If `X= A xx B and Delta X Delta A and Delta B` are maximum absolute error in X ,A and B respectively , then the maximum relative in X is given by

Options

A. `DeltaX = DeltaA + DeltaB`
B. `DeltaX = DeltaA - DeltaB`
C. `(DeltaX)/X = (DeltaA)/A - (DeltaB)/B`
D. `(DeltaX)/X = (DeltaA)/A + (DeltaB)/B`

Answer

To solve the problem, we need to derive the maximum relative error in \( X \) when \( X = A \times B \). The maximum absolute errors in \( X \), \( A \), and \( B \) are denoted as \( \Delta X \), \( \Delta A \), and \( \Delta B \) respectively.

### Step-by-Step Solution:

1. **Understanding the relationship**:
We start with the equation given:
\[ X = A \times B \]
Here, \( A \) and \( B \) are the variables, and \( X \) is their product.

2. **Finding the fractional error**:
The fractional error in a product can be expressed in terms of the fractional errors of the individual components. The formula for the fractional error in \( X \) is given by:
\[ \frac{\Delta X}{X} = \frac{\Delta A}{A} + \frac{\Delta B}{B} \]
where \( \Delta X \), \( \Delta A \), and \( \Delta B \) are the absolute errors in \( X \), \( A \), and \( B \) respectively.

3. **Rearranging for maximum relative error**:
To express the maximum relative error in \( X \), we can rewrite the equation:
\[ \Delta X = X \left( \frac{\Delta A}{A} + \frac{\Delta B}{B} \right) \]
This shows that the maximum relative error in \( X \) is the sum of the relative errors in \( A \) and \( B \).

4. **Conclusion**:
Therefore, the maximum relative error in \( X \) can be summarized as:
\[ \text{Maximum relative error in } X = \frac{\Delta X}{X} = \frac{\Delta A}{A} + \frac{\Delta B}{B} \]

### Final Answer:
The maximum relative error in \( X \) is given by:
\[ \frac{\Delta X}{X} = \frac{\Delta A}{A} + \frac{\Delta B}{B} \]