The average of eight numbers is 38 . 4 and the average of seven of them is 39 . 2 . What is the eighth number ?

Options

A. `0.8`
B. `32 . 8`
C. `34. 8`
D. `33 . 8`

Answer

To find the eighth number when the average of eight numbers is 38.4 and the average of seven of them is 39.2, we can follow these steps:

### Step 1: Calculate the sum of the eight numbers.
The formula for average is given by:

\[ \text{Average} = \frac{\text{Sum of observations}}{\text{Total observations}} \]

For the eight numbers, we have:

\[ 38.4 = \frac{\text{Sum of 8 numbers}}{8} \]

To find the sum of the eight numbers, we rearrange the formula:

\[ \text{Sum of 8 numbers} = 38.4 \times 8 \]

Calculating this gives:

\[ \text{Sum of 8 numbers} = 307.2 \]

### Step 2: Calculate the sum of the seven numbers.
Using the same formula for the seven numbers, we have:

\[ 39.2 = \frac{\text{Sum of 7 numbers}}{7} \]

Rearranging gives:

\[ \text{Sum of 7 numbers} = 39.2 \times 7 \]

Calculating this gives:

\[ \text{Sum of 7 numbers} = 274.4 \]

### Step 3: Find the eighth number.
Now, we can find the eighth number by subtracting the sum of the seven numbers from the sum of the eight numbers:

\[ \text{Eighth number} = \text{Sum of 8 numbers} - \text{Sum of 7 numbers} \]

Substituting the values we calculated:

\[ \text{Eighth number} = 307.2 - 274.4 \]

Calculating this gives:

\[ \text{Eighth number} = 32.8 \]

### Final Answer:
The eighth number is **32.8**.

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