The arithmetic mean of the first 15 numbers is 82. The mean is reduced by 6 when one of the numbers is removed. Find the number t…

Question

The arithmetic mean of the first 15 numbers is 82. The mean is reduced by 6 when one of the numbers is removed. Find the number that is excluded?

1. A. 142
2. B. 100
3. C. 166 (Correct answer)
4. D. 164

Correct Answer

Option C — 166

Detailed Solution & Explanation

The correct answer is 166.

Shortcut Trick

• Removing a number drags the mean down by 6 across the 14 that remain, so the excluded value exceeds the new mean by its own share plus the deficit it leaves behind.
• Excluded $=$ new mean $+ 15 \times$ (drop) $= 76 + 90 = \textbf{166}$.

Alternate Method

• Sum of all 15 numbers $= 15 \times 82 = 1230$
• New mean $= 82 - 6 = 76$, over 14 numbers $\Rightarrow$ sum $= 14 \times 76 = 1064$
• Excluded number $= 1230 - 1064 = \textbf{166}$

Additional Information

• Always convert averages to totals. The mean itself cannot be added or subtracted directly, but sums can — which is why $\text{sum} = \text{mean} \times \text{count}$ is the first line of nearly every average question.
• The removed value being well above the old mean of 82 is what pulls the average down; removing a below-average value would raise it.
• General result: removing one item from $n$ changes the mean by $\dfrac{\text{mean} - x}{n-1}$, which rearranges to the shortcut used above.