Average Calculation Problem Explained

This problem requires calculating the average of a subset of numbers after a specific number is excluded. We are provided with the average of the first 16 numbers and the average of the last 15 numbers from a set of 32 numbers. The goal is to determine the new average once the 17^th number is removed.

Given Information Breakdown

Here's a summary of the data given in the question:

• Total count of numbers: 32
• Average of the first 16 numbers: 54
• Average of the last 15 numbers: 57
• Number to be excluded: The 17^th number

The structure of the 32 numbers is considered as:

(Numbers 1 to 16) + (Number 17) + (Numbers 18 to 32)

The "last 15 numbers" are interpreted as the numbers from index 18 through 32.

Average Calculation Steps

Step 1: Sum of First 16 Numbers

To find the sum of a group of numbers, we multiply the average by the count of numbers in that group.

Sum$_{1-16}$ = Average$_{1-16}$ $\times$ Count$_{1-16}$

Sum$_{1-16}$ = $54 \times 16$

Sum$_{1-16}$ = 864

Step 2: Sum of Last 15 Numbers

Similarly, we calculate the sum for the last 15 numbers (indices 18 to 32):

Sum$_{18-32}$ = Average$_{18-32}$ $\times$ Count$_{18-32}$

Sum$_{18-32}$ = $57 \times 15$

Sum$_{18-32}$ = 855

Step 3: Remaining Numbers Identified

After the 17^th number is excluded, the remaining set consists of:

• The first 16 numbers (indices 1-16)
• The last 15 numbers (indices 18-32)

The total count of these remaining numbers is $16 + 15 = 31$.

Step 4: Sum of Remaining Numbers

The total sum of the numbers left in the set is obtained by adding the sums calculated in Step 1 and Step 2, as the 17^th number is not part of either calculated sum.

Sum of remaining numbers = Sum$_{1-16}$ + Sum$_{18-32}$

Sum of remaining numbers = $864 + 855$

Sum of remaining numbers = 1719

Step 5: New Average Calculation

The average of the remaining numbers is found by dividing their total sum by their count.

Average$_{remaining}$ = $\frac{\text{Sum of remaining numbers}}{\text{Count of remaining numbers}}$

Average$_{remaining}$ = $\frac{1719}{31}$

Step 6: Division and Rounding

Performing the division gives the precise average:

$$ \frac{1719}{31} \approx 55.4516... $$

The question asks for the result correct to one decimal place. Rounding $55.4516...$ involves looking at the second decimal place (5). Since it is 5 or greater, we round up the first decimal place.

Rounded Average = 55.5

Final Average Result

Therefore, the average of the remaining numbers after excluding the 17^th number is 55.5.