Calculating the Average of Remaining Numbers after Exclusions

When dealing with a set of numbers, understanding how to find the average of certain subsets can be quite useful. This guide will walk you through the steps to calculate the average of remaining numbers after excluding specific elements from a larger set. We will explore a scenario where the average of 52 numbers is 48.5, and we need to exclude the 21st and 22nd numbers to find the new average.

Step-by-Step Guide

Step 1: Calculate the Total Sum of the 52 Numbers

The first step is to calculate the total sum of the 52 numbers, given that their average is 48.5. The formula for the total sum, S, of these numbers is:

S Average × Number of values

Substituting the given values:

S 48.5 × 52

Calculating this:

S 48.5 × 52 2522

Step 2: Calculate the Total Sum of the First 22 Numbers

The average of the first 22 numbers is 59.8. The total sum, S?, of the first 22 numbers can be calculated as:

S? Average × Number of values

Substituting the given values:

S? 59.8 × 22

Calculating this:

S? 59.8 × 22 1316

Step 3: Calculate the Total Sum of the Last 32 Numbers

The average of the last 32 numbers is 41.7. The total sum, S?, of the last 32 numbers can be calculated as:

S? Average × Number of values

Substituting the given values:

S? 41.7 × 32

Calculating this:

S? 41.7 × 32 1334.4

Step 4: Verify the Sums

The sum of the first 22 numbers and the last 32 numbers should equal the total sum of the 52 numbers. However, we need to account for the overlap of the 21st and 22nd numbers. The sum of the first 22 numbers can also be expressed as:

S? Sum of first 20 numbers 21st number 22nd number

Let x be the 21st number and y be the 22nd number. The sum of the last 32 numbers can also be expressed as:

S? Sum of last 30 numbers 21st number 22nd number

Using these expressions, we can find the sum of the first 20 and last 30 numbers:

Let S? be the sum of the first 20 numbers and S? be the sum of the last 30 numbers. Thus:

S? x y 1316

S? x y 1334.4

Step 5: Calculate the Sum of the 21st and 22nd Numbers

From the above equations, we can find the sum of the 21st and 22nd numbers:

S? S? 2(x y) 1316 1334.4 2650.4

Subtracting the total sum of the 52 numbers:

2650.4 - 2522 128.4

Therefore, the sum of the 21st and 22nd numbers, x y, is 128.4.

Step 6: Calculate the Total for the Remaining 50 Numbers

Since we need to exclude x and y, the total sum of the remaining 50 numbers is:

Total remaining 2522 - (128.4) 2393.6

The average of the remaining numbers is:

Average Total remaining / Number of remaining numbers

Average 2393.6 / 50

Average 47.872

Final Answer

The average of the remaining numbers after excluding the 21st and 22nd numbers is 47.872. This detailed approach helps in understanding how to systematically work with different subsets of numbers to find desired averages.

Note: This calculation assumes no further information or assumptions about the values of the specific numbers are provided.