20 numbers averaged to 40, if 25 and 55 are two numbers which are subtracted from the list, the new average become

1. Break Down the Problem

To find the new average after removing two numbers (25 and 55) from a list of 20 numbers that averaged to 40, we can break the problem down into steps:

1. Calculate the total sum of the original set of numbers.
2. Subtract the two numbers from the sum.
3. Calculate the new average based on the new total sum and the number of remaining numbers.

2. Relevant Concepts

• The average of a set is calculated as: $\text{Average} = \frac{\text{Total Sum}}{\text{Number of Items}}$
• When removing items from the dataset, the new average is recalculated with the new total sum and the reduced number of items.

3. Analysis and Detail

1. Calculate the Total Sum: The original average of the 20 numbers is 40, so: $\text{Total Sum} = \text{Average} \times \text{Number of Items} = 40 \times 20 = 800$

2. Subtract the Two Numbers: The sum of the two numbers to be subtracted is: $25 + 55 = 80$ Therefore, the new total sum after subtraction becomes: $\text{New Total Sum} = 800 - 80 = 720$

3. Calculate the New Average: The new number of items left is: $20 - 2 = 18$ The new average can now be calculated as: $\text{New Average} = \frac{\text{New Total Sum}}{\text{New Number of Items}} = \frac{720}{18} = 40$

4. Verify and Summarize

The calculations of the total sum, the subtraction of the two numbers, and the new average confirm that the operations were carried out correctly as the average remains unchanged.

Final Answer

The new average, after removing the numbers 25 and 55, is 40.