The average of 15 numbers Is 7. If the average of the first 8 numbers be 6.5 and the average of last 8 numbers be 9.5, then the middle number is:

Options

A. 20
B. 21
C. 23
D. 18

Answer

To solve the problem step by step, let's break it down:

### Step 1: Calculate the total sum of all 15 numbers.
The average of 15 numbers is given as 7.
Using the formula for average:
\[ \text{Average} = \frac{\text{Sum of all observations}}{\text{Number of observations}} \]
We can rearrange this to find the sum:
\[ \text{Sum of all observations} = \text{Average} \times \text{Number of observations} \]
Substituting the values:
\[ \text{Sum of all observations} = 7 \times 15 = 105 \]

### Step 2: Calculate the total sum of the first 8 numbers.
The average of the first 8 numbers is given as 6.5.
Using the same formula:
\[ \text{Sum of first 8 numbers} = 6.5 \times 8 = 52 \]

### Step 3: Calculate the total sum of the last 8 numbers.
The average of the last 8 numbers is given as 9.5.
Using the formula:
\[ \text{Sum of last 8 numbers} = 9.5 \times 8 = 76 \]

### Step 4: Identify the middle number.
The middle number is the 8th number in the sequence of 15 numbers.
Since the first 8 numbers and the last 8 numbers overlap at the 8th number, we can express the total sum of all numbers as:
\[ \text{Sum of all observations} = \text{Sum of first 8 numbers} + \text{Sum of last 8 numbers} - \text{Middle number} \]
Substituting the known sums:
\[ 105 = 52 + 76 - \text{Middle number} \]
Simplifying this gives:
\[ 105 = 128 - \text{Middle number} \]
\[ \text{Middle number} = 128 - 105 = 23 \]

### Final Answer:
The middle number is **23**.

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