Question

The mean of $5$ numbers is $24$. If one number is excluded, the mean becomes $20$. What is the excluded number?

• A. $24$
• B. $36$
• C. $40$
• D. $44$

Hint:

If the mean decreases after a number is removed (like going from $24$ to $20$), it means the excluded number was higher than the original mean. This helps you quickly eliminate smaller options!

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Mean (or average) is the sum of all values divided by the count of those values.
When a number is removed from the data set, both the total sum and the count of numbers change.
Step 2: Key Formula or Approach:
We can find the sum of observations using the formula:
\[ \text{Sum} = \text{Mean} \times \text{Number of observations} \] The excluded number is the difference between the original sum and the new sum.
Step 3: Detailed Explanation:
Initial state:
Mean of 5 numbers = 24.
\[ \text{Sum of 5 numbers} = 24 \times 5 = 120 \] State after excluding one number:
Number of observations remaining = 4.
New Mean = 20.
\[ \text{Sum of 4 numbers} = 20 \times 4 = 80 \] The excluded number is the value that was removed from the set:
\[ \text{Excluded number} = \text{Sum of 5 numbers} - \text{Sum of 4 numbers} \] \[ \text{Excluded number} = 120 - 80 = 40 \] Step 4: Final Answer:
The excluded number is 40.