Sachin Tendulkar has a certain batting average N (a whole number) in his career of 86 innings. In the 87th inning, he gets out after scoring 270 runs which increases his batting average by a whole number. The batting average is calculated by dividing the total number of runs scored by the total number of innings played by the player. How many values of his new average is/are possible?

Options

A. 0
B. 1
C. 2
D. None of these

Answer

To solve the problem, we need to find how many possible new batting averages Sachin Tendulkar can have after scoring 270 runs in his 87th inning, given that this increases his batting average by a whole number.

### Step-by-Step Solution:

1. **Define the Variables**:
Let \( N \) be Sachin's original batting average after 86 innings. The total runs scored after 86 innings can be expressed as:
\[ \text{Total Runs} = 86N \]

2. **Calculate the New Total Runs**:
After scoring 270 runs in the 87th inning, the new total runs become:
\[ \text{New Total Runs} = 86N + 270 \]

3. **Calculate the New Average**:
The new average after 87 innings is given by:
\[ \text{New Average} = \frac{86N + 270}{87} \]

4. **Set Up the Condition for Whole Number Increase**:
We know that the new average must be greater than the old average \( N \) by a whole number \( k \):
\[ \frac{86N + 270}{87} = N + k \]
Rearranging gives:
\[ 86N + 270 = 87N + 87k \]
Simplifying this leads to:
\[ 270 = N + 87k \]
Therefore, we can express \( N \) in terms of \( k \):
\[ N = 270 - 87k \]

5. **Determine Valid Values of \( k \)**:
Since \( N \) must be a non-negative whole number, we require:
\[ 270 - 87k \geq 0 \]
This implies:
\[ k \leq \frac{270}{87} \approx 3.1 \]
Thus, \( k \) can take values \( 0, 1, 2, \) or \( 3 \).

6. **Calculate New Averages for Each Valid \( k \)**:
- For \( k = 0 \):
\[ N = 270 - 87 \times 0 = 270 \quad \Rightarrow \quad \text{New Average} = 270 + 0 = 270 \]
- For \( k = 1 \):
\[ N = 270 - 87 \times 1 = 183 \quad \Rightarrow \quad \text{New Average} = 183 + 1 = 184 \]
- For \( k = 2 \):
\[ N = 270 - 87 \times 2 = 96 \quad \Rightarrow \quad \text{New Average} = 96 + 2 = 98 \]
- For \( k = 3 \):
\[ N = 270 - 87 \times 3 = 9 \quad \Rightarrow \quad \text{New Average} = 9 + 3 = 12 \]

7. **List the Possible New Averages**:
The possible new averages are:
- 270 (for \( k = 0 \))
- 184 (for \( k = 1 \))
- 98 (for \( k = 2 \))
- 12 (for \( k = 3 \))

8. **Count the Unique New Averages**:
The unique new averages are: 270, 184, 98, and 12. Thus, there are **4 possible new averages**.

### Conclusion:
The answer to the question is that there are **4 possible values** of Sachin's new average.