The Removal of a Number to Change the Average
In mathematics, the process of determining a number that, when removed from a group, changes the average of the remaining numbers, is a commonly encountered problem. This article explores a straightforward example where the average of five numbers is 28. When one number is removed, the new average of the remaining four numbers is 26. We will solve for the removed number using various methods and highlight the underlying mathematical principles.Sum of the Numbers
Let's start by calculating the sum of the five numbers. Given that the average of the five numbers is 28, the sum of these numbers is calculated as follows:28 × 5 14028times 5 140
New Average and Sum
When one number is removed, the new average of the remaining four numbers becomes 26. Therefore, the sum of the four remaining numbers can be calculated as:26 × 4 10426times 4 104
Determining the Removed Number
To find the removed number, we subtract the sum of the four remaining numbers from the sum of the original five numbers:140 - 104 36140 - 104 36
Verification Using Equations
We can also use equations to verify our findings. Let the removed number be denoted by ( X ). The equation for the original set of numbers is:28 × 5 - X/4 2628 times 5 - X / 4 26
Simplifying the equation:140 - X 104140 - X 104
From which we can solve for ( X ):X 140 - 104 36X 140 - 104 36
Additional Verification Methods
Another method to solve this problem involves direct calculation of the sums of the original and remaining sets of numbers. The sum of the five original numbers is 140, and the sum of the remaining four numbers after removal is 104. Therefore, the removed number ( X ) is:140 - 104 36140 - 104 36
Conclusion
In conclusion, the process of determining the removed number when a new average is given involves simple arithmetic operations and understanding the relationship between the original sum and the sum of the remaining numbers. By using the formulas and methods explained above, we can accurately determine the removed number without any ambiguity.140 - 104 36140 - 104 36