Calculating Increase in Average

This question asks us to find out how much the average of a set of numbers increases when one number is replaced by another larger number.

We are given the following information:

• The original average of 40 numbers is 71.
• One number, 100, is removed from the set.
• A new number, 140, is added to the set in its place.
• The total count of numbers remains 40.

Step-by-Step Solution

Step 1: Find the Original Sum

The average of a set of numbers is calculated by dividing the sum of the numbers by the count of the numbers. We can use this relationship to find the original sum.

Formula for Average:

\[ \text{Average} = \frac{\text{Sum of numbers}}{\text{Number of numbers}} \]

Given the original average is 71 and the number of numbers is 40:

\[ 71 = \frac{\text{Original Sum}}{40} \]

To find the Original Sum, we multiply the average by the number of numbers:

\[ \text{Original Sum} = 71 \times 40 \]

\[ \text{Original Sum} = 2840 \]

Step 2: Calculate the Change in the Sum

When the number 100 is replaced by 140, the sum of the numbers changes. The change in the sum is the difference between the new number and the number being replaced.

Change in Sum = New Number - Replaced Number

\[ \text{Change in Sum} = 140 - 100 \]

\[ \text{Change in Sum} = 40 \]

This means the total sum of the numbers increases by 40.

Step 3: Find the New Sum

The new sum is the original sum plus the change in the sum.

New Sum = Original Sum + Change in Sum

\[ \text{New Sum} = 2840 + 40 \]

\[ \text{New Sum} = 2880 \]

Step 4: Calculate the New Average

Now that we have the new sum and the number of numbers (which is still 40), we can calculate the new average.

New Average = New Sum / Number of numbers

\[ \text{New Average} = \frac{2880}{40} \]

\[ \text{New Average} = 72 \]

Step 5: Determine the Increase in Average

The increase in average is the difference between the new average and the original average.

Increase in Average = New Average - Original Average

\[ \text{Increase in Average} = 72 - 71 \]

\[ \text{Increase in Average} = 1 \]

Alternatively, we can directly calculate the increase in average from the change in sum:

Increase in Average = Change in Sum / Number of numbers

\[ \text{Increase in Average} = \frac{40}{40} \]

\[ \text{Increase in Average} = 1 \]

Therefore, the average is increased by 1.