Solving for the Removed Number in Averaged Sets

The Importance of Understanding Averaged Sets and Removing Numbers

Mathematically, averages play a critical role in analyzing and understanding numerical data. In this article, we'll explore the concept of finding the removed number when the average of a set of numbers changes. This is a common problem that requires a systematic approach to solve accurately. Whether you're dealing with a set of 5 numbers or a larger set, the principles remain the same. This knowledge is highly valuable in fields ranging from basic mathematics to complex data analysis.

The Scenario

Let's delve into a scenario where we have a set of numbers. Suppose we have a series of numbers, and the average is a certain value. Then, if one number is removed, the new average changes. Our task is to find out which number was removed. This requires a bit of algebraic manipulation.

Example 1: Removing a Number from a Set of 5 Numbers

Given: The average of 5 numbers is 80. When one number is removed, the average of the remaining 4 numbers becomes 50.

To Find: The number that was removed.

Solution: Let the five numbers be a1, a2, a3, a4, a5. The average of these 5 numbers is 80: (a1 a2 a3 a4 a5) / 5 80 Multiplying both sides by 5 gives: a1 a2 a3 a4 a5 400 When one number, say x, is removed, the average of the remaining 4 numbers is 50: (a1 a2 a3 a4 a5 - x) / 4 50 Multiplying both sides by 4 gives: a1 a2 a3 a4 a5 - x 200 We already know that a1 a2 a3 a4 a5 400, so we can substitute this into the equation: 400 - x 200 Solving for x: x 400 - 200 200 Therefore, the removed number is 200.

Example 2: Removing a Number from a Set of 6 Numbers

Given: The average of 6 numbers is 40, and the sum is 40 × 6 240. After removing one number, the average of the remaining 5 numbers becomes 30, and the sum is 30 × 5 150.

To Find: The removed number.

Solution: The sum of the original 6 numbers is 240. The sum of the remaining 5 numbers is 150. The removed number is: 240 - 150 90

Example 3: Removing a Number from a Set of 5 Numbers with Given Averages

Given: The average of 5 numbers is 35, and the sum is 35 × 5 175. The average of 4 numbers is 25, and the sum is 25 × 4 100.

To Find: The removed number.

Solution: The sum of the original 5 numbers is 175. The sum of the remaining 4 numbers is 100. The removed number is: 175 - 100 75

Example 4: Removing a Number with Summation Analysis

Given: The sum of 4 numbers is 232, and the sum of 3 numbers is 150. The difference gives the value of the removed number.

Solution: The sum of 4 numbers is 232. The sum of 3 numbers is 150. The removed number is: 232 - 150 82

Example 5: Analyzing the Sum with Given Averages and Products

Given: The average of 5 numbers is 42, and the sum is 42 × 5 210. The average of 4 numbers is 35, and the sum is 35 × 4 140.

To Find: The removed number.

Solution: The sum of the original 5 numbers is 210. The sum of the remaining 4 numbers is 140. The removed number is: 210 - 140 70

Conclusion

By following these steps, we can solve for the removed number in scenarios where the average changes after removing one number. This method is not only helpful in mathematical problems but also in real-world applications such as data analysis, budgeting, and more. Understanding these principles can be a significant asset in various fields where averages and data manipulation are critical.