Question

The arithmetic mean of 10 numbers is 65. If one number is removed and 25 is included in the list. The new arithmetic mean is 60. The removed number is

• A. 75
• B. 65
• C. 60
• D. 80

Hint:

Observe the change in average: The average drops from 65 to 60, a decrease of 5 units.
For 10 numbers, a drop of 5 units in average means the total sum decreased by:
\[ 10 \times 5 = 50 \text{ units} \]
This means the new number (25) is 50 units smaller than the removed number ($x$).
\[ x = 25 + 50 = 75 \]
This simple reasoning avoids long calculations!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question relates to averages (arithmetic means).
We are told the mean of 10 numbers is 65.
Then, one number is removed, and a new number (25) is added in its place (keeping the total count of numbers at 10).
The new average drops to 60. We need to find the value of the removed number.

Step 2: Key Formula or Approach:
The basic formula for arithmetic mean is:
\[ \text{Mean} = \frac{\text{Sum of all observations}}{\text{Number of observations}} \implies \text{Sum} = \text{Mean} \times \text{Number of observations} \]
Let $S_1$ be the initial sum of the 10 numbers, $x$ be the removed number, and $S_2$ be the new sum after replacing $x$ with 25.

Step 3: Detailed Calculation:
1.

Find the initial sum ($S_1$):
\[ S_1 = 10 \times 65 = 650 \]
2.

Express the relationship for the new sum ($S_2$):
The new sum is obtained by subtracting the removed number ($x$) and adding 25:
\[ S_2 = S_1 - x + 25 \]
\[ S_2 = 650 - x + 25 = 675 - x \]
3.

Find the actual value of $S_2$ from the new mean:
Since the count remains 10 and the new mean is 60:
\[ S_2 = 10 \times 60 = 600 \]
4.

Equate the two expressions for $S_2$ to solve for $x$:
\[ 675 - x = 600 \]
\[ x = 675 - 600 \]
\[ x = 75 \]
Thus, the removed number is 75.

Step 4: Final Answer:
The removed number is 75, corresponding to Option (A).