Question:hard

In a group of 10 students, the mean of the lowest 9 scores is 42 while the mean of the highest 9 scores is 47. For the entire group of 10 students, the maximum possible mean exceeds the minimum possible mean by

Updated On: Jan 15, 2026
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The Correct Option is A

Solution and Explanation

Let:

  • \( x_1 \) denote the smallest observation
  • \( x_{10} \) denote the largest observation

Step 1: Apply the given average conditions

The average of the 9 largest numbers is: \[ \frac{x_2 + x_3 + \dots + x_{10}}{9} = 47 \Rightarrow x_2 + x_3 + \dots + x_{10} = 423 \tag{1} \]

The average of the 9 smallest numbers is: \[ \frac{x_1 + x_2 + \dots + x_9}{9} = 42 \Rightarrow x_1 + x_2 + \dots + x_9 = 378 \tag{2} \]

Step 2: Subtract equation (2) from equation (1)

Subtracting (2) from (1): \[ (x_2 + \dots + x_{10}) - (x_1 + \dots + x_9) = 45 \Rightarrow x_{10} - x_1 = 45 \]

Step 3: Determine the total sum of the 10 observations

From equation (1): \[ x_2 + x_3 + \dots + x_{10} = 423 \Rightarrow \text{Total sum} = x_1 + 423 \Rightarrow \text{Average} = \frac{x_1 + 423}{10} \]

Step 4: Calculate the minimum and maximum possible averages

  • For the minimum average: Use the smallest possible value for \( x_1 = 2 \). \[ \text{Average} = \frac{423 + 2}{10} = \frac{425}{10} = 42.5 \]
  • For the maximum average: Use the largest possible value for \( x_1 = 42 \). \[ \text{Average} = \frac{423 + 42}{10} = \frac{465}{10} = 46.5 \]

Step 5: Calculate the final result

\[ \text{Required difference} = 46.5 - 42.5 = \boxed{4} \]

Final Answer:

✅ The correct answer is: \[ \boxed{4} \] (Option A)

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