Average

The new average is then the new total divided by the number of numbers, which remains 8:

New Average (8A 14) / 8

Final Simplification

Thanks to the distributive property, the expression can be simplified as:

New Average A 14/8

Furthermore, 14/8 can be simplified to 1.75, so the final answer is:

New Average A 1.75

Key Points to Remember

The original total for 8 numbers is calculated as the average (A) times the number of elements (8). When replacing a number in a set, adjust the total sum by subtracting the old number and adding the new number. The new average is calculated by dividing the adjusted total by the number of elements in the set.

Practical Applications

This concept is widely applicable in various fields, including finance, data analysis, and engineering. For instance, in finance, it can be used to adjust stock averages, and in engineering, it can help in recalibrating measurements and calculations.

Example Replacements and Their Effects

Consider a scenario where the average of 8 numbers is 25. If one of the numbers (14) is replaced with 28, the new average can be calculated as follows:

Original Average (A) 25 New Total (8 × 25) 14 New Average (200 14) / 8 214 / 8 26.75

In summary, replacing a number in a set of numbers can significantly impact the average, and the new average can be systematically calculated using the steps outlined above.