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
Show 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!
Step 1: Find the original total sum. Mean of 10 numbers = 65, so Total sum = 10 x 65 = 650. Step 2: Set up the equation for the new mean. One number (x) is removed and 25 is included; the count stays 10. \[ \frac{650 - x + 25}{10} = 60 \] Step 3: Solve for the removed number. \[ 675 - x = 600 \Rightarrow x = 75 \] \[ \boxed{75} \]