Find the approximate value which should replace the question mark (?) in each of the following question .
The average of 17 number is 45 . The average of first 9 of these numbers is 51 and the last 9 of these numbers is 36 . What is the ninth number ?

Options

A. 14
B. 16
C. 22
D. None of these

Answer

To find the ninth number in the given problem, we will follow these steps:

### Step 1: Calculate the total sum of all 17 numbers.
Given that the average of 17 numbers is 45, we can calculate the total sum (S) of these numbers using the formula:

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

So,

\[ S = \text{Average} \times \text{Number of items} = 45 \times 17 \]

Calculating this gives:

\[ S = 765 \]

### Step 2: Calculate the sum of the first 9 numbers.
The average of the first 9 numbers is given as 51. Therefore, the sum of the first 9 numbers (let's call it S1) can be calculated as:

\[ S1 = \text{Average} \times \text{Number of items} = 51 \times 9 \]

Calculating this gives:

\[ S1 = 459 \]

### Step 3: Calculate the sum of the last 9 numbers.
The average of the last 9 numbers is given as 36. Therefore, the sum of the last 9 numbers (let's call it S2) can be calculated as:

\[ S2 = \text{Average} \times \text{Number of items} = 36 \times 9 \]

Calculating this gives:

\[ S2 = 324 \]

### Step 4: Set up the equation to find the ninth number.
Let the ninth number be \( x \). The sum of the first 8 numbers (S1 without the ninth number) can be expressed as:

\[ \text{Sum of first 8 numbers} = S1 - x = 459 - x \]

The sum of the last 8 numbers (S2 without the ninth number) can be expressed as:

\[ \text{Sum of last 8 numbers} = S2 - x = 324 - x \]

### Step 5: Use the total sum to find the ninth number.
From the total sum of all 17 numbers, we know:

\[ \text{Sum of first 8 numbers} + \text{ninth number} + \text{Sum of last 8 numbers} = S \]

Substituting the values we have:

\[ (459 - x) + x + (324 - x) = 765 \]

Simplifying this gives:

\[ 459 + 324 - x = 765 \]

Combining the constants:

\[ 783 - x = 765 \]

### Step 6: Solve for \( x \).
Rearranging the equation gives:

\[ x = 783 - 765 \]

Calculating this gives:

\[ x = 18 \]

### Conclusion:
The ninth number is **18**.

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