An experiment measures quantities a, b and c and X is calculated from `X= ab^(2)//c^(3)`. If the percentage error in a, b and c are `pm 1%, pm 3% and pm 2%` respectively, the percentage error in X will be-

Options

A. `pm 13 % `
B. `pm 7 % `
C. `pm 4 % `
D. `pm 1 %`

Explanation

To find the percentage error in the quantity \( X \) calculated from the formula \( X = \frac{a b^2}{c^3} \), we can use the rules of propagation of errors. The formula for the percentage error in a derived quantity is given by:

\[ \frac{\Delta X}{X} \times 100 = \frac{\Delta a}{a} \times 100 + 2 \frac{\Delta b}{b} \times 100 - 3 \frac{\Delta c}{c} \times 100 \]

Where:
- \( \Delta a \), \( \Delta b \), and \( \Delta c \) are the absolute errors in \( a \), \( b \), and \( c \) respectively.
- The coefficients (1, 2, and -3) correspond to the powers of \( a \), \( b \), and \( c \) in the formula for \( X \).

### Step-by-Step Solution:

1. **Identify the given percentage errors**:
- Percentage error in \( a \) = \( \pm 1\% \)
- Percentage error in \( b \) = \( \pm 3\% \)
- Percentage error in \( c \) = \( \pm 2\% \)

2. **Write the formula for percentage error in \( X \)**:
\[ \text{Percentage error in } X = \text{Percentage error in } a + 2 \times \text{Percentage error in } b + 3 \times \text{Percentage error in } c \]

3. **Substitute the values into the formula**:
\[ \text{Percentage error in } X = 1\% + 2 \times 3\% + 3 \times 2\% \]

4. **Calculate each term**:
- For \( a \): \( 1\% \)
- For \( b \): \( 2 \times 3\% = 6\% \)
- For \( c \): \( 3 \times 2\% = 6\% \)

5. **Add the contributions together**:
\[ \text{Percentage error in } X = 1\% + 6\% + 6\% = 13\% \]

### Final Result:
The percentage error in \( X \) is \( \pm 13\% \).