In an experiment four quantities a,b,c and d are measure with percentage error `1% , 2% , 3%`,and `4%` respectively quantity is P is calculate as follow
`P = (a^(3)b^(2))/(cd) %` error in `P` is

Options

A. `14%`
B. `10%`
C. `7%`
D. `4%`

Explanation

To find the percentage error in the quantity \( P \) defined as

\[ P = \frac{a^3 b^2}{cd} \]

we will use the rules of error propagation. The percentage error in a product or quotient of quantities is given by the sum of the percentage errors of each quantity, weighted by their respective powers in the formula.

### Step-by-Step Solution:

1. **Identify the Powers of Each Quantity**:
- In the expression for \( P \):
- \( a \) is raised to the power of 3.
- \( b \) is raised to the power of 2.
- \( c \) and \( d \) are raised to the power of 1 (since they are in the denominator).

2. **List the Given Percentage Errors**:
- The percentage error in \( a \) is \( 1\% \).
- The percentage error in \( b \) is \( 2\% \).
- The percentage error in \( c \) is \( 3\% \).
- The percentage error in \( d \) is \( 4\% \).

3. **Apply the Formula for Percentage Error**:
- The formula for the percentage error in \( P \) is given by:
\[ \text{Percentage error in } P = \left(3 \times \text{Percentage error in } a\right) + \left(2 \times \text{Percentage error in } b\right) + \left(1 \times \text{Percentage error in } c\right) + \left(1 \times \text{Percentage error in } d\right) \]

4. **Substitute the Values**:
- Substitute the known percentage errors into the formula:
\[ \text{Percentage error in } P = \left(3 \times 1\%\right) + \left(2 \times 2\%\right) + \left(1 \times 3\%\right) + \left(1 \times 4\%\right) \]

5. **Calculate Each Term**:
- Calculate each term:
\[ = 3\% + 4\% + 3\% + 4\% \]

6. **Sum the Contributions**:
- Add the contributions together:
\[ = 3 + 4 + 3 + 4 = 14\% \]

7. **Conclusion**:
- Therefore, the percentage error in \( P \) is \( 14\% \).

### Final Answer:
The percentage error in \( P \) is \( 14\% \).