The mean of the 6 smallest numbers from a group is 15 while the mean of all the 14 members of the group taken together is 17. What is the mean of the 8 largest numbers?

Options

A. 18.25
B. 17.75
C. 18.75
D. `18.50`

Answer

To find the mean of the 8 largest numbers in the group, we can follow these steps:

### Step 1: Calculate the sum of the 6 smallest numbers.
Given that the mean of the 6 smallest numbers is 15, we can use the formula for the mean:
\[ \text{Mean} = \frac{\text{Sum of values}}{\text{Number of values}} \]
Thus, we have:
\[ 15 = \frac{a_1 + a_2 + a_3 + a_4 + a_5 + a_6}{6} \]
Multiplying both sides by 6 gives:
\[ a_1 + a_2 + a_3 + a_4 + a_5 + a_6 = 15 \times 6 = 90 \]

### Step 2: Calculate the sum of all 14 numbers.
The mean of all 14 members is given as 17. Using the same mean formula:
\[ 17 = \frac{a_1 + a_2 + \ldots + a_{14}}{14} \]
Multiplying both sides by 14 gives:
\[ a_1 + a_2 + \ldots + a_{14} = 17 \times 14 = 238 \]

### Step 3: Calculate the sum of the 8 largest numbers.
We already know the sum of the 6 smallest numbers is 90. Therefore, the sum of the 8 largest numbers (from \(a_7\) to \(a_{14}\)) can be calculated as:
\[ a_7 + a_8 + \ldots + a_{14} = \text{Total sum} - \text{Sum of the smallest numbers} \]
Substituting the values we have:
\[ a_7 + a_8 + \ldots + a_{14} = 238 - 90 = 148 \]

### Step 4: Calculate the mean of the 8 largest numbers.
Now, we can find the mean of the 8 largest numbers:
\[ \text{Mean} = \frac{a_7 + a_8 + \ldots + a_{14}}{8} \]
Substituting the sum we found:
\[ \text{Mean} = \frac{148}{8} = 18.5 \]

### Conclusion
The mean of the 8 largest numbers is **18.5**.

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