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. 0.14
B. 0.1
C. 0.07
D. 0.04

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 can be calculated by summing the relative errors (percentage errors) of the individual quantities, each multiplied by their respective powers.

### Step-by-Step Solution:

1. **Identify the powers of each variable in the expression for \( P \)**:
- For \( a \): power = 3
- For \( b \): power = 2
- For \( c \): power = 1
- For \( d \): power = 1

2. **Write down the percentage errors for each variable**:
- Percentage error in \( a \) = 1%
- Percentage error in \( b \) = 2%
- Percentage error in \( c \) = 3%
- Percentage error in \( d \) = 4%

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

4. **Substituting the values**:
\[ \text{Percentage error in } P = \left( 3 \times 1\% \right) + \left( 2 \times 2\% \right) + \left( 1 \times 3\% \right) + \left( 1 \times 4\% \right) \]

5. **Calculating each term**:
- From \( a \): \( 3 \times 1\% = 3\% \)
- From \( b \): \( 2 \times 2\% = 4\% \)
- From \( c \): \( 1 \times 3\% = 3\% \)
- From \( d \): \( 1 \times 4\% = 4\% \)

6. **Summing these contributions**:
\[ \text{Percentage error in } P = 3\% + 4\% + 3\% + 4\% = 14\% \]

7. **Expressing the final answer**:
The percentage error in \( P \) is \( 14\% \). In decimal form, this can be written as \( 0.14 \).

### Final Answer:
The percentage error in \( P \) is \( 0.14 \).

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