Question

The average of 8 numbers is 24. If two numbers 18 and 30 are removed, the average of the remaining numbers is:

• A. 22
• B. 23
• C. 24
• D. 25

Hint:

You can solve this instantly by checking the average of the removed items! The average of the two removed numbers is $\frac{18 + 30}{2} = \frac{48}{2} = 24$. Since the average of the elements leaving the group is exactly equal to the overall group average (24), their removal will have zero impact on the rest of the group. The remaining average stays 24!

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
An average is the total sum of all values divided by the total count of those values.
To find the new average after removing elements, we must find the original total sum, subtract the values of the removed elements to find the new sum, and then divide by the new count of elements.
Step 2: Key Formula or Approach:
1. Total Sum = Average \( \times \) Count
2. New Average = \( \frac{\text{Original Sum} - \text{Sum of Removed Numbers}}{\text{Original Count} - \text{Removed Count}} \)
Step 3: Detailed Explanation:
Initial data analysis:
Original Count (\(n_1\)) = 8 numbers
Original Average (\(A_1\)) = 24
\[ \text{Original Sum} = 8 \times 24 = 192 \]
Numbers to be removed: 18 and 30.
Sum of removed numbers = \( 18 + 30 = 48 \)
Number of elements removed = 2
Calculating the new state:
New Count (\(n_2\)) = \( 8 - 2 = 6 \) numbers
New Sum = \( 192 - 48 = 144 \)
Now, calculate the revised group average:
\[ \text{New Average} = \frac{144}{6} \]
Dividing 144 by 6:
\[ 14 \div 6 = 2 \text{ remainder } 2 \]
\[ 24 \div 6 = 4 \]
\[ \text{New Average} = 24 \]
Notice that the average stayed exactly the same. This happens because the average of the two numbers being removed was \( \frac{18 + 30}{2} = 24 \), which is identical to the group's overall average. Removing the "average weight" does not change the balance of the remaining items.
Step 4: Final Answer:
The average of the remaining numbers is 24.