The mean of 8 numbers is 22. If one number is excluded, then their mean is 16. The excluded number is :

Question

Accepted Answer

To solve this problem, we need to determine the excluded number based on the given information about the means of the numbers.

1. **Understanding the Problem:**
   - The mean of 8 numbers is 22. This means the sum of these 8 numbers is $8 \times 22 = 176$.
   - When one number is excluded, the mean of the remaining 7 numbers is 16. This means the sum of these 7 numbers is $7 \times 16 = 112$.

2. **Finding the Excluded Number:**
   - The sum of the original 8 numbers is 176.
   - The sum of the remaining 7 numbers is 112.
   - Therefore, the excluded number is the difference between these two sums: $176 - 112 = 64$.

3. **Explanation of Options:**
   - **Option 1 (16):** If the excluded number were 16, the sum of the remaining 7 numbers would be $176 - 16 = 160$, and the mean would be $160/7 \approx 22.86$, not 16. Therefore, this option is incorrect.
   - **Option 2 (22):** If the excluded number were 22, the sum of the remaining 7 numbers would be $176 - 22 = 154$, and the mean would be $154/7 \approx 22$, not 16. Therefore, this option is incorrect.
   - **Option 3 (32):** If the excluded number were 32, the sum of the remaining 7 numbers would be $176 - 32 = 144$, and the mean would be $144/7 \approx 20.57$, not 16. Therefore, this option is incorrect.
   - **Option 4 (64):** As calculated, if the excluded number is 64, the sum of the remaining 7 numbers is 112, and the mean is $112/7 = 16$, which matches the given condition. Therefore, this option is correct.

In conclusion, the correct answer is Option 4 (64), as it satisfies the condition that the mean of the remaining numbers is 16 after excluding this number.