If `Y=a+b`, the maximum percentage error in the measurement of Y will be

Options

A. `((Delta a)/(a)+(Delta b)/(b))xx100`
B. `((Delta a)/(a+b)+(Delta b)/(a+b))xx100`
C. `((Delta a)/(a)-(Delta b)/(b))xx100`
D. `((Delta a)/(a-b)-(Delta b)/(a-b))xx100`

Answer

To find the maximum percentage error in the measurement of \( Y \) where \( Y = a + b \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Variables and Errors**:
- Let \( a \) and \( b \) be two measured quantities.
- Let \( \Delta a \) be the absolute error in the measurement of \( a \).
- Let \( \Delta b \) be the absolute error in the measurement of \( b \).

2. **Express the Total Error in \( Y \)**:
- The total error in \( Y \) can be expressed as:
\[ \Delta Y = \Delta a + \Delta b \]
- This is because when adding quantities, the absolute errors add up.

3. **Calculate the Percentage Error**:
- The percentage error in \( Y \) can be calculated using the formula:
\[ \text{Percentage Error} = \frac{\Delta Y}{Y} \times 100\% \]
- Substituting \( Y = a + b \) and \( \Delta Y = \Delta a + \Delta b \):
\[ \text{Percentage Error} = \frac{\Delta a + \Delta b}{a + b} \times 100\% \]

4. **Express the Percentage Error in Terms of Individual Errors**:
- We can express the percentage error as:
\[ \text{Percentage Error} = \left( \frac{\Delta a}{a + b} + \frac{\Delta b}{a + b} \right) \times 100\% \]
- This shows how the individual errors contribute to the total percentage error.

5. **Final Result**:
- Therefore, the maximum percentage error in the measurement of \( Y \) is given by:
\[ \text{Percentage Error} = \left( \frac{\Delta a}{a + b} + \frac{\Delta b}{a + b} \right) \times 100\% \]