Finding the Removed Number Using Averages

In this article, we explore a fascinating problem involving the average of a set of numbers. We will use algebraic and statistical methods to determine a specific number that was removed from a dataset. This problem not only tests your understanding of averages but also your problem-solving skills.

Problem Statement

Consider a list of 10 numbers with an average of 28. When one number is removed, the average of the remaining 9 numbers becomes 18. Our task is to find the number that was removed.

Solution Using Algebra

Let's denote the number removed as x.

The sum of the 10 numbers can be expressed as:

Total sum of 10 numbers 10 times; 28 280

After removing the number x, the average of the remaining 9 numbers is 18, so:

Total sum of 9 numbers 9 times; 18 162

Therefore, we can set up the following equation to solve for x:

280 - x 162

Solving for x gives:

x 280 - 162 118

Thus, the number that was removed is 118.

Verification Using Different Sets of Numbers

Let's verify our method with different sets of numbers to ensure it works universally.

Case 1: Average of 6 Numbers is 25

The sum of 6 numbers with an average of 25 is:

Sum of 6 numbers 6 times; 25 150

After removing one number, let’s call the average of the remaining 5 numbers 20. The sum of the 5 numbers is:

Sum of 5 numbers 5 times; 20 100

The number removed can be found by subtracting:

Removed number 150 - 100 50

Case 2: Average of 12 Numbers is 28

The sum of 12 numbers with an average of 28 is:

Sum of 12 numbers 12 times; 28 336

After removing one number, let’s call the average of the remaining 11 numbers 26. The sum of the 11 numbers is:

Sum of 11 numbers 11 times; 26 286

The removed number can be found by subtracting:

Removed number 336 - 286 50

Case 3: Using the Formula for Average

Let's use the formula for average, where:

S a times; n

Given:

S_12 28 times; 12 336

S_11 26 times; 11 286

Now we can write:

S_12 - r S_11 where r is the number removed.

Solving for r gives:

r 336 - 286 50

Therefore, the removed number is 50.

Another Example

Let's solve another similar problem:

Given that the sum of 6 numbers is 150 and the average of 5 numbers is 15, we need to find the removed number:

Sum of 6 numbers 150

Since the average of 5 numbers is 15, their total sum is:

Sum of 5 numbers 5 times; 15 75

The removed number can be found by subtracting:

Removed number 150 - 75 75

Thus, the removed number is 75.

Conclusion

By understanding and applying the concept of averages and algebra, we can effectively find the removed number in a dataset. This method can be applied to various problems involving the most common statistical functions like average, sum, and count.

Keywords: average, numbers, removal, calculation