The mean of the 5 smallest numbers from a group is 18 while the mean of all the 13 members of the group taken together is 20. What is the mean of the 8 largest numbers?

Options

A. 22.5
B. 22.75
C. 21.25
D. 21.75

Answer

To solve the problem step by step, we will follow the information given in the question:

### Step 1: Calculate the sum of the 5 smallest numbers
The mean of the 5 smallest numbers is given as 18. The formula for the mean is:

\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]

So, we can rearrange this to find the sum:

\[ \text{Sum of 5 smallest numbers} = \text{Mean} \times \text{Number of observations} = 18 \times 5 = 90 \]

### Step 2: Calculate the sum of all 13 numbers
The mean of all 13 members of the group is given as 20. Using the same formula:

\[ \text{Sum of all 13 numbers} = \text{Mean} \times \text{Number of observations} = 20 \times 13 = 260 \]

### Step 3: Calculate the sum of the 8 largest numbers
We know the sum of the 5 smallest numbers and the sum of all 13 numbers. To find the sum of the 8 largest numbers, we subtract the sum of the 5 smallest numbers from the sum of all 13 numbers:

\[ \text{Sum of 8 largest numbers} = \text{Sum of all 13 numbers} - \text{Sum of 5 smallest numbers} = 260 - 90 = 170 \]

### Step 4: Calculate the mean of the 8 largest numbers
Now that we have the sum of the 8 largest numbers, we can find their mean:

\[ \text{Mean of 8 largest numbers} = \frac{\text{Sum of 8 largest numbers}}{\text{Number of observations}} = \frac{170}{8} = 21.25 \]

### Final Answer:
The mean of the 8 largest numbers is **21.25**.

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